Djamel Benredjem

Mohammad F Gharaibeh

Jean-Marc M Bizau
email: jean-marc.bizau@u-psud.fr

D Cubaynes

S Guilbaud

El Hassan

M M Al Shorman

C Miron

C Nicolas

E Robert

C Blancard

B M Mclaughlin
email: b.mclaughlin@qub.ac.uk

N El

I Sakho

Photon energy (eV)

Keywords: Photoionization, Cross section, Synchrotron radiation, FTICR, ECRIS, SCUNC, R-matrix, MCDF, MAIA, PLEIADES, DESIRS, SOLEIL photoionization, K-shell, nitrogen ions, cross sections

In this thesis work, the K-shell photoionization of multi-charged ions has been studied as well as the valence photoionization of atomic and small molecular ions. The K-shell photoionization cross sections were measured for nitrogen iso-nuclear series, from N + to N 4+ ions using the ion-photon merged beam technique and the valence photoionization cross sections for Xe + and Kr + ions and the small molecular ions CO and N using both the merged beam and ion trap techniques at the SOLEIL synchrotron radiation facility in Saint-Aubin, France. Combination of the two techniques allows for the measurement of the pure ground state ionization cross section on an absolute scale. The experimental K-shell photoionization cross sections are compared with theoretical results obtained from the multi-configuration Dirac-Fock (MCDF), R-matrix and the Screening Constant by Unit Nuclear Charge (SCUNC) methods. The interplay between experiment and theory enables the identification and characterization of the strong 1s→2p and 1s→3p resonances observed in the spectra. The experimental valence photoionization cross sections for Xe + and Kr + ions are compared with MCDF calculations results obtained for the direct photoionization process.

The quality of the absolute cross section measurements using the merged beam techniques is strongly dependent on the performance of Electron Cyclotron Resonance Ion Source (ECRIS). In order to improve the current of ions in the interaction region, the ions extraction system and transport was simulated by using IGUN program and ECRopt.

Quatre-vingt-dix-neuf pourcents de la matière dans l'Univers se trouvent sous forme de plasma, une soupe d'ions, d'électrons et de photons en interaction forte, souvent appelé le quatrième état de la matière. L'étude expérimentale de l'interaction entre les ions et les photons présente plusieurs intérêts. D'un point de vue fondamental en premier, c'est le seul moyen de tester la validité des différents modèles décrivant ces processus. Les multiples effets des interactions entre électrons impliqués dans les structures atomiques et moléculaires doivent être pris en compte dans ces modèles. Une bonne compréhension de ces processus dans les systèmes simples comme des atomes ou des ions est une condition préalable pour leur compréhension dans les états les plus complexes de la matière. D'un point de vue appliqué, la spectroscopie est un outil pratique permettant de décrire la composition chimique et l'état thermodynamique des plasmas. La plupart de nos informations sur les plasmas astrophysiques et de laboratoire sont amenées par les photons. Obtenir des données de laboratoire sur les processus d'interaction entre les photons et les ions multichargés atomiques qui habitent les plasmas chauds, ou les ions moléculaires positifs et négatifs impliqués dans la chimie des ionosphères planétaires, est essentiel pour l'interprétation des spectres enregistrés par les observatoires spatiaux. Dans les plasmas denses et chauds, la connaissance des sections efficaces de photoabsorption en couches internes est également essentielle pour la description du transfert radiatif dans le plasma. Ce mécanisme est très important pour la vi fusion par confinement inertiel, ainsi qu'en astrophysique où il contrôle, par exemple, l'équilibre radiatif-gravitationnel dans les étoiles comme le Soleil. L'étude des processus de photoionisation donne un merveilleux outil pour étudier les effets relativistes et l'interaction dynamique produite par les corrélations électron-électron à l'intérieur des systèmes atomiques. Les études spectroscopiques dans la gamme d'énergie des rayons X mous des processus de photoionisation en couche K dans les ions C, N, O, Ne, S et Si et en couche L des ions Fe et Ni sont d'intérêt primordial pour la description de nombreux systèmes physique, y compris les atmosphères planétaires, les plasmas de laboratoire impliqués dans les applications industrielles, les plasmas astrophysiques (étoiles, milieu interstellaires...) Des calculs extensifs de section efficace de photoionisation, utilisant des codes sophistiqués tels que la R-matrice et MCDF (Multi-Configurational Dirac-Fock), ont été effectués près des seuils d'ionisation en couche K le long de séquences isoélectroniques et iso-nucléaires de ces ions pour interpréter la richesse des spectres de rayons X actuellement fournis par les satellites Chandra et XMM-Newton.

Jusqu'à présent, peu de données expérimentales sont disponibles pour comparer avec ces calculs. Tout d'abord, les sections efficaces de photoabsorption ont été mesurées dans des plasmas produits par laser. Avec l'avènement des sources de rayonnement synchrotron de troisième génération, des expériences ion-photon avec des faisceaux colinéaires se sont développés. Cette technique permet de déterminer en absolu des sections efficaces de photoionisation sur des ions multiplement chargés, ouvrant la voie à des études systématiques sur des séquences isoélectroniques. Cependant, la plupart des données ont été obtenues jusqu'à maintenant dans la gamme d'énergie de photons inférieure à 200 eV, excluant la couche K de la plupart des ions. Des résultats expérimentaux sur la photoionisation en couche K des ions légers ont été effectuées par différents groupes à l'aide de la technique de vii faisceaux colinéaires: Li + , B 2+ , C 3+ , C 2+ , C + , O + , Ne + , en plus de N et O neutres. Avant ce travail, aucune section efficace de photoionisation en couche K n'a été mesurée pour les ions d'azote.

Presque aucune expérience n'a été réalisée sur la photoionisation des ions moléculaires. Très récemment, une nouvelle technique est en développement rapide dans les installations de rayonnements synchrotron, l'utilisation de pièges à ions, ouvrant la possibilité d'étudier des ions froids et, grâce à une résolution en masse élevée, tous les types d'ions moléculaires, y compris les agrégats et les ions d'intérêt biologique.

B-Photoionisation en couche K des ions azote

Les sections efficaces de photoionisation en couches K ont été mesurées pour les ions de la série iso-nucléaire de l'azote, de N + à N 4+ ions en utilisant le montage MAIA de faisceaux colinéaires sur la ligne de lumière PLEIADES de SOLEIL, sur une gamme d'énergies de photon allant de 398 à 406 eV pour l'ion N + , trois gammes d'énergie pour explorer l'ion N 2+ de 404 à 409 eV , 439 à 442 eV et 452 à 458 eV, deux gammes d'énergie explorées pour l'ion N 3+ de 410 à 415 eV et 560 à 460.5 eV, et à partir de 240 à 426 eV pour l'ion N 4+ . Les sections efficaces expérimentales de photoionisation en couche K sont comparées aux résultats théoriques réalisés par des théoriciens pour ce travail, utilisant différentes méthodes : MCDF, la R-matrice et la Constante d'Écran par Unité de charge Nucléaire (SCUNC). L'interaction entre l'expérience et la théorie permet l'identification et la caractérisation des résonances 1s→2p et 1s→3p observées dans les spectres.

-1 ions N + La configuration électronique de l'azote ionisé (N + ) est 1s 2 2s 2 2p 2 3 P 0 dans son niveau fondamental, En plus du niveau fondamental, l'ion N + a cinq niveaux métastables N + viii (1s 2 2s 2 2p 2 3 P1, 2, 1 D2, 1 S0) and N + (1s 2 2s2p 3 5 S ) susceptibles d'être peuplés dans la source d'ions. La présente étude fournit des valeurs absolues pour les sections efficaces de photoionisation qui se produisent dans l'interaction entre un photon et l'ion N + dans l'état fondamental et ces cinq états métastables. Les structures de photoionisation les plus intenses sont les photoexcitation1s → 2p: un électron 1s est excité sur la sous-couche 2p partiellement rempli, laissant un ion N + dans un état très excité (avec un trou dans la sous-couche 1s) qui se désexcitera par autoionisation en laissant un ion N 2+ . A titre d'exemple pour N + dans le terme 3 P, et en suivant les règles de sélection dipolaires électriques, ce procédé peut être représenté par : hν + N + (1s 2 2s 2 2p 2 3 P) → N +* (1s2s 2 2p 3 3 S o , 3 P o , 3 D o ) ↓ N 2+ (1s 2 2s 2 2p 2 P o ) + e-Pour les ions N + dans le niveau 1s 2 2s2p 3 5 S , ce processus devient: hν + N + (1s 2 2s2p 3 5 S o ) → N + [1s2s[ 1,3 S]2p 4 ( 3 P, 1 D, 1 S)] 5 P ↓ N 2+ (1s 2 2s2p 2 4 P) + e-.

Dans toute la gamme d'énergie de 388 à 430 eV de photons que nous avons explorée, seules trois structures intenses ont été observées autour de 400 eV. La figure 1 montre les spectres expérimentaux comparés aux calculs théoriques (MCDF et R-matrice) pour la gamme d'énergie de photons de 398 eV à 406 eV. Les résultats expérimentaux ont été pris avec une résolution en énergie de photon de 300 meV. Afin de comparer directement avec les mesures expérimentales, les résultats MCDF et R-matrice ont été convolutés avec une fonction Gaussienne pour simuler les mesures.

2-ions N 2+

Pour les ions N 2+ , le processus le plus intense de promotion d'un électron de la couche K de l'ion N 2+ dans l'état fondamental 1s 2 2s 2 2p 3 P est la photoexcitation 1s → 2p, hν + N 2+ (1s 2 2s 2 2p 2 P 0 ) → N 2+ (1s 2s 2 2p 2 [ 3 P, 1 D, 1 S] 2 S , 2 P, 2 D) ↓ N 3+ (1s 2 2s 2 1 S) + e - ou N 3+ (1s 2 2s np 1 P 0 ) + e - À partir du terme métastable 4 P aussi peuplé dans la source d'ions, les processus de photoexcitation 1s → 2p deviennent:

x hν + N 2+ (1s 2 2s 2p 2 4 P) → N 2+ (1s 2s[ 1,3 S] 2p 3 [ 4 S 0 , 2 D 0 , 2 P 0 ] 4 P, 4 D, 4 S) ↓ N 3+ (1s 2 2s np 3 P 0 ) + e -.

La figure 2 présente la variation de la section efficace de photoionisation en fonction de l'énergie des photons dans la région des photoexcitation 1s → 2p utilisant trois largeurs de bande d'excitation différentes, 90 meV, 60 meV and 30 meV, respectivement. Les plus forts processus d'excitation en couche K dans l'interaction d'un photon avec l'état fondamental 1s 2 2s 2 1 S et l'état métastable 1s 2 2s2p 3 p 0 de N 3+ est le processus de photoexcitation 1s → np, respectivement hν + N 3+ (1s 2 2s 2 1 S) → N 3+ (1s2s 2 np [ 1 P 0 ]) ↓ N 4+ (1s 2 2s 2 S) + e -, ou N 4+ (1s 2 2p 2 P) + e -, n=2, 3 et hν + N 3+ (1s 2 2s2p 3 p 0 ) ↓ N 3+ (1s2s [[ 1,3 S] 2p 2 ( 3 P, 1 D, 1 S) 3 S, 3 P, 3 D) ↓ N 4+ (1s 2 2s 2 S) + e -, ou N 4+ (1s 2 2p 2 P) + e -.

La figure 3 montre la variation des sections efficaces de photoionisation expérimentale et théoriques avec l'énergie du photon de 412 eV à 415 eV, enregistrés avec 111 meV de résolution de photons. xiii 5-Conclusions pour les ions azote J'ai présenté les premières mesures des sections efficaces de photoionisation la couche K des ions iso-nucléaires d'azote dans le fondamental et les états métastables. Les mesures ont été comparées aux résultats des méthodes théoriques SCUNC, MCDF et R-matrice, obtenus en collaboration avec plusieurs groupes de théoriciens. Les mesures de l'énergie d'excitation absolue, les largeurs naturelles et de la force d'oscillateur des plus intenses raies de photoexcitations 1s → np (n = 2, 3) ont été extraites des spectres de sections efficaces de photoionisation. Ces mesures sont essentielles pour améliorer les résultats des codes théoriques, comme MCDF et R-matrice, comme il a été démontré ici, un excellent accord trouvé entre les théories et expérience étant obtenu après l'obtention de mes résultats. Ces résultats aboutissent à l'amélioration des codes décrivant l'opacité des plasmas comme X STAR ou OPAS.

C-Photoionisation en couche de valence d'ions atomiques et de petits ions moléculaires en utilisant un piège FTICR

1-Introduction

Un problème rencontré en utilisant la technique de faisceaux colinéaires est la contribution des ions dans des états métastables, qui rend plus difficile la comparaison des données expérimentales avec les données théoriques car la population relative dans les différents états est la plupart du temps inconnue. Nous avons étudié un moyen d'éviter ce problème, piéger les ions pour permettre aux états excités de décroitre radiativement à l'état fondamental. Les sections efficaces de photoionisation en couche de valence ont été étudiées pour les ions Xe + et Kr + et pour des petits ions moléculaires CO et N , en utilisant à la fois les techniques de xiv faisceaux colinéaires et un piège à ions de type FTICR (Fourier Transform Ion Cyclotron Resonance). La combinaison des deux techniques permet la mesure de la section efficace de photoionisation des ions dans l'état fondamental pur sur une échelle absolue. Les sections efficaces expérimentales de photoionisation en couche de valences masurées pour les ions Xe + et Kr + ont été comparées aux résultats de calculs MCDF obtenus pour le processus de photoionisation directe.

Les expériences avec le piège (MICRA) ont été effectuées sur la ligne de faisceau DESIRS de SOLEIL, et les expériences avec la technique de faisceaux colinéaires ont été effectuées précédemment en utilisant le montage d'ASTRID (Aarhus STorage Ring in Denmark).

2-Photoionisation des ions Xe + et Kr + Les figures 5 et 6 montrent la comparaison entre la section efficace photoionisation des ions Kr + et Xe + , utilisant à la fois la technique de faisceaux colinéaires (panneau supérieur) et le piège à ions. Les spectres de Kr + ont été mesurés avec un retard différent (Δt = 0 ms, panneau du milieu, et 520 ms, panneau du bas) entre la production des ions et l'irradiation par le rayonnement synchrotron. Comme la densité des ions dans la cellule et le facteur de forme ne sont pas connues, les sections efficaces sont obtenues avec la trappe seulement en valeurs relatives. Ils ont été normalisés sur les données du panneau supérieur en supposant que la surface sous les spectres obtenus avec le piège et le montage en faisceaux colinéaires sont les mêmes. Bien que le spectre enregistré avec le piège directement après la production des ions (Δt = 0 ms) montre une grande dispersion statistique, il reproduit la section obtenue avec le montage de faisceau colinéaire, montrant que nous avons la contribution des deux 2 P3/2 et 2 P1/2 niveaux avec, probablement, une population d'ions dans l'état métastable plus faible dans le piège. Une attente de 520 ms, un temps plus long que la durée de vie des états métastables (350 ms), permet aux ions dans ce niveau excité de décroître radiativement vers xv l'état fondamental. On observe en effet que l'intensité des structures en-dessous du premier seuil d'ionisation (24,36 eV) n'est plus observable, ce qui démontre que tous les ions qui restent à l'intérieur du piège sont dans le niveau fondamental.

Figure 5: Comparaison de la section efficace de photoionisation de l'ion Kr + en fonction de l'énergie des photons à l'aide du montage en faisceaux colinéaires d'ASTRID (en haut) et du piège FTICR sur la ligne de lumière DESIRS de SOLEIL (au milieu et en bas), obtenus à des délais différents entre le temps de production et d'irradiation des ions.

Pour les ions Xe + , La figure 6 présente les résultats obtenus avec la technique des faisceaux colinéaires (panneau supérieur) et le piège à ions (panneau du milieu), après convolution après convolution par un profil Gaussien de la section efficace obtenue à ASTRID pour obtenir la même résolution totale que celle utilisée pour enregistrer la section efficace avec le piège (100 meV). Un retard de 1 s été utilisé avant l'irradiation des ions, un temps beaucoup plus long que la durée de vie de l'état 2 P1/2 métastable (50 ms). De toute évidence, tout le signal sous l'énergie du premier seuil d'ionisation (20,98 eV) observée sur le spectre du haut a totalement disparu dans le spectre enregistré avec le piège, montrant que tous les ions irradiés étaient dans l'état fondamental 2 P3/2. xvi De la différence des deux spectres, il est possible d'estimer la section efficace pour les ions dans les niveaux métastables en supposant une population statistique des ions dans l'état fondamental et l'état métastable dans l'expérience de faisceau colinéaire. Le résultat est représenté sur le panneau du bas des figures 6.

Figure 6: Comparaison de la section efficace de photoionisation de l'ion Xe + en fonction de l'énergie des photons mesurées à l'aide du montage en faisceau colinéaire d'ASTRID (en haut) et du piège FTICR (au milieu). La différence entre les deux spectres permet la détermination de la section efficace pour les ions dans le pur état métastable (en bas). xvii passante équivalente d'environ 2 eV. En raison de l'utilisation de l'ordre zéro, la section efficace n'a pu être déterminée qu'en valeur relative avec MAIA, la contribution relative des différentes harmoniques de l'onduleur étant inconnue.

Certaines structures observées sur le spectre obtenu avec MICRA à 27,4 eV, 29,0 eV, 31,2 eV et de 33,8 eV peuvent être associés à des seuils d'ionisation connus de N2 + (lignes verticales dans le panneau supérieur), correspondant respectivement aux états excités de N2 2+ X 1 Σg + , A 1 Πu, d 3 Πg et 1 Πg. Ces structures ne sont pas observées sur le spectre enregistré en utilisant le montage MAIA. Une explication pourrait être que les deux techniques ne mesurent pas exactement la même chose. Pour l'expérience réalisée avec MICRA, de l'azote isotopique 14 N 15 N + a été utilisé, permettant de distinguer le processus de photoionisation de la photodissociation, au contraire du spectre enregistré avec MAIA qui a été obtenu avec de l'azote naturel 14 N2 + . Deuxièmement, les durées de vie typiques pour la détection des dications sont très différentes, de l'ordre de la μsec pour MAIA, et de l'ordre de la ms pour MICRA, résultant en différents états finaux possibles qui peuvent être détectés pour le dication dans le processus de photoionisation. Enfin, les cations sont produits avec une énergie interne élevée dans l'ECRIS, pouvant résulter en la présence de cations dans des états excités dans la région d'interaction avec les photons.

Dans le cas du cation CO2 + , la figure 8 montre la variation de la section efficace mesurée avec le piège entre 23 eV et de 25,8 eV d'énergie des photons. Elle a été obtenue avec les photons diffractés en premier ordre par le monochromateur de DESIRS et une bande passante de 130 meV. La détection des dications a été effectuée dans la fenêtre de temps de 10 à 1000 ms après leur production par les photons, de telle manière que principalement les dications à l'état fondamental sont présents. Comme dans le cas du cation N2 + , des structures sont observées et peuvent être associées à des états excités observés précédemment du dication CO2 + , indiquées par les barres verticales de couleur verte sur la figure. I would like to thank the PLEIADES beamline staff for providing us with a good environment and facilities to complete this project. Also, I would like to take the opportunity to thank the "Centre National de la Recherche Scientifique (CNRS)" for its financial support through the whole work. Ninety nine percent of the matter in the Universe is in the form of plasma, a "soup" of ions, electrons and photons in strong interaction, often called the fourth state of matter. The experimental study on the interaction between ions and photons presents several interests.

From a fundamental point of view first, it is the only way to test the validity of the different models describing these processes. The multiple effects of electron-electron interactions involved in atomic and molecular structures have to be taken into account. A good understanding of these processes in simple systems like atoms or ions is a prerequisite for their understanding in more complex states of matter. From a more applied point of view, spectroscopy is the preferred tool for describing the thermodynamic and chemical composition of plasmas. Most of our information on astrophysical and laboratory plasmas is brought by photons. To obtain laboratory data on the interaction processes between the photons and atomic multiply charged ions which inhabit hot plasmas, and the positive and negative molecular ions involved in the chemistry of planetary ionospheres, it is essential for the interpretation of the spectra recorded by space observatories. In dense and hot plasmas, knowledge of the inner shell photoabsorption cross sections is also essential for the description of radiative transfer inside the plasma. This mechanism is very important for inertial confinement fusion. It also occurs in astrophysics where it controls, for example, the radiation-gravitational equilibrium in stars like the Sun.

The study of photoionization processes gives a wonderful tool to investigate the dynamical interplay produced by the electron-electron correlations inside the many body atomic systems and the relativistic effects [1]. Spectroscopic studies in the soft x-ray energy range of the K-shell photoionization processes in C, N, O, Ne, S and Si ions and L-shell photoionization of Fe and Ni ions are essential for the description of many physical systems, including the upper planetary atmospheres, laboratory plasmas involved in industrial applications, bright blazar, extreme environments in the active galactic nuclei (AGNs), x-ray binary systems, cataclysmic variable star (CVs), Wolf-Rayet stars, interstellar media (ISM), novae, supernovae... Extensive photoionization cross-section calculations, using sophisticated codes such as Rmatrix and MCDF (Multi-Configurational Dirac-Fock), have been performed near the Kshell ionization threshold along isoelectronic and isonuclear sequences of these ions to interpret the wealth of x-ray spectra currently provided by Chandra and XMM-Newton satellites [2][3][4][5][6][7][8].

Up to now, few experimental data are available to benchmark these calculations. First, photoabsorption cross sections were measured in laser produced plasma [9]. With the advent age of the third generation synchrotron radiation sources, ion-photon merged beam experiments were developed [10]. This technique allows for the determination of absolute photoionization cross sections for singly and multiply-charged ions, opening the way to systematic studies along isoelectronic sequences. However, most of the data have been obtained up to now in the photon energy range below 200 eV, excluding the K-shell for most of the ions, and almost no experiments were performed on the photoionization of molecular ions. Very recently, a new technique is in fast development in the synchrotron facilities, the use of ion traps [11,12], it opens the possibility to study cold ions and, thanks to their high mass resolution, all kind of molecular ions, including clusters and ions of biological interest.

In this thesis, I will present results we have obtained on the K-shell photoionization cross sections for the nitrogen isonuclear sequence using a merged beam set up, as well as on valence shell of small molecular ions (N2 + , CO2 + ) found in planetary ionospheres using an ion trap. Both have been obtained at SOLEIL, the French synchrotron radiation facility. In this introduction chapter, it recalls first what the main photoionization processes involved in these studies are. My experimental work has been performed in close collaboration with several groups of theoreticians, and I will present very shortly the theoretical codes they used. Then I will present what was known before this work on the photoionization processes for these ions, at the end, to introduce the experiments using the ion traps.

Selected Types of Interaction Processes between an Atomic Target and Photons

The various processes can be ordered following increasing photon energy.

Photoabsorption

For the photon energy below the first ionization threshold, the only (I will not discuss here scattering of the photon) allowed process is photoabsorption

ʋ + → *
If the target X (atom or ions) is initially in its ground state, this process occurs when the photon energy hʋ is equal to the energy of an unoccupied bound excited level (boundbound transition). It is a resonant process, leaving the target in an excited state X * which will decay via fluorescence.

Direct Photoionization

If the photon energy equals or exceeds the binding energy of an electron of the target, the photoelectric effect can take place,

ʋ + → + +
This single direct photoionization process has a threshold, and the behavior of the photoionization cross section as a function of photon energy appears like a step function starting from zero at the ionization threshold, then usually smoothly decreasing with increasing photon energy.

Resonant Photoionization

If the photon energy exceeds the first ionization threshold and is equal to the difference in energy of a bound unoccupied state with a bound occupied state, resonances are observed above the direct photoionization cross section at discrete photon energies corresponding to the photoexcitation process:

ʋ + ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ * ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ +
The target is left in a highly excited state X * which will mainly decay (in the energy range considered in this work) by autoionization, resulting by electron correlations in the ejection of a valence electron. The resonance photoionization is often described like a two-step process, photoexcitation and autoionization (although it must be treated theoretically like a single process). As the direct and resonant photoionization channels have the same final state, they interfere and produce asymmetric profiles called Fano profiles [13].

Double Photoionization

If the photon energy is greater than the energy needed to remove two (or more) electrons from the target, double (or multiple) photoionization process occurs. Two scenarios may be distinguished.

The Direct Double Photoionization

Two electrons are ejected simultaneously:

ʋ + → + 1.1.4.

The Auger Decay

After direct single photoionization in an inner-shell, the target is left in a highly excited state which can decay under the effect of electron correlations by the Auger effect: an electron from the valence shell fills the hole in the inner shell and the excess energy generated is used to eject a second valence electron to the continuum (called the Auger electron)

ʋ + ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ * + ⎯⎯⎯ +

Theoretical Background

After the theory of the photoelectric effect is known since the beginning of 20 th century [14], the modern theories on atomic photoionization developed rapidly in the 60's with new experimental observations in the XUV range at the emerging synchrotron radiation facilities [15]. It became clear that independent particle models were insufficient to describe observed phenomena like inner shell resonant photoexcitation. Nowadays, many approaches to calculate the electron correlation effects have been developed but in the following section, I will only discuss the fundamental concept of R-matrix and MCDF calculations, the two models have been used in this work to interpret the experimental observations.

R-Matrix Calculations

The origin of R-matrix theory was derived by Eugene P. Wigner to calculate the resonance formula for nuclear reactions [16], and then the theory was extended and developed for calculating atomic and molecular collisions [17][18][19][20][21]. The idea of R-matrix theory is to divide the configuration space into two regions: a sphere of radius r where electron correlations are strong (Figure 1) and an outer region where the escaping electron sees a pure Coulomb field and the final wave function can be calculated analytically. The boundary radius a0 is chosen to be large enough for the ionic core wave function to be zero for r > a0.

MCDF (Multi Configurational Dirac-Fock) Calculations

The MCDF formalism is an extension of the Hartree-Fock model [22]. It approximates the electronic N-body wave function of a relativistic atomic and molecular system, resulting in a coupled system of nonlinear eigenvalue equations. The MCDF wavefunctions, ΨMCDF, is written as a linear combination of Slater determinants associated with the occupied atomic orbitals,

=

The determinants are constructed on the basis of one-electron relativistic functions written in the form of Dirac spinors Θ (r, θ, φ):

Θ ( , , ) = 1 ( ) Ω ( , ) ( ) Ω ( , )
These wavefunctions are defined by the (principal) quantum numbers n, l (orbital angular momentum), j (total angular momentum) and mj (j projection of quantization axis) associated with the electronic configuration of the system. The relativistic-electron wavefunctions are divided into a radial part (Pnlj (r) and iQnlj (r)) and an angular part Ωljm (θ, φ) built on the spherical harmonic and geometric Clebsch-Gordan. Pnlj(r) radial functions and Qnlj (r) are the unknowns of the system, with the coefficients ci. The total wavefunction ΨMCDF (normalized), which minimizes the total energy E of the system of n electrons associated with a configuration, should therefore satisfy the following differential equations:

= 〈 | | 〉 = 〈 | | 〉 = 〈 | | 〉
The energy is calculated to minimize the following energy functional,

= ∑ (2 + 1) 2 ∑ (2 + 1) + ∑ (2 + 1) 2 ∑ (2 + 1)
where α and β run over all the initial and final states, respectively [23][24][25]. In contrast to the R-matrix, the MCDF approximation calculates separately the direct and resonant photoionization processes, thus neglecting the interference term between the two channels.

1.3 What is Known on K-shell Photoionization of Nitrogen Ions?

K-shell photoionization of Li-like to C-like ions plays a particular role in astrophysics.

Absorption and emission lines from these ions have been observed in hot plasmas like active galaxy nuclei or quasars. For example, the description of the formation in the blue compact dwarf galaxy Mrk 209 requires the knowledge of K-shell photoionization cross sections of carbon, oxygen, and nitrogen in their various stages of ionization [26,27].

Due to the lack of experimental data, the photoionization cross sections of nitrogen ions were known mainly from theoretical works. Reilman and Manson [28] calculated the photoionization cross sections for inner shell C-like ions using the Hartree-Slater wavefunctions of Herman and Skillman [29], and Verner and co-workers [30] by using Dirac-Slater potentials within a central field approximation. These methods have proven to give accurate photoionization cross section data at high photon energy, far from the thresholds, but give poor results near thresholds where resonant structure, not described by these simple models, are dominating. More recently, significant efforts have been made to improve the quality of the theoretical methods using more sophisticated approaches. P.

Quinet and co-workers reviewed the recent advances in the determination of atomic parameters for modeling K lines in cosmically abundant elements [7]. Garcia and coworkers [31] calculated the photoionization cross sections along the Nitrogen isonuclear series using the optical potential method within the Breit-Pauli R-matrix formalism [32][33][34]. This work has included the effect of radiation damping, but takes only into account the photoionization from ground state, thus neglecting any contribution from the metastable states.

These theoretical results remain largely untested by experiments. Up to now, experimental results on K-shell photoionization for light ions have been performed by different groups using the merged beam technique: He-like Li + [35][36][37]. Li atoms [38], Li-like B 2+ [39], C 3+

[40] C 2+ [41], B-like C + [42], C-like [2], N-like O + [43], F-like Ne + [44], neutral nitrogen [45] and oxygen [46,47]. along with valence shell studies on B-like ions, N 2+ , O 3+ and F 4+ [48], C-like Ar + [49] As-like Se + [50][51][52] and Br-like Kr + [53]. Before this work, no Kshell photoionization cross sections were measured for nitrogen ions.

Only Electron-Beam-lon-Trap (EBIT) measurements have been performed at the Lawrence Livermore National Laboratory (LLNL) for the inner-shell ls → 2p transitions in He-like and Li-like nitrogen ions [54], data used in plasma modelling to determine impurity transport properties [55]. Let's note that, in EBIT experiments, the spectra are contaminated and blended with ions in multiple stages of ionization, making the spectral interpretation fraught with difficulties, unlike the cleaner spectra obtained with the merged beam setups at synchrotron radiation facilities.

Experiments with Ion Traps

One difficulty, suffered both by DLPP (Dual Laser Produced Plasma) and merged beam techniques, relates to the composition of the ionic target. Very often, it is made of a mixture of ions in the ground and metastable states, most of the time in unknown relative populations. One way to avoid this problem is the use of ion traps. The first coupling of an ion trap with synchrotron radiation for the study of photoionization processes on ions was performed by S. D. Kravis and his group [56,57]. They measured the distribution of photoion charge stages after K-shell photoionization of Ar 2+ ions stored in a Penning ion trap using broadband synchrotron radiation [56,57]. Nevertheless, the use of ions traps has developed only recently with the demonstration at ELETTRA by Thissen et al [11] of the capability to cool down the ions. Using a Fourier Transform Ion Cyclotron Resonance (FTICR) trap, they measured the photoionization cross section of Xe + ions in the pure ground state. Following these pioneering works, different ion traps are now used at synchrotron radiation facilities: EBIT for the study of photoionization of multiply-charged ions (N 3+ and Ar 8+ [58], Fe 14+ [59]); linear traps for the study of size-selected cobalt cluster ions [60], and metallic singly-charged ions [61] or large molecular ions [62,63]. We are trying to develop this technique on the DESIRS beam line of SOLEIL, using the FTICR trap used previously at ELETTRA [11]. I will present the first results we have obtained on Kr + and Xe + ions, as well as small molecular ions (N 2+ and CO 2+ ).

Concerning Xe + and for Kr + ions, most of the previous experiments have been performed using the merged beam technique in the energy range of 4d photoexcitation.

Photoionization of 4d-eIectrons in single charged Xe ions was measured by M. Sano and his group [64]. T Koizumi and co-workers extended this study to multiply charged Xe q+ (q=l-3) ions [65]. Absolute cross sections for the photoionization of 4d electrons in Xe + and Xe 2+ ions were explored by P. Andersen et al [65], and absolute photoionization cross section measurements of Xe + ions in the 4d threshold energy region have been measured

by Yoh Itoh and co-workers [66,67,11,12].

On the theoretical side, again only the 4d excitation has aroused interest for Xe ions.

Zhifan Chen and Alfred Z. Msezane calculated the 4d giant resonance of Xe + ion using the random-phase approximation with exchange (RPAE) method [68]. Very recently, following our measurements, B M McLaughlin and C P Ballance have published R-matrix calculations for photoionization cross section of halogen-like ions Kr + and Xe + ions, in very good agreement with our experimental results [69].

Up to now, very few study on photoionization for small molecules have been performed.

Only the cross section for the CO + ion has been obtained using the merged beam technique [69,70].

This manuscript is organized as following: In chapter II, I will describe the merged beam setup and the experimental procedure used to extract the absolute cross sections. The results on nitrogen ions will be presented in chapter III. Chapter IV will describe the ion trap and the results obtained. In the last chapter, I will present the results of ion transport simulations I did to improve the merged beam setup.

Chapter 2: Merged Beam Setup and Experimental Procedure

Introduction

Since a few decades, photoionization studies have drawn attention of atomic and molecular physicists, because they offer a powerful tool to understand the electron correlation effects in many electrons systems. They allow also to obtain spectroscopic information, such as ionization thresholds, excitation energy of the levels... as well as dynamical information on the ions, such as photoionization cross section, oscillator strength... The data which can be obtained in absolute values, are of great use for diagnostics and modeling of plasmas.

Achievement of such studies on atomic and molecular ions requires a powerful technique, that gives high quality data and the freedom to choose the target, and so appeared the merged beam technique.

The principle of the merged beam technique is to overlap a photon beam of a given energy with a beam of ions on the same axis in an interaction region. Absolute photoionization cross section as low as 10 -19 cm 2 can be measured [1]. Synchrotron radiation is the ideal source of photons for these studies due to its high brilliance and stability. The first merged beam setup was developed by Lyon et al [2] at Daresbury in the 80's. It was derived from the Peart et al [3] setup used for the study of electron impact processes on atomic ions. Now there are several setups of this type in activity around the world in Denmark (ASTRID), US (ALS), Germany (PETRA) and France (SOLEIL).

A scheme of the merged beam setup of Lyon et al is shown on Figure 1. It consists of three main parts. The first one is the ion production. The parent ions were collected in a Faraday cup C1 while the doubly charged ions produced in the interaction with the photons (the so-called photoions) were counted by the detector D. The ion beams were separated by a magnetic dipole (M2) and the kinetic energy of the photoions analyzed by an electrostatic deflector (E) fields. The photon flux was monitored by a calibrated diode (PD). The inset shows the collimator C and the biased cylinder on a larger scale [2].

The ions were produced using a surface ionization ion source S, allowing to produce singly-charged atomic ions. The ions were extracted by applying a positive bias on the ion source, then analyzed in charge to mass ratio by a magnetic dipole M1 before reaching the interaction region I. At the entrance of the interaction region, the ion beam was collimated to match the size of the photon beam (2×3 mm). The second part is the interaction region I.

Here the ion beam is overlapped with a photon beam of monochromatic VUV radiation produced in a dipole magnet of the Daresbury synchrotron radiation source (SRS) storage ring. The photoions, the ions which have gained one or several charges in the interaction with the photons, were produced in this region. In the final stage, a second magnet M2 demerged the photoions, which were counted using a calibrated Johnston electron multiplier D, from the parent ions which were collected in a Faraday cup C1.

For our experiments, I have used the merged beam setup MAIA (Multi Analysis Ions Apparatus) which is installed on one branch of the PLEIADES beam line of SOLEIL. In the following, I will describe successively the photon source, the MAIA setup and the procedure used to extract the absolute photoionization cross sections from the measurements.

The Light Source

Synchrotron Radiation

Synchrotron radiation is the light produced when charged particles are accelerated.

Generally, it is produced in storage rings that run charged particles (most of the time electrons) on a confined, near-circular path by means of electric and magnetic fields.

Because the charged particles are relativistic (their velocity is close to the speed of light), the radiation is emitted in a small cone around the forward direction of the particles, with an opening angle 1/γ (typically between 0.1 mrad to 1 mrad) where γ is the particle energy relative to its rest mass (m0), with v the particle velocity and c the speed of light [4].

1 γ = 1 -⁄ = m c
The particles can be accelerated in two ways, by changing the magnitude of their speed or changing the direction of their velocity. It is the second way which is used in storage ring, using either the dipole magnets that bend the trajectory of the charged particles on a quasicircular path or insertion devices, such as undulators or wigglers that are installed on the straight sections between the bending magnets.

Bending Magnet

The bending magnets were the first used radiation sources. The photons are emitted tangentially to the orbit (Figure 2). The emission spectrum of a bending magnet is continuous, extending from infra-red to X-ray.

Figure 2. Scheme of bending magnet deflecting the fast electrons onto a curved path, thereby causing them to emit synchrotron radiation.

Undulator and Wiggler

An undulator and wiggler are a periodic structure of magnets (figure 3). A sinusoidal magnetic field is produced by permanent magnets or coils. The strength of the magnetic field inside the undulator depends on the undulator gap (g).

Figure 3. Scheme of an undulator with magnetic period λu.

The magnetic field causes electrons to oscillate and to emit synchrotron radiation at each oscillation. The emission spectrum is composed of lines (harmonics). The position of the lines can be adjusted by varying the strength of the magnetic field (varying the current in the coils or the gap in the case of permanent magnets). Compared to bending magnet emission, the photon flux at the energy of the maximum of the first harmonic is multiplied by N 2 , where N is the number of periods of the undulator. The energy width of the first harmonic is of the order of the energy of the 1 st harmonic divided by the number of periods N.

The difference between an undulator and a wiggler is the strength of the magnetic field. As for bending magnets, the emission spectrum of a wiggler is continuous, but with a maximum flux at maximum increased by a factor N.

SOLEIL Storage Ring

SOLEIL is the French national synchrotron radiation facility. The storage ring supports now twenty six beamlines, using photons emitted from bending magnets, undulators and wigglers (Figure 4). Each beam line was constructed for a particular field in science. We can see some technical data of the SOLEIL in table 1.

Component Electron bunch

Nominal energy E0

PLEIADES Beamline

PLEIADES is a high resolution soft X-ray beamline. It covers the energy range from 10 eV to 1 KeV with a resolving power up to 10 5 at 50 eV. It was built for studying spectroscopy of diluted samples (ions, atoms, molecules, nanoparticles…). It uses two undulators, a permanent magnet 80mm period Apple II type undulator to cover the energy range 80 to 1000 eV, and an electromagnetic undulator of 256 mm period for the range 10-80 eV. Both have a quasi-periodic design to enhance the spectral purity. They also provide to the users all types of photon polarization (horizontal, vertical, tilted linear, circular and elliptical).

For our studies, we used the horizontal linear polarization mode of the HU80 undulator which gives the highest photon flux.

The Monochromator:

Most applications require a well-defined photon energy. This is realized using monochromators, delivering photons of adjustable energy and spectral bandwidth. They are necessary when the synchrotron radiation is produced by bending magnets or wigglers, or when a bandwidth smaller than the width of the harmonics of the undulators is required.

Following the energy range, gratings (for visible light to soft X-rays) or crystals (X-rays) are used.

On the PLEIADES beamline, a plane grating monochromator is used. Four gratings with 400, 600, 1600 and 2400 lines/mm, with varied groove depth (VGD) and varied line spacing (VLS), can be selected as a function of the desired resolution. This kind of monochromator, schematically shown on Figure 5, is composed of a plane grating diffracting the light in the vertical plane, and a compensating plane mirror to keep the exit beam horizontal. Focalization on the fixed in position exit slit, with adjustable width horizontal blades, is performed by upstream spherical mirrors. The spectral purity is provided, in addition of the quasi periodic design of the undulators, by the use of the VGD for the gratings, reducing the diffraction efficiency of higher diffraction orders as compared to the first order.

Figure 5: Scheme of PLEIADES monochromator. The undulator source is located 26 m upstream the grating. The monochromator uses a plane grating and a plane mirror in fixed exit configuration [5].

MAIA setup is installed on one of the three optical branches of the beam line, each offering different focusing properties of the beam. We can see some technical data on the PLEIADES beamline in table 2. Table 2: Some technical data on the PLEIADES beamline.

Figure 6 gives the photon flux available on the MAIA branch for the different gratings and a resolving power of 10000 (7000 for grating 600 l/mm (G2)), and for a fixed slit width of 200 m for grating 2 (black curve). Figure 7 shows a scheme of the MAIA set-up. As the merged beam setup of Daresbury shown on Figure 1, it consists of three main parts described down: the production of the ionic target, the interaction region with the photons and the detection of the photoions.

Ion Production

Ion Source

The trip of the ions starts when molecular nitrogen is injected inside the ion source (Figure 8). We used an Electron Cyclotron Resonance Ions Source (ECRIS), especially designed for this set-up at the CEA (Commissariat à l'Energie Atomique et aux Energies Alternatives) in Grenoble and previously tested at LURE (Laboratoire pour l'Utilisation de Rayonnement Electromagnétique) in Orsay. The nitrogen gas is heated and ionized by a 12.6 GHz radio frequency wave to produce a nitrogen plasma that contains several charged stages of nitrogen ions. The plasma is confined by a permanent magnet producing a 1 T magnetic field. The heating frequency is equal to the cyclotron frequency of the electrons in the plasma. By this way, the electrons are accelerated, producing by sequential ionization the multiply-charged ions. The ECRIS is designed to produce high currents of multiply-charged ions (up to 1 mA total current). The current is stable and continuous during the acquisition of spectra, which takes often several hours to be achieved. The main disadvantage of the ECRIS is it produces ions not only in the ground state, but also ions in all the excited states, and in particular long living metastable states which can contribute to the photoionization signal.

Ion Extraction

The multiply-charged ions are extracted from the plasma by a two electrode extraction system. A 2 kV positive voltage is applied on the first electrode (the plasma electrode, which is at the same potential as the ion source). The second electrode is grounded, in such a way that the ions of charge q are accelerated to 2 x q keV final kinetic energy. The ion beam is focused by an einzel lens located directly after the extraction electrodes.

In the simulation chapter, we will see more details about the effect of the gap and the voltages on the electrodes on the extracted ion beam characteristics.

Ion Selection

All charge stages of the nitrogen ions are present in the extracted beam, as well as impurities present in the plasma chamber (water, oxygen, carbon,…). A dipole magnet (MAGNET 1 on Figure 7) is used to select a given charge stage as mass per charge. It is a 0.4 m radius magnet producing up to 0.35 T vertical magnetic fields. A mass spectrum of the ions extracted from the ECRIS is performed by scanning the current in the coils of the magnet, and measuring the ion current collected in the Faraday cup FC2 located behind the slit S2. These scans are used to determine the correct values of the magnetic field needed to select the ion that we are interested in. Let's note that a magnetic device does not allow us to distinguish ions which have the same mass per charge ratio, like for example and N + ions. The figure 9 shows an example of a mass spectrum measured at the beginning of the experiments on nitrogen ions. After selection of the correct ions with the dipole magnet, we have to transport them to the interaction region. The dipole magnet focuses the ions at the position of the slit number two (S2 on figure 7), which is used also to control the size of the ion beam and to collimate it. Then an einzel lens (E2) focuses the ion at the position of slit S3, also used as a

collimator.

An einzel lens is a charged particle lens. It consists of three cylinders that focuses without changing the energy of the beam at the exit. The two outer cylinders are grounded, while the middle one is biased as shown in the figure 10. Horizontal and vertical steerers (St on Figure 7), placed at different places along the beam transport line, allow us to move the ion beam in the horizontal and/or vertical direction. Steerers are parallel plate capacitors, composed of two plates biased to potentials +V and -V, respectively. After passing slit S3, the selected ions are deflected by a 45 • spherical electrostatic deflector (ED1) to be merged with the photon beam inside the interaction region (IR).

Figure 10: Scheme of an einzel lens.

The Interaction Region

The interaction region is a 50 cm long tube kept in ultra-high vacuum (of the range 10 -10 torr). The overlap between ion beam and photon beam occurs inside this tube. Photoions are produced inside this region, but also outside where the 2 beams overlap. In order to tag the ions produced inside the tube, the tube is polarized (typically to 1 kV) in such a way the ions which have gained one (or several) charges by interaction with the photons inside the tube have at the exit of the tube a different velocity than the same ions produced outside the tube.

The intensity of the photoions signal depends on the quality of the overlap between the two beams. To optimize and quantify the overlap, we use three sets of xy scanners placed at each end and in the middle of the interaction region. Each scanner is a 0.2 mm width slit moved across the ion and photon beams. The current of primary ion passing through the slit is measured in the Faraday cup FC4 at each position of the slit. In the same way, the current of photons passing through the slits is measured simultaneously by the photodiode PD. Figure 11 gives an example of profiles measured in the horizontal and vertical planes for both the ion (blue curves) and the photon (red curves) beams at the three places in the tube. Ultra-high vacuum is required in order to minimize the contamination of the photoions signal by collisional ionization between the parent ion beam and the residual gas inside the vacuum chamber. Let's note that the collisional ionization cross sections are often 2 to 3 orders of magnitude higher than the photoionization cross sections.

Photoions Detection

After the interaction region, the charge of the ions is analyzed by the second dipole magnet (MAGNET2). The dipole magnet separates the photoions from the parent ions, which are collected in a Faraday cup (FC4). The current in the coils of the magnet is adjusted to bend by 90° the trajectory of the photoions produced inside the interaction region. Then the velocity of the photoions is further analyzed by a 90° cylindrical electrostatic analyzer (ED2), and finally the transmitted photoions are counted by channelplates.

Determination of the Photoionization Cross Section

In this section, I will describe the procedure used to determine the data from our measurements: the photon energy and the absolute photoionization cross sections.

Absolute Photoionization Cross Section

During the interaction of the photon beam with the ion beam, in the case where one photon can interact with one ion to produce one photoelectron, the photoionization yield (S) obeys the Lambert-Beer law:

= (1 - )
Where F is the photon flux, n is the target density (the density of ions inside the interaction region), σ is the photoionization cross section, and Ɩ is the interaction length. In our experiment, typical values are l = 50 cm, σ ≈ 10 -18 cm 2 , and n ≈ 10 -5 cm -3 , in such a way the factor σnƖ « 1, and, after expansion of the exponential, the yield can be approximated by: ≈ To connect the cross section to the quantities we measure experimentally, let's consider the photoion yield dSxyz emitted by a small volume δxδyδz of the interaction region. If we assume the density of the target and the pressure inside this small interaction region remains constant, and if i and j are the ion and photon currents passing through the surface δxδy (we suppose the beams propagating in the z-direction), respectively, then the photon flux can be written:

= where e is the electron charge and η is the photodiode efficiency. The target density becomes:

= with v the ion velocity and q the ion charge stage, and:

= δ

Integration over the xy-plane gives:

= δ δ

Let's introduce the form factor F(z):

( ) = ∬ ∬ ∬ = ∬
which defines the overlap of the two current densities in the small volume δxδyδz, with the currents measured by the Faraday cup and the photodiode are = ∬ and J = ∬ . After substitution of F(z) and integration along the interaction region of length l, we get the expression for the photoionization cross section:

= * * * * * * * ∫ 1 ( )
In order to determine the absolute photoionization cross sections, all the parameters in the equation above have to be measured in absolute value:

-The photoion yield S is given by the counting rate of the channelplates. A chopper, placed at the exit of the photon beam line, allows us to subtract the false signal not produced by the photons. Main causes of noise are collisional ionization of the ions with the residual gas, charge stripping on the ion beam by the slits, autoionization during their flight of long living highly excited ions produced inside the ion source.

Typical frequency for the chopper is 0.1 Hz;

-The velocity v is obtained from the voltages applied on the ion source and on the tube in the interaction region;

-The efficiency η of the photodiode was calibrated in the 10-1000 eV energy range at the Physikalisch-Technische Bundesanstalt (PTB) beam line at BESSY in Berlin. It is verified by using a Samson type gas cell [6,7]. The gas cell, shown in the figure 12, is a chamber in which is injected a known pressure of a noble gas, typically helium, whose photoionization cross section is known with a high accuracy. The ion current produced on the electrodes by the photoionization of the gas allows to determine the photon flux (ϕ) passing through the cell. Using the Lambert-Beer law, it is given by:

= + (- )[1 - (-(2 )]
Where and are the current collected on the two collectors, n is determined from the pressure inside the cell, 2l is the total length of the two collectors, and σ is the photoionization cross section taken from literature, corrected for double photoionization [8].

Figure 12. Scheme of the ionization chamber used to calibrate the photodiode efficiency and the photon energy. The length of the collectors is l = 20 cm.

-The ion current I is measured with the Faraday cup FC4.

-The photon current J is measured with the calibrated photodiode PD.

-The efficiency of the channelplates Ω is determined by sending a weak ion beam (of the order of 10 -14 A) alternatively on the detector and the Faraday cup FC4, changing the field in MAGNET2. Comparison of the counting rate of the detector and the current measured in FC4 allows to directly determine Ω.

-The form factor is measured by using the xy scanners distributed along the interaction region, doing two approximations: the two dimensional currents can be replaced by a product of one dimension currents:

( , ) ≅ ( ) ( ) and ( , ) ≅ ( ) ( ) and we can replace ∫ by ∑ in the form factor equation, giving:

( ) = ∑ ( ) ∑ ( ) ∑ ( ) ( ) ∑ ( ) ∑ ( ) ∑ ( ) ( ) = ( ) * ( )
where the currents i and j are obtained at three different z positions from the profiles scans as the one shown on the figure 11, δx and δy are the steps used to do the scans in the horizontal and vertical planes. The interpolation and integration along the z axis is performed by fitting of the three measurements by a second order polynomial. Table 3 gives an example of magnitude for the various parameters measured at 412.5 eV photon energy for N 3+ ion, as well as their associated relative uncertainty. The total uncertainty on the cross sections is of the order of 15 %.

In another mode of data acquisition, the voltage is not applied on the interaction region. In this mode, only relative photoionization cross section can be determined, but with a better statistics since the full length of the interaction region between the photons and the ions is used. Table 3: Typical values for the experimental parameters involved in evaluating the absolute cross section measured at photon energy of 412.5 eV.

To test the validity of our experimental procedure, we measure regularly the photoionization cross section of the He + ion. As this ion is hydrogenic, its photoionization cross section can be calculated analytically accurately [9]. Figure 13 shows the curves of theoretical and experimental cross section. We note the very good agreement between the two curves.

The dispersion of the experimental points around the analytical curve is within the error bars (about 10%), confirming the systematic uncertainty on our measurements of the absolute cross sections, estimated to be between 10 and 15%. For the spectrum of figure 13, the statistical uncertainty on the number of photoions detected, given by the error bars on each point, is of the order of 5%.

Figure 13: Variation of theoretical [9], green curve, and experimental photoionization cross section of He + ion, points.

Determination of Photon Energy

The energy of the photons is determined using the gas cell. Ion yield spectra are recorded in regions where resonant ionization lines in rare gases are intense and have an excitation energy accurately known. For this work, we have used the transitions 2p → 4s in Ar at 244.39 eV excitation energy and 1s → π * in N2 at 400.865 eV [10]. The photon energy is also corrected for the Doppler shift produced by the velocity of the ions using the formula: where c is the speed of light, v is the ion speed inside the interaction region, E and E0 are the observable photon energy during the data acquisition and the actual photon energy respectively. For N ions, the Doppler Effect produces an accelerating shift between 0.1 and 0.2 eV.

Chapter 3: K-shell Photoionization Study of Nitrogen Isonuclear Series

In this chapter, I will present the results we have obtained on the K-shell photoionization of the first ions of the nitrogen iso-nuclear series N + to N 4+ . All these results have been obtained by using the merged beam set up MAIA on the PLEIADES beam line. Because the plasma produced in the ECRIS is very hot (electronic temperature of several hundreds of keV), the ions are produced not only in the ground state but also in all the excited states.

Most of these excited states have time to decay during the time of flight of the ions up to the interaction region (a few sec), except the metastable states with lifetime longer than 1 sec. This is in particular the case for the states with the same configuration as the ground state, whose contribution is observed in our spectra in addition to the ground state. The electronic configuration of singly ionized nitrogen (N + ) is 1s 2 2s 2 2p 2 and the ground level is 3 P0. In addition to the ground level, N + ion has five metastable levels, N + (1s 2 2s 2 2p 2 3 P1, 2, 1 D2, 1 S0) and N + (1s 2 2s2p 3 5 S ). The present investigation provides absolute values (experimental and theoretical) for the photoionization cross sections occurring in the interaction between a photon and N + ion in the ground level and these five metastable levels. Table 1 lists the configuration and excitation energy for all these levels [1]. In the photon energy range considered here, the most intense photoionization structures are expected to be the 1s → 2p photoexcitations: a 1s electron is excited to the 2p partially filled subshell, leaving a N + ion in a highly excited state (with a hole in the 1s subshell)

which will decay via autoionization to leave a N 2+ ion. As an example, for N + ions in the 3 P term, and following the electric dipolar selection rules, this process can be represented by:

hν + N + (1s 2 2s 2 2p 2 3 P) → N +* (1s2s 2 2p 3 3 S o , 3 P o , 3 D o ) ↓ N 2+ (1s 2 2s 2 2p 2 P o ) + e-
For the N + ions in the 1s 2 2s2p 3 5 S level, this process becomes:

hν + N + (1s 2 2s2p 3 5 S o ) → N + [1s2s[ 1,3 S]2p 4 ( 3 P, 1 D, 1 S)] 5 P ↓ N 2+ (1s 2 2s2p 2 4 P) + e-.

Theory

Multi-Configurational Dirac-Fock (MCDF) Calculations

The MCDF calculations have been performed based on full intermediate coupling in a jj basis using the code developed by Bruneau [2]. The photoexcitation cross sections involving the 1s→np transitions have been considered for all levels of the N + ground configuration, namely 1s 2 2s 2 2p 2 3 P0,1,2, 1 D2 and 1 S0.The metastable 1s 2 2s2p 3 5 S o 2 level has also been considered. The calculations were limited up to principal quantum number n = 4.

Only the electric dipole transitions were computed using Couloumb and Babushkin gauges.

The oscillator strengths obtained from the two gauges differ by less than 10%.

The average autoionization rate Гav has been calculated for the 1s2s 2 2p 3 configuration (with 1s vacancy) to be equal to 96 meV (which corresponds to a lifetime τ of 3.43 fs from the uncertainty principle ΔE Δt= h/4π). The MCDF spectra are constructed as a sum of Lorentzian profiles using the full width at half maximum (FWHM) autoionization rate Гav.

In order to compare the experimental results and the MCDF calculations, the MCDF results were convoluted with Gaussian functions of FWHM equal to the experimental width given by the monochromator band pass.

R-matrix

The non-relativistic R-matrix code written by C P Balance and D C Griffin [3] was used to determine all photoionization cross sections in LS coupling, for the ground 3 P and metastable 1 S, 1 D and 5 S terms. First, the bound state energies of N + were calculated. 390 levels were used in the close-coupling expansion. As for the MCDF calculations, in order to compare with the experimental measurements, the calculated cross sections were convoluted with Gaussian functions of FWHM equal to the experimental band pass to simulate the measurements.

Results and Discussion

In the photon energy range between 388 to 430 eV, three intense lines are observed, both in the single and double ionization channels. Figure 1 shows the experimental and theoretical calculations (MCDF and R-matrix) results for the photon energy range 398 eV to 406 eV. The experimental results were taken with a photon energy resolution of 300 meV. The black triangles display the single photoionization cross section measurements recorded in the absolute mode and the yellow circles the measurements obtained in the relative mode, normalized assuming the same area under the lines as for the absolute measurements. The double photoionization cross section obtained in the relative mode is given by the blue triangles. It was normalized assuming that the length of the interaction region was the same through collecting the data for single and double photoionization cross sections. Its contribution is only 4% of the single photoionization cross section. For the theoretical results, the dash line gives the results of the MCDF calculations and the red line of the R-matrix calculations. The errors bars on the absolute cross section give the total uncertainty on the cross section. et al [4]. Considering the lines one by one, it can be seen that:

1-1s2s 2 2p 3 3 D 0 peak (about 399.85 eV) is primarily resulting from photo-excitation of the ground 1s 2 2s 2 2p 2 3 P term.

2-1s2s2p 4 5 P resonance is resulting from the photo-excitation of the metastable state 1s 2 2s2p 3 5 S 0 .

3-1s2s 2 2p 3 1 P 0 resonance is resulting from the photo-excitation of the 1s 2 2s 2 2p 2 1 S metastable state.

4-The strong resonance (at about 400.63 eV) results from the overlap of resonances from ground 1s 2 2s 2 2p 2 3 P and 1s2s 2 2p 3 1 D 0 metastable terms.

5-This resonance is resulting from the photo-excitation of the 1s 2 2s 2 2p (eV), and the natural linewidths (meV) for the core photoexcited n = 2 states of N + ions in the photon energy region 399-403 eV. The experimental uncertainty is ± 40 meV for the resonance energies. The earlier MCDF calculations of Chen and co-workers [4] are included for comparison purposes.

The MCDF and R-matrix calculations give an overall good agreement with the experimental measurements, with a slight preference to the more sophisticated R-matrix calculations which gives a closer agreement with the experiment. The resonance energies of all peaks deviates at most by 45 meV (cf the average for lines 1+2), which is just outside the experimental error of 40 meV). All other resonance energies from the R-matrix work are within the 40 meV experimental uncertainty. Discrepancies occur in the position and autoionization linewidth of the resonances from our work when compared to the early MCDF calculations of Chen et al [4]. This was also highlighted by Garcia and coworkers [6], where outstanding discrepancies were found with that MCDF work, in particular wavelengths and A-coefficients for this C-like ion, which those authors believed to be due to numerical error by Chen et al [4].

K-shell Photoionization of Boron-like N 2+ Ion

Introduction

The ground state electronic configuration of doubly ionized nitrogen (N 2+ ) is 1s 2 2s 2 2p 2 P / . In addition to the ground state, the configuration of N 2+ ion has four metastable levels, N 2+ (1s 2 2s 2 2p 2 P3/2) and N 2+ (1s 2 2s2p 2 4 P1/2, 3/2, 5/2). The present investigation provides absolute values (experimental and theoretical) for the single photoionization cross sections occurring in the interaction between a photon with N 2+ ions both in the ground level (1s 2 2s 2 2p 2 P / ) and (1s 2 2s 2 2p 2 P / ) and (1s 2 2s2p 2 4 P1/2, 3/2, 5/2) metastable levels.

Table 3 presents a list of configurations for all ground and metastable levels.

In this study, the photoionization cross sections measurements of N 2+ are presented in the three photon energy ranges 404 → 410 eV which covers the 1s→2p photoexcitations, 439→442 eV which covers 1s→3p photoexcitations, and the photon energy range between 452→457 eV around the 1s direct photoionization threshold.

Theory

The experimental results were compared with the three theoretical approaches, MCDF, Rmatrix and Screening Constant by Unit Nuclear Charge Calculations (SCUNC), in order to identify the strongest 1s→2p and 1s→3p transition resonances observed in the experimental spectra.

MCDF Calculations

The MCDF calculations have been performed using the code developed by Bruneau [2].

Photoionization cross sections have been calculated for B-like atomic nitrogen ions in the region of the K-edge. Only electric dipole transitions have been computed using the length form. The following initial configurations have been considered:

1 2 2 2 2 and 1 2 2 2 2 .
In order to take into account correlation and relaxation effects on final states, multiple orbitals with the same quantum number have been used. Then, the following final configurations have been considered: 1s2s 2 2p 2 , 1s2s 2 2p np, 1s2s2p 3 , and 1s2s2p 2 np. Such notation means that radial functions with a principal quantum number n = 1 or 2 are not the same for initial and final configurations. Radial functions with principal quantum number up to n = 6 have been considered. Photoexcitation cross sections from the 1s 2 2s 2 2p and 1s 2 2s2p 2 configurations have been calculated separately. A synthetic spectrum has been constructed as a weighted sum of photoexcitation cross sections from 1s 2 2s 2 2p ( 2 P1/2, 3/2) levels and 1s 2 2s 2p 2 ( 4 P1/2, 3/2, 5/2) metastable levels. Each electric dipole transition has been dressed by a Lorentzian profile with a full width at half maximum given by the autoionization width of the upper level. Autoionization widths have been evaluated from MCDF calculations. For 1s 2s 2 2p 2 and 1s 2s 2p 3 photoexcited configurations, average autoionization widths are, 64 meV and 105 meV, respectively.

R-matrix Calculations

For this B-like system, the R-matrix method was used to determine all the photoionization cross sections for the ground state 2 P o and 4 P metastable states. Photoionization cross section calculations were accomplished in LS-coupling with 390-levels retained in the close-coupling expansion using the R-matrix with pseudo states method (RMPS). Because the existence of the metastable states in the parent ion beam, the photoionization crosssection calculations are required for ground, 1s 2 2s 2 2p 2 P 0 , and metastable, 1s 2 2s2p 2 4 P, states of N 2+ in order to simulate the experiment. The Hartree-Fock tabulated orbitals 1s, 2s, and 2p by Clementi and Roetti were used with residual ions n=3 and n=4 pseudoorbitals of N 3+ ions [7]. Using the atomic structure code CIV3 [8], all n=4 pseudo-orbitals were energy optimized on appropriate 1s -1 hole-shell states. These n=4 so called pseudoorbitals are incorporated into the scattering basis set to accommodate for core relaxation and electron correlation effects in the multi-reference-configuration interaction wavefunctions used to describe the atomic ion states. All the N 3+ residual ion states were then represented by using multi-reference-configuration-interaction (MRCI) wave functions. The nonrelativistic R-matrix approach was used to calculate the energies of the N 2+ bound states and the subsequent PI cross sections.

The R-matrix cross-sections calculations of admixture of ground and metastable states were convoluted with a Gaussian function of width 60meV and 67meV to simulate the experiment.

All peaks in the photoionization cross-section spectrum were fitted to Fano profile for overlapping resonance. [9][10][11][12][13][14].

SCUNC Calculations

In the framework of the Screening Constant by Unit Nuclear Charge formalism [15,16,17], total energies of (Nl, nl ' ) 2S+1 L π excited states are expressed in the form (in Rydberg units):

                    2 1 2 2 2 2 1 2 ; ; ' 1 1 1 ; ' Z L n N n N Z L n N E S S        (1)
In this equation, the principal quantum numbers N and n are the inner and the outer electron of the helium-isoelectronic series, respectively. The β-parameters are screening constant by unit nuclear charge expanded in inverse powers of Z and given by (2) where are parameters to be evaluated empirically with q the parameter (q) is related to the degree of accuracy in the calculation, in the case of nitrogen ions, q = 2 .In the case of 1s → 2p photoexcitation in the N 2+ ion, from Eq.( 1) and ( 2), we express the total energies of the (1sNs 2 np 2 2 L) excited states of N 2+ as follows

                q k k Z k f Z L S n N 1 1 ; 1 2 ; '        L n N f f S k k 1 2 ; '                                         2 3 0 2 2 2 2 2 2 2 3 0 2 2 2 2 2 2 2 ) ( ) 1 ( ) ( ) ( 1 2 ) ( ) 1 ( ) ( ) ( 1 2 1 Z Z Z n n L f Z L f n Z Z Z N N L f Z L f N Z E (3)
For the (1s2s 2 2p 2 2 L) levels, we get

                     2 3 0 2 2 2 2 2 2 2 ) ( 36
) ( ) ( 1 1 ) 2 2 1 ( Z Z Z L f Z L f Z L p s s E . (4)
In Eq. ( 4), Z0 denotes the nuclear charge of the atomic ion used in the empirical determination of the f( 2 L) -screening constants. To evaluate f( 2 L) in equation 4, the energies measured by Schlachter et al. [18] on the isoelectronic C + ion (Z0 = 6) for the 1s2s 2 2p 2 2 D, 2 P, 2 S levels, measured from the ground state of C + ion at -1018.8467 eV according to NIST tables [1] were used. Using these experimental data, equation 2 gives f

( 2 D) = 1.7903  0.0003, f ( 2 P) = 1
.7944  0.0003 and f ( 2 S) = 1.8075  0.0003, the given uncertainties resulting from the uncertainties on the reference energies [18].

Results and Discussion

Promotion of a K-shell electron in B-like nitrogen (N 2+ ) ions to the outer np-valence shell from the ground state 1s 2 2s 2 2p 3 P is the 1s→2p photoexcitation process,

hν + N 2+ (1s 2 2s 2 2p 2 P 0 ) → N 2+ (1s 2s 2 2p 2 [ 3 P, 1 D, 1 S] 2 S , 2 P, 2 D) ↓ N 3+ (1s 2 2s 2 1 S) + e - or N 3+ (1s 2 2s np 1 P 0 ) + e -
Starting from the 4 P metastable term also populated in the ECRIS, the 1s→2p photoexcitation process becomes: N 3+ (1s 2 2s np 3 P 0 ) + e -.

Figure 3 presents the variation of the photoionization cross section as a function of photon energy in the region of the 1s→2p photoexcitations using three different excitation band widths, 90 meV, 60 meV and 30 meV, respectively. Each cross section is obtained from the weighted mean of more than twenty individual sweeps. (in eV), natural linewidths Г (meV) and resonance strengths σ PI (in Mb eV), for the dominant core photoexcited n=2 states of the N 2+ ion, in the photon energy region 404 eV to 409 eV with previous investigations.

  2 1 ) ( PI       h h dh h
The uncertainty given on the experimental oscillator strength does not include the uncertainty on the relative populations.

MCDF code calculates the photoexcitation oscillator strength. The experimental and Rmatrix values give the photoionization cross section, including the autoionization decay of the intermediate excited states but not the fluorescence decay. According to previous Rmatrix calculations by Garcia et al [6] who calculated fluorescence yields separately, fluorescence decay contributes to 18 % of the total cross section for the 1s → 2p transitions, and up to 90 % for the 1s → 3p transitions. This is not confirmed by the present R-matrix calculations which found a negligible contribution of the fluorescence decay for these low n 1s → np excitations.

Compared to the C + isoelectronic ion [18], we note a good agreement between the measured widths of the corresponding transitions in the two ions, within the experimental uncertainties. In contrast, taking into account the different relative populations in the two experiments, we observe for N 2+ ion a strong increase of the measured oscillator strength, by a factor of about 2 for the transition from the ion in the ground state.

Finally, Figure 6 shows the variation of the measured cross section recorded in the relative mode in the 452 -460 photon energy range. It was observed in the double photoionization channel, corresponding to the detection of N 4+ photoions. Despite a low statistic, an onset is observed at 454.4 ± 0.4 eV. One possible assignment for this structure is the direct photoionization of 1s electron followed by the Auger decay of the 1s hole: hν + N 2+ (1s 2 2s 2 2p 2 P) → N 3+ (1s 2s 2 2p 3 P) + e - ↓ N 4+ (1s 2 2s 2 S) + e - Auger. Such onset could be observed in this experiment because the higher members of the Rydberg series converging to the thresholds, whose autoionizing decay usually hides the onset in case of outer shells ionization threshold, decay mainly here via fluorescence. The relative cross section has been normalized to the value at threshold of the PI cross section calculated by Verner and Yakovlev [21]. They have calculated the threshold to be at 447.3 eV, 7 eV below the measured value. The result of the R-matrix calculations of Garcia et al [6] is also displayed as the continuous line. These authors give the 3 P threshold to be at 458.9 eV, and the 1 P at 462.5 eV.

Another possible assignment is suggested by our R matrix calculations shown on Figure 7. This structure could be a resonant feature in the cross section corresponding to theoretically predicted belonging to Rydberg series converging to the 1s2s 2 2p 3 P o K-edge calculated to be around 457.4 eV.

Figure 6: Experimental cross section measured for N 2+ ion in the region of 1s threshold recorded in the relative mode (dots). It was normalized at threshold on the cross section calculated by Verner and Yakovlev [21] (open triangles). R-matrix photoionization cross section of Garcia et al (continuous line [6]). line-widths and resonance strengths, occurring for the interaction of a photon with the 1s 2 2s 2 1 S, and 1s 2 2s2p 3 P 0 states of the N 3+ ion.

One of the strongest excitation processes in the interaction of a photon with the 1s 2 2s 2 1 S ground state of the Be-like nitrogen ion is the 1s→np photoexcitation process; hν + N 3+ (1s 2 2s 2 1 S) → N 3+ (1s2s 2 np [ 1 P 0 ]) ↓ N 4+ (1s 2 2s 2 S) + e -( ), or N 4+ (1s 2 2p 2 P) + e -( ), n=2, 3 where is the outgoing energy of the continuum electron with angular momentum l. In our setup, the ions are produced in the gas phase by ECRIS, so the long life metastable term 1s 2 2s2p 3 p 0 is also present in the parent ion beam, giving additional lines in the energy range of the 1s→2p photoexcitation, corresponding to the process:

hν + N 3+ (1s 2 2s2p 3 p 0 ) ↓ N 3+ (1s2s [[ 1,3 S] 2p 2 ( 3 P, 1 D, 1 S) 3 S, 3 P, 3 D) ↓ N 4+ (1s 2 2s 2 S
) + e -( ), or N 4+ (1s 2 2p 2 P) + e -( ).

Theory

Theoretical photoionization cross section calculations for inner-shell processes in Be-like atomic nitrogen ions have be performed in order to compare with our experimental works.

SCUNC calculations

As already mentioned in the N 2+ results, equations ( 1) and ( 2) are used to calculate the total energy of the core-excited states. For N 3+ ion, from these equations the total energies of the (1s2s 2 np 1 P°) and (1s2s2p 2 3 L) levels are calculated as follows

                                         2 3 0 1 2 2 1 2 2 2 3 0 1 2 1 1 1 2 1 2 ) ( 36
) ( ) ( 1 1 ) ( 12 ) ( ) ( 1 4 2 1 2 1 Z Z Z P f Z P f n Z Z Z P f Z P f Z P np s s E . (5.a)                         2 3 0 3 3 2 3 2 ) ( ) ( ) ( 1 4 2 4 1 1 2 2 1 Z Z Z L f Z L f Z L p s s E . (5.b)
In a similar way, one may get the Auger widths Г in Rydbergs (1 Rydberg = 13.605698 eV) from the formula

Г( ) = 1 -× -× ( ) -× ( ) (6)
The fi-screening constants in Eq. (5.a) are evaluated from the data of Müller and coworkers on Be-like carbon C 2+ [22] for the (1s2s 2 2p 1 P°) and (1s2s 2 3p 1 P°) levels respectively at (in eV) 293.94  0.03 and 322.93  0.04. Using these experimental data along with the ground state value of C 2+ equal to -994.4615 eV [1], Eq. (5.a) gives

f 1( 1 P°) = 1.1726  0.0012 f 2( 1 P°) = 2.0832  0.0033
In the same way, the screening constants in Eq. (5.b) are determined for the (1s2s2p 2 3 D), (1s2s2p 2 3 P) and (1s2s2p For Be-like carbon, we note the labeling of the 1s2s2p 2 3 D and 1s2s2p 2 3 P states of Müller and co-workers should be reversed as pointed out in the recent calculations on the carbon iso-nuclear sequence [23].

MCDF Calculations

As before, MCDF calculations have been performed using the code developed by Bruneau [2], taking into account a full intermediate coupling regime in a jj-basis and only electric dipole transitions. Here the cross sections have been computed using the Babushkin gauge.

The following states have been considered as initial configurations, 1s 2 2s 2 , 1s 2 2s̅ 2p , and 1s 2 2p 2 , where the bar over the orbital is to indicate that they are different for initial and final states. In order to describe the correlation and relaxation effects, multiple orbitals with the same quantum number have been used. Then, the following final configurations have been considered: 1s2s 2 2p, 1s2s2p 2 , 1s2p 3 , 1s2s 2 3p, 1s2s2p3p, and 1s2p 2 3p.

The MCDF calculations spectrum is constructed as a weighted sum of photoexcitation cross sections from the ground state and metastable levels 1s 2 2s 2 ( 1 S0), 1s 2 2s2p( 3 P0,1,2)

respectively. Each dipole transition has been dressed with a Lorentzian profile, assuming a full width at half maximum (FWHM) equal to 66 meV and 80 meV for 1s→2p and 1s→3p transitions, respectively. A mixture of 60% of ground state and 40% of metastable state is used to fit the experimental measurements, after that the MCDF calculations were convoluted with Gaussian functions to simulate the experimental results resolution.

R-matrix Calculations

The R-matrix method was used to determine all the photoionization cross sections for the ground state 1 S and 3 P 0 metastable states. Cross section calculations were accomplished in LS-coupling with 390-levels retained in the close-coupling expansion using the R-matrix with pseudo states method (RMPS). The Hartree-Fock orbitals, 1s, 2s and 2p, tabulated on working of Clementi and Roetti [7] were used with n=3 physical and n=4 pseudo orbitals of the residual N 4+ ion. The n=4 pseudo orbitals were determined by energy optimization on the ground state of the N 4+ ion, using the atomic structure code CIV3 [8]. The n=4 pseudo-orbitals are included to account for core relaxation and additional electron correlation effects, in the multi-configuration interaction target wave functions. All the 390 N 4+ residual ion states were then represented by using multi-configuration interaction target wave functions.

The non-relativistic R-matrix approach was used to calculate the energies of the N 3+ bound states and the subsequent photoionization cross sections. Photoionization cross sections out of the 1s 2 2s 2 1 S ground term and the 1s 2 2s2p 3 P 0 metastable term were then obtained for all total angular momentum scattering symmetries. The scattering wave functions were generated by allowing three-electron promotions out of selected base configurations of N 3+ into the orbital set employed. Scattering calculations were performed with twenty continuum functions and a boundary radius of 8.2 Bohr radii. From our RMPS calculations, the 1 S ground state gave a bound state ionization potential of 5.69195

Rydbergs compared to the experimental value of 5.69420 Rydbergs [1]. In the case of the 3 P 0 metastable state, the ionization potential from the RMPS calculations gave 5.08202

Rydbergs, compared to the experimental value of 5.08200 Rydbergs. The discrepancy is approximately for the ground state of 31 meV and for the metastable state 0.3 meV.

Results and Discussion

The experimental cross sections were measured with different bandwidths, respectively, 38 meV, 56 meV, 111 meV and 133 meV. Each cross section is obtained from the weighted mean of several sweeps (from 3 to 8 following increasing resolution). Figure 8 shows the variation of the photoionization cross section with the photon energy from 412 eV to 415 eV, recorded with 111 meV of photon resolution. Figure 9 and 10 illustrates the results obtained with higher resolutions, 56 meV and 38 meV, respectively. On the figure 10 the two peaks around 412 eV are more separated. The theoretical cross sections were convoluted with a Gaussian function to simulate the results taking into account the 60% contribution of the ground state, N 3+ 1s 2 2s 2 1 S 0 , and 40% for the metastable state, N 3+ 1s 2 2s2p 3 P 0 . It is clear from the figures 8 to 10 that the R-matrix and MCDF calculations give excellent agreement with the experimental results on the absolute cross sections and the resonance energy positions. The MCDF calculations give less agreement comparing with the R-matrix calculations for the resonance due to the N 3+ 1s 2 2s2p 3 P 0 metastable term which is calculated 1.46 eV below the experimental measurements.

Table 5 displays the values of resonance excitation energy, Auger width and strength for all n=2 observed lines. The extracted experimental widths were determined by fitting each individual sweep by a Voigt profiles to avoid the shift in the energy delivered by the monochromator. The experimental oscillator strengths were obtained from the area under the lines. Our experimental values are compared with the calculated values using the three codes, as well as with previously published results [5]. Estimates for the resonance energies of Be-like nitrogen made using the SCUNC empirical fitting approach [15] show satisfactory agreement with the more sophisticated theoretical methods and experiment. In the case of the Auger widths, apart from the first 1s2s2p 2 [ 4 P] 3 P resonance, we see they are all in respectable agreement with experiment. We note that the more sophisticated theoretical methods such as R-matrix and the Saddle Point Method (SPM, [24,25,26],

consistently give values in agreement with each other and in general with experiment. We point out that in recent K-shell measurements for Be-like boron (e.g., B + [27] the Auger width for the 1s2s2p 2 [ 4 P] 3 P gave a value of ~ 10 meV, consistent with theory. Even higher resolution than the present work would be required to fully resolve the double peak structure in the N 3+ spectrum at about 412 eV.

Figure 10 shows the experimental date in the range of 1s→3p photoexcitation, around 460 eV photon energy. Due to the low statistic of the experimental data in that region, the Auger widths cannot be extracted by the fitting procedure. The experimental value for the energy of the resonance was found at 460.280 ± 0.04 eV and the strength equal to 6.30 ± 2.6. The R-matrix with a pseudo-state method (RMPS), SCUNC and MCDF calculations found the positions of the resonance energy respectively at 460.107 eV, 460.019 ± 0.045eV and 460.189, and the Auger width to be 72 meV, 87 meV and 80 meV, respectively. Furthermore, the RMPS method calculates the resonance strength to be 6.59.

Compared to the C 2+ isoelectronic ions [28], we note again in the case of N 3+ ion an increase by about 40 % of the oscillator strength of the 1s → 2p transition for the ions in the ground state. b LS-coupling, R-matrix [29], R-matrix, + Berrington and co-worker [30].

c MCDF [29]. d MCDF, chen and co-workers [31]. e Saddle-point with complex-rotation method (SPM-CR) [24,32]. f (SPM-CR) [25]. g R-matrix, intermediate coupling, level averaged [6]. h SCUNC approximation. [15,16,17] Table 5: Experimental and theoretical results for the resonance energies, natural line widths and the average photoionization cross sections for the K-shell photoexcitation within the photon energy range between 410 eV to 415 eV. In this work we have explored the photon energy around 460 eV with a photon resolution of 125 meV.

Theory

SCUNC Calculations

For the total energies of the (1s2s2p 1 P) 2 P° and (1s2s2p 3 P) 2 P° levels of N 4+ , we obtain from Eqs.( 1) and ( 2)

                          2 3 0 3 , 1 2 3 , 1 2 3 , 1 ) ( 36 ) ( ) ( 1 2 1 1 ) 2 2 ( 1 Z Z Z P f Z P f Z P p s s E . ( 6
)
To evaluate empirically the screening constants 6), we use the data of Müller et al. [33] on C 3+ for the (1s2s2p 1 P) 2 P° and (1s2s2p 3 P) 2 P° levels respectively (in eV) at 303.44 ± 0.03 and 299.98 ± 0.03. These energies are measured from the ground state of C 3+ at -946.57562 eV according to NIST [1]. Using these experimental data, Eq. ( 6) gives (with Z0 = 6) respectively f ( 1 P) = 1.2524  0.0005 f ( 3 P) = 1.1992  0.0005

) ( 3 , 1 P f in Eq. (

MCDF Calculations

A similar procedure as for the previous N ions has been used for the Li-like atomic nitrogen ion, except that (i) the following configurations have been considered: 1s 2 2s, 1s2s2p, and 1s2s3p, and (ii) Lorentzian FWHM equal to 45 meV and 60 meV have been used respectively for the 1s-2p and 1s-3p transitions. Only the ground state contribution has been retained to compare with experimental data.

R-matrix

For this Li-like system intermediate-coupling photoionization cross section calculations were performed using the semi-relativistic Breit-Pauli approximation which allows for relativistic effects to be included in a similar manner to the previous work on Li-like, boron [22] and carbon [33] ions. An appropriate number of N 5+ residual ion states (19 LS, 31 LSJ levels) were included in our intermediate coupling calculations. The n=4 basis set of N 5+ orbitals obtained from the atomic-structure code CIV3 [8] were used to represent the wave functions. Photoionization cross-section calculations were then performed in intermediate coupling for the 1s 2 2s 2 S 1=2 initial state of the N 4+ ion in order to incorporate relativistic effects via the semi-relativistic Breit-Pauli approximation.

In the calculations the He-like LS states were retained: 1s 2 1 S, 1sns 1;3 S, 1snp 1;3 P, 1snd 1;3 D, and 1snf 1;3 F, n ≤ 4, of the N 5+ ion core which give rise to 31 LSJ states in the intermediate close-coupling expansions for the J=1/2 initial scattering symmetry of the Lilike N 4+ ion. The use of the n=4 pseudo states is an attempt to account for core relaxation, electron correlations effects and the infinite number of states (bound and continuum) left out by the truncation of the close coupling expansion in our work. For the structure calculations of the residual N 5+ ion, all n=3 physical orbitals and n=4 correlation orbitals were included in the multiconfiguration-interaction target wave functions expansions used to describe the states.

The Hartree-Fock 1s and 2s orbitals of Clementi and Roetti [7] together with the n=3 orbitals were determined by energy optimization on the appropriate spectroscopic state using the atomic structure code CIV3 (A. Hibbert 1975). The n=4 correlation (pseudo) orbitals were determined by energy optimization on the ground state of this ion. All the states of the N 5+ ion were then represented by using multi-configuration interaction wave functions. The Breit-Pauli R-matrix approach was used to calculate the energies of the N 4+ (LSJ) bound states and the subsequent photoionization cross sections. A minor shift (< 0.1 %) of the theoretical energies for the N 5+ ion core states to experimental values (NIST) was made so that they would be in agreement with available experimental thresholds. Both double and triple promotion models for the scattering wave functions were investigated.

The R-matrix method [34,35] was used to determine all the photoionization cross sections for the initial ground state in LS and intermediate-coupling. Finally, in order to compare directly with experiment, the theoretical cross section was convoluted with a Gaussian function of appropriate full-width half-maximum to simulate the energy resolution of the measurements.

Results and Discussion

Figure 12 shows the variation of the photoionization cross sections in the photon energy range 421 to 426 eV. Two intense lines were observed at 421.472 ± 0.03 eV and 425.499 ± 0.03 eV respectively. Table 6 presents the resonance parameters for the two resonances.

Our experimental values are compared to previous measurements [5] as well as the prediction of our and previous calculations [5]. The fitting of the first peak by a Voigt profile has allowed to extract its Auger width to be 11 ± 8 meV. Both the R-matrix, MCDF and Saddle point methods agree with each other and the experimental value for the location of this resonance. Theoretical predictions from R-matrix calculations and other methods for the Auger width all lie within the experimental error estimate for this resonance.

The second peak observed had limited experimental data due to the low signal to noise ratio which doesn't allow to extract its natural width. The natural line width of this 1s [2s2p 1 P] 2 P 0 resonance was calculated by R-matrix, MCDF and Saddle point methods to be respectively 42 meV, 45 meV and 42 meV. It is also seen that the R-matrix calculations yield a resonance strength of 6.5 Mb eV, MCDF estimates a value of 9 Mb eV and both theoretical values are in suitable agreement with the experimental measurement of 7.3 ± 2.5. For the positions of both resonances, the theoretical values from the RMPS calculations are in more favorable agreement with our measurements than either the earlier EBIT [36], the Laser produced plasmas (LPP) [37] or those determined from electron spectroscopy in ion -atom collisions [38] measurements. Table 6 indicates that estimates for the resonance energies and Auger widths of Li-like nitrogen made using the SCUNC empirical fitting approach [15][16][17] are in satisfactory agreement with the more sophisticated theoretical methods and experiment. j Complex scaled multi-reference configuration interaction method (CMR-CI) [25]. l Saddle-point method with R-matrix (SPM-RM) [43]. m SCUNC approximation. [15,16,17] Table 6: Experimental and theoretical results for the resonance energies, natural line widths and the average photoionization cross sections for the K-shell photoexcitation within the photon energy range between 420 eV to 426 eV.

Along the lithium isoelectronic series [22,33] we observe an increase of the splitting of the 3 P and 1 P resonances, from 1.5 eV for neutral to 4 eV for N 4+ ion. An increase is observed also for the 1 P/ 3 P branching ratio, going from 3 % for Li to 15 % for N 4+ , resulting mainly from the increase of the oscillator strength of the 1 P resonance, the one for the 3 P line staying constant around 0.46 from B 2+ to N 4+ ions (measurements on Li were performed in relative values only). In opposite, we determine a width for the 3 P resonance in N 4+ ion (11±8 meV) significantly larger than in B 2+ ion (4.8±0.6 meV) [22].

Conclusion

I have presented the first measurements of K-shell photoionization cross sections for C-like, B-like, Be-like and Li-like nitrogen ions in the ground and metastable states. The fields. FTICR offers potentially ultrahigh mass resolution (mass resolution down to 1 part in 10 9 or better), as well as the capability to cool the ions, allowing to suppress the contribution of ions produced in short lived metastable states [1].

In this chapter, I will present first the principle of FTICR ion trap. Then I will describe the first results we have obtained on the DESIRS beam line of SOLEIL on Xe + and Kr + atomic ions and compare them with results previously obtained on the same ions with a merged beam setup. Finally, I will present the preliminary results we have obtained on N2 + and CO2 + molecular ion, using both the MAIA setup and the FTICR ion trap.

Principle of the FTICR Ion Trap

Usually, in FTICR ion traps the ions are produced outside the trap then introduced by some transfer line. The ions are trapped by a combination of magnetic and electric fields.

Measurement of the mass spectrum of the ions inside the trap is performed in situ by excitation of the ions by a radio-frequency wave in resonance with the cyclotron frequencies of the ions.

Interaction between the Ions and a Magnetic Field

An ion moving in a uniform magnetic field ⃗ is subject to the magnetic force (Lorentz force) given by

⃗ = ⃗ × ⃗
where q and v are the ionic charge, and the velocity of the ion, respectively, and the direction of the force is perpendicular both to the velocity and the magnetic field.

When the velocity of the ion is perpendicular to the uniform magnetic field, the ion trajectory bends to a circular path (the ion cyclotron orbit), and the radius r of the path depends on the mass to charge ratio m/q, velocity, and magnitude of the uniform magnetic field is following:

= * ( )
The cyclotron frequency of the motion and the kinetic energy of the ion are given by:

= . . = 2
The cyclotron frequency is independent of the ion velocity, and all ions with the same mass to charge ratio have the same frequency. To fix the orders of magnitude, a singly charged ion of mass 100 a.m.u excited to an ICR orbital radius of 1 mm in a magnetic field of 1

Tesla has a translational velocity equal to 9.65 10 5 m/s, and a cyclotron frequency of 1036 kHz. Figure 1. Incoherent ion cyclotron orbital motion is converted to coherent (and, therefore, detectable) motion by the application of a rotating electric field, which rotates in the same sense and at the ICR frequency of the ions of a given m/q value [2].

Excitation and Detection

The cyclotron motion of the ions in the middle of the trap does not generate an observable electric signal on the walls of the trap which are used as an antenna because the radius of motion is too small. To increase the radius, the ions are coherently accelerated to a large radius by using an oscillating electric field E with a frequency equal to the cyclotron frequency of the ions, ω=ωc:

= cos scanning the frequency of the oscillating wave, it includes the contribution of all the ions as a function of time. The Fourier transform (FT) of this signal converts to the frequency domain and then to the mass spectrum after calibration on a reference ion (figure 2).

Figure2. Fourier transform is performed on time domain data to convert it to the frequency domain, and the resulting spectrum is then calibrated to m/q values.

In FTICR traps, all steps (ion selection, excitation, detection, time domain data storage and Fourier transformation) are made at the same spatial place but separated in time (figure 3). We systematically checked that no doubly charged ions were produced when the electron pulse and/or the synchrotron radiation pulse were missing. The energy of the photons or the electrons alone is not sufficient to produced doubly-charged ions (thresholds for double ionization of Xe and Kr atoms are 33.1 eV and 38.36 eV respectively [4]).

4.2 Results and Discussion:

Kr + and Xe + Ions

The electronic configuration for Kr + and Xe + are [Ar] 3d 10 4s 2 4p 5 and [Kr] 4d 10 5s 2 5p 5 respectively, and because the two ions have a hole in the outer p subshell, the ground state level is 2 P3/2 for both. The 2 P1/2 level of the ground state configurations is a long life metastable state, with measured lifetime of 340 ms and 49 ms for Kr + and Xe + ions, respectively [5]. Thus we expect the contribution of these metastable states to appear in the merged beam spectra, but not in the trap spectra if we succeed to trap the ions before their interaction with the photons for a time long enough to allow them to radiatively decay to the 2 P3/2 ground level.

The results obtained with the trap have been performed at the DESIRS beam line using the MICRA set up, and the results obtained with the merged beam technique have been performed previously using the ASTRID (Aarhus STorage RIng in Denmark) merged beam set up, at the University of Aarhus in Denmark [6].

Merged Beam Experiment

Figure 6 shows the variation of the photoionization cross section of Kr + and Xe + ions as a function of photon energy (red line). The black curves give the variation of the direct photoionization cross section as calculated using the same MCDF code as used for the results on nitrogen ions. These results have been obtained with a fixed slit width of the monochromator, resulting in the ASTRID setup in a band pass increasing from 50 meV at the threshold to 100 meV at 15 eV above the threshold for the Kr + ion spectrum, and from 30 to 100 meV for the Xe + ion spectrum. The relative statistical uncertainties on the data at 30 eV are 6% and 16% for Kr + and Xe + ions respectively. A Mg filter was used to suppress the contribution of higher orders diffracted by the monochromator, but below 25 eV the filter is not efficient and the magnitude of the photoionization cross sections might be underestimated since secondorder radiation is not filtered out and contributes in an unknown proportion to the measured photon flux.

We note the close similarity of the two spectra in shape and intensity. Resonant structures are observed above the threshold and are followed by a smoothly decreasing continuum.

Such similarity was already observed for Br and I isoelectronic atoms [7,8]. The integrated oscillator strengths for the two ions are also very close, 1.40 and 1.49 for Kr + and Xe + ions respectively.

For the Kr + ion, at photon energy 31.38 and 31.69 eV respectively, weak window resonances are observed, and can be interpreted as the n=5 term of the Kr + 4s4p 5 ( 3 P2,1) np Rydberg series that have threshold limits at 38.73 and 39.16 eV [9].

The MCDF calculations have been performed using the code developed by [10] to calculate the direct photoionization cross section. All the levels of the p 5 configuration have been considered in the initial state and all levels of the p 4 configuration in the final state. The energy interval used for the calculations was 0.1 eV close to threshold and 0.5 eV above. The calculations reproduce correctly the variation of the experimental cross sections, but with a magnitude lower by 15% to 30%.

Figures 7 and8 give an enlargement of the region close to threshold for Kr + and Xe + ions respectively. These cross sections have been obtained with an improved resolution compared to the spectrum on figure 6. For Kr + ion the bandpass was 20 meV up to 40 meV and for Xe + ion from 20 meV to 30 meV. The relative statistical uncertainties of the photoionization cross section are 7% and 4% for Kr + and Xe + ions respectively. The long vertical lines on figure 7 and 8 indicate the position of the np 4 3 P2, 1, 0 and 1 D2 ionization thresholds [4] (n=4 for Kr + and n=5 Xe + ) for the ions in the np 5 2 P3/2 ground level and np 5 2 P1/2 first excited level. For both ions, strong signal is observed below the first ionization threshold for ions in the ground level, clearly demonstrating the contribution of ions in the 2 P1/2 metastable level.

The observed lines on the spectra can be attributed to np 5 →np 4 ms, md photoexcitations of the valence electron, corresponding to the allowed electric-dipole transitions. The np 4 ms, md excited states decay to the ground state of the doubly charged ion by autoionization, following the scheme given, figure 9, in the case of the Xe + ion: Figure 9: simplified energy levels illustrating the general scheme of the two steps photoionization process observed on Xe + ion.

Rydberg states both of ms and md configurations, converging to each of the four upper np 4 levels, are expected to be observed in the photoionization spectra, resulting in reach structures of overlapping Rydberg series. It is even more complicated once considering the contribution of the ions in the metastable level, because the number of expected lines is doubled. For halogen atoms, considering the moderate resolution used in this work to compensate the low density of ionic targets, it is not possible to discuss in detail all the observed lines, because most of the structures are composed of unresolved lines. I have analyzed the spectra using the formula

= - ( -)
where R, E∞, Z and δ are the Rydberg constant, the converging limit, the electric charge of the core ion and the quantum defect, respectively. The E∞ threshold is taken from the NIST energy levels tables and we assume that Z=2. The result of this analysis is given in the tables 2 and 3, for Kr + and Xe + respectively. For both ions, the dominant structures are associated with the Rydberg series converging to the 1 D2 limit, with the principal quantum number n up to 13 and a quantum defect approximately constant. The position of the Rydberg series lines are indicated as vertical 97 lines on the figure 7 and8. For the Kr + ion, the Rydberg series converging to the 1 S0 threshold can also be identified up to n = 8 by a constant quantum defect.

Recently, R-matrix calculations have been performed by G. Hinojosa and co-workers [11] allowing the identification of the lines. The results are shown on Figure 10 together with new measurements performed at the ALS. They have been normalized to the top panel data by assuming the area under the spectra on FTICR and merged beam results to be the same. Although the spectrum recorded with the trap directly after the ion production (Δt=0 ms) shows large statistical scatters, it reproduced the cross section obtained with the merged beam set up, showing we have contribution of both 2 P3/2 and 2 P1/2 levels with, maybe, different populations. Waiting for 520 ms, a time longer than the metastable states lifetime (350 ms), allows ions in this excited level to radiatively decay to the ground state. We observe in effect the intensity of the structures below the first ionization threshold (24.36 eV) are strongly suppressed, demonstrating that none of the remaining ions inside the trap are in the ground level.

The main process for the loss of ions inside the trap is charge exchange with the water molecules present in the residual gas. In the case of the Xe + ion, as the ionization potential of water is greater than that of Xe atom, this process is not possible and the trapping time can be increased. Figure 12 presents the results obtained for the Xe + ion with the merged beam technique (upper panel) and the ion trap (middle panel). As previously, the cross section measured with the merged beam set up is the one shown on figure 8 after convolution to obtain the same total resolution as the one used to record the cross section with the trap (100 meV). A delay of 1 s was used before irradiation of the ions, a time much longer than the lifetime of the 2 P1/2 metastable state (50 ms). Clearly, all the signal below the energy of the first ionization threshold (20.98 eV) observed on the top spectrum has totally disappeared in the spectrum recorded with the trap, showing all the irradiated ions were in the 2 P3/2 ground state. We note that the strong trapping magnetic field (1.24 T) has no apparent effect on the position of the resonance lines. Calculations with the MCDF code [10] have shown that the shift produced by such a field on the Xe + 5p 5 levels has maximum amplitude of 0.15 meV, and 0.21 meV for the Xe 2+ 5p 4 levels, values much lower than our experimental resolution. Here the cross section determined with the trap has been normalized to the merged beam cross section by assuming a statistical population of the ground and metastable levels.

From the difference of the two spectra, it is possible to estimate the cross section for the ions in pure metastable levels, assuming once again a statistical population of the ground and metastable levels in the merged beam experiment. The result is shown on the bottom panel of figure 12.

Let's note this procedure is made approximate here because the cross sections have been measured on different beam lines, with different resolution. In addition, the spectrum recorded at ASTRID is probably affected by the contribution of light diffracted in second order by the monochromator. This can explain the low value of the cross section in the high energy part on the bottom panel spectrum. In this energy region, it is the direct photoionization cross section which dominates, and its intensity is expected to be quite close for the ions in the ground and metastable states.

Photoionization of N2 + Ion

The high mass resolution of FTICR ion traps makes them an ideal tool for the study of photoionization processes on molecular ions. Up to now, very few studies using merged beam technique have been devoted to the study of molecular ions, on CO + and Fullerene ions [12][13][14][15]. Nevertheless, studies on molecular ions using ion traps are starting now as for our knowledge.

One of the difficulties of these studies is you have to detect doubly charged molecules (dication). These species are generally thermodynamically unstable and tend to dissociate very fast into fragments due to the Coulombic repulsion that dominates the long range interaction between the nuclei. Nevertheless, short range covalent bonding can overcome the repulsive Coulombic forces and stable it. Metastable molecular dications have been observed [16]. In the case of the N2 2+ dication, lifetimes up to 3 sec have been measured [17].

Figure 13 presents the cross section measured for the N2 + cation using both MAIA (bottom panel) and MICRA (top panel). Because of low signal to noise ratio in both experiments, measurements could be performed using zero order light only. Both PLEIADES and DESIRS beam lines have undulators with almost the same number of periods, resulting in an equivalent excitation bandpass of about 2 eV. Some structures observed on the spectrum obtained with MICRA at 27.4 eV, 29.0 eV, 31.2 eV and 33.8 eV may be associated with known ionization thresholds for N2 + (vertical lines), corresponding to X 1 Σg + , A 1 Πu, d 3 Πg and 1 Πg states of N2 2+ , respectively [17].

Figure 13: Variation of photoionization cross section as a function of photon energy for N2 + molecular ion measured with the FT-ICR ion trap (upper panel) and the merged beam setup MAIA (lower panel). These structures are not observed on the spectrum recorded using the MAIA setup. One explanation could be the two techniques don't measure exactly the same thing. For the MICRA experiment, isotopic N2 + ( 14 N 15 N + ) was used, allowing to resolve the photoionization process from the photodissociation, while the MAIA spectrum was obtained with natural N2 + ( 14 N2 + ). Second, typical lifetimes for detection of the dications are quite different, of the order of μsec for MAIA, while of the order of msec for MICRA, resulting in possible different final states that can be detected for the dication in the photoionization process. Finally, the cations are produced with a higher internal energy in the ECRIS, resulting in presence of cations in different states in the interaction region with the photons.

Because of the use of zero order radiation, the cross section could be determined in relative value only with MAIA, the relative contribution of the different harmonics of the undulator is unknown.

Photoionization of CO2 + Ion

In the case of the CO2 + cation, we have been able to measure with MICRA the photoionization cross section using the light diffracted in first order by the monochromator of the DESIRS beam line. Figure 14 shows the variation of the cross section between 23 eV and 25.8 eV photon energy. It was obtained with a band pass of 130 meV. The lifetime of CO2 2+ ground state was measured by D Mathur et al to be 4 s [18], and of the order of 1μs for the metastable excited states [16]. Detection of the dications was performed in the 10-1000 ms time window after their production by the photons, in such a way only dications in the ground state can be detected. As in the case of the N2 + cation, structures are observed and might be associated with previously observed excited states of the CO2 2+

Study of extraction of a high quality ion beam from an ECR ion source

The quality of the absolute cross section measurements with the merged beam techniques is strongly dependent on the performance of the ion source (i.e. in this case the ECRIS).

The extraction of high quality ion beams from the ECRIS is very difficult due to the poor emittance of these sources resulting from the large aperture size. The size and divergence of the extracted beam is affected by space charge effects, particularly important in our experiments due to the low extraction voltage (2 to 4 kV) needed to find a good compromise between the signal and noise intensities in our spectra, as well as aberrations produced by the strong magnetic fields in the extraction region. The extraction of ions from ECRIS is violently focusing unless strong Coulomb repulsion occurs between the extracted ions. The intensity of the extracted ions is affected by the applied RF power, geometry and the polarity of the extracted electrodes [1][2][3].

Ion Beam Properties

A particle moving in three dimensional space can be represented by three position coordinates x, y, z and three velocity components vx, vy, vz, or alternatively by the six dimensional phase space coordinates x, px, y, py, z, pz, where px, py, pz are the momentum of the particle. In the same way, a set of ions moving together can be represented as points in a six dimensional hyper-volume space, every point referring to one ion.

The transverse components of phase space are taken to be elliptical, given by:

= , =
For a group of ions moving in the z-direction the momentum in z-direction is approximately constant then,

= ≅ , = ≅
where tan φ ≅ x ' and tan φ ≅ y ' are the angles between the x and y components of ion

ECRopt Program

ECRopt program uses a charged particle beam as an optical system which is composed of electromagnetic elements. It combines the integration of the equation of motion to the multiplication of transfer matrices, without space charge effect. Usually, the calculation of the behavior of beams in an optical system, such as an electrostatic lens, requires solving the Laplace equation using a complicated computational code. ECRopt uses instead on analytical expression for the distribution of ions and the electrodes geometry in the cylindrical coordinates. The FORTRAN code of the program is given in appendix A.

Basic Equations of ECRopt Code

The force ( ⃗ ) that acts on a charged particle of mass m and charge q moving with velocity ( ⃗) under a combination of electric ( ⃗ ) and magnetic ( ⃗ ) field in x, y, z coordinate is:

⃗ = ⃗ = ⃗ + ⃗ × ⃗
We can separate this equation into three equations with respect to x, y, z coordinates as the following,

= + ( - ) = + ( - )
By substituting the electric field components on the x and y equations, we can solve the behavior of the beam. Note that the initial positions and angles of the charged particles are obtained from the IGUN output file.

Results and Discussions

Two Electrodes System

This study has been performed using IGUN software. The potential produced by the electrodes and the ions trajectories in this potential are calculated for a 2D cylindrical symmetry using a finite steps (1mm) integration of the differential equation of motion.

Effects of space charge are also included.

The system consists of a fixed plasma electrode, Ep, and one extraction electrode Eext which can be longitudinally translated relative to the plasma electrode to optimize the electric field strength during ion extraction, between minimum gap (18 mm) and maximum gap (55 mm). The plasma electrode is fixed. The ion beam is extracted from the plasma electrode aperture of radius 2.5mm using 2 kV or 4 kV extraction voltages applied on the plasma electrode, while the potential on the second electrode is kept at the ground potential. The IGUN input file requires several input plasma parameters such as the initial energy of ions, ions mass, ion charge, the current density, the initial location of the starting ions and the initial positions of starting fields [5,6].

Ions Extraction

The first step of the extraction study is to determine the optimal value for the distance between the electrodes when applying 2KV or 4KV on the plasma electrode (Ep) and 0V on the extraction electrode, through changing of the gap (g) between the two electrodes from 18 mm to 55 mm with steps equal to 5 mm. The optimum gaps are located at a position equal to 40 mm (maximum angle of 24.2 mrad and emittance equal to 1.9 pi*cm*mrad) for 2KV and 45 mm (39.8 mrad and 3.4 pi*cm*mrad) for 4KV. Let's note that, in our experiment, the ion beams must fit the size of the photon beam over a long distance in the interaction region, so we need an ion beam not too focused, i.e. we need more a parallel beams than a focused beam. In this step of the simulations, the voltage applied on the einzel lens (E1) is varied to focus the ion beam at the position of the slit S2, at 80 cm after the magnet M1, corresponding to the focusing point of the magnet which is symmetrical.

The simulation of the trajectories in the horizontal and vertical planes is shown on Figures 7 and8 for 2KV and 4KV extraction voltages, respectively. To focus the ion beam at the position of slit number two, we have to put -11760V and -32000V (but the -32000V

voltage is not realistic) on E1, respectively. Figure 11 shows the emittance measured after slit S2 for Ar 5+ ions extracted at 2 KV in the horizontal plane while figure 12 in the vertical plane. The current density was 1.07×10 -5 A/cm 2 . We used for this measurement the Pantechnik emittance meter composed of two sets of vertical and horizontal wires. The gap of the electrodes was at its minimum value (18 mm), giving the maximum current after slit two. This is a first difference with the simulations which give an optimum gap of 40 mm. A second difference is the emittance in the horizontal plane, the measurements showing strong aberration. The quality of the ion beam depends strongly on the shape of the plasma emission surface and the position of this surface depends on the source parameters (gas pressure, heating power...). Using a three electrodes extraction system, two possibilities are offered to extract the ions from the ECRIS plasma: achieve two successive accelerations (accel-accel system), or achieve an acceleration followed by a deceleration (accel-decel system). In this study we will focus only on the accel-accel system.

Three Electrodes System of Extraction

As previously, the first step of the extraction study is to determine the optimal values of g1 and g2 inter-electrode distances. For this, I have followed the procedure:

1. Fix g1+ g2= 25mm while moving the Eext electrode and varying its voltage, the plasma electrode is fixed to 2 KV and 4 KV. All data (maximum angle and emittance) are collected at 70mm from the plasma electrode. The optimal value of g1 is the one that corresponds to the minimum divergence and possibly emittance, for this step, I will use IGUN. It should be noted that, for optimum transport through the einzel lens, the maximum diameter of the beam must be less than onethird of the diameter of the einzel lens. [3,7,8].

2. Find the optimal values found for several voltages applied to Eext electrode.

3. Transport the ions beam throughout the einzel lens E1 and the first dipole magnet M1 to focus the ions beam at the position of slit number two, using ECRopt code.

Ions extraction

The figures 14

Conclusions

In this chapter, we have determined the geometry and potentials to apply to the two electrode extraction system in order to obtain the optimum ion extraction and transportation in the first dipole magnet for 2KV and 4KV extraction voltages, table 1. These simulations were achieved by optimizing the initial conditions obtained using IGUN and ECRopt programs. It has been shown that some ions are lost in the chamber of the first magnet.

With the three electrodes system, the ellipse emittance and maximum divergence are greatly improved. No ions are lost in the magnet both at 2 and 4 KV extraction voltages, allowing to expect a higher current in the interaction region with the photons. The position of data acquisition In the future, the emittance of the beam at the position of slit S2 will be measured more systemically, in particular with the three electrodes system and the simulations will be performed up to the interaction region using SIMION, allowing to optimize all transportation parameters to make an intense parallel ions beam inside that region.

Introduction

Photoabsorption and photoionization (PI) are fundamental atomic processes that play important roles in many physical systems, including a broad range of astrophysical objects as 8 Authors to whom any correspondence should be addressed. diverse as quasi-stellar objects, the atmosphere of hot stars, protoplanetary nebula, HII regions, novae and supernovae. Satellites Chandra and XMM-Newton currently provide a wealth of x-ray spectra of astronomical objects; the lack of high-quality atomic data hammers the interpretation of these spectra [1][2][3][4][5]. Studies recently carried out on carbon and its ions showed it was necessary to have good quality data to model the observations in the x-ray spectrum of the bright blazar Mkn 421 observed by the Chandra LETG+HRC-S [6].

Spectroscopy in the soft x-ray region (5-45 Å) including K-shell transitions of C, N, O, Ne, S and Si, in neutral, and low stages of ionization, L-shell transitions of Fe and Ni, provide a valuable tool for investigating the extreme environments in active galactic nuclei, binary systems, cataclysmic variable stars and Wolf-Rayet Stars [7] as well as the interstellar media (ISM) [8]. One may expect similar results concerning the chemical composition of the ISM if accurate data are available on neutral [2,5,9,10], and various charge stages of ionization of atomic nitrogen K-edge cross sections [9,11]. PI models of the brightest knot of star formation in the blue compact dwarf galaxy Mrk 209 required abundances for ions of oxygen and nitrogen [12]. Therefore PI cross section data and abundances for carbon, nitrogen, and oxygen in their various stages of ionization are essential for PI models applied to the plasma modelling in a variety of planetary nebulae [13]. Nitrogen abundance in particular plays a fundamental role in η Car studies, because it is a key tracer of CNO processing [14]. Li-like atomic nitrogen ions (N V) are used in the determination of PI structures of pressure-supported gas clouds in gravitationally dominant dark matter mini-halos in the extended Galactic halo or local group environment [15] with the modelling code CLOUDY [16,17]. X-ray spectra obtained by Chandra from sources such as Capella, Procyon, and HR 1099 are used as standards to benchmark plasma spectral modelling codes. He-like nitrogen (N VI) lines have been observed with Chandra and XMM-Newton in the x-ray spectra of Capella and Procyon [18,19], the M dwarf binary YY Gem [20], and the recorded outburst of the recurrent nova RS Oph [21], in the wavelengths region 28.7-29.6 Å (420 eV) that are attributed to 1s → 2 transitions. XMM-Newton observations of the fast classical nova V2491 Cyg [22] have also indicated the N VI, K α and K β lines are present at about 28.78 and 24.90 Å. The He-like series lines of N VI and O VII are detected up to 1s → 5p, and for O VII, the recombination/ionization continuum at 16.77 Å (739.3 eV) is present between 16.6-16.8 Å. He-like N VI has also been observed in the Chandra-LETGS x-ray spectroscopy of NGC 5548 [23].

In the x-ray community, electron-beam-ion-trap (EBIT) measurements (used for calibrating resonance energies), have been carried out for the inner-shell 1s → 2 transitions in He-like and Li-like nitrogen ions [24] (used in plasma modelling to determine impurity transport properties [25]). He-like and Li-like x-ray lines energies of atomic nitrogen have been measured at the Lawrence Livermore National Laboratory EBIT [24]. In EBIT experiments, the spectrum is contaminated and blended with ions in multiple stages of ionization, making spectral interpretation fraught with difficulties, unlike the cleaner higher-resolution spectra obtained from third generation synchrotron radiation facilities such at the Advanced Light Source, BESSY II, SOLEIL, ASTRID II and Petra III.

PI cross sections used for the modelling of astrophysical phenomena have mainly been provided by theoretical methods, due to limited experimental data being available. As a consequence, significant effort has been put into improving the quality of calculated data using state-of-the-art theoretical methods. Recent advances in the determination of atomic parameters for modelling K lines in cosmically abundant elements have been reviewed by Quinet and co-workers [4]. Until recently, many of these calculations have not been severely tested by experiment, and this remains an urgent task [26]. Experimental K-shell PI cross section measurements have been made by various groups on a variety of atoms and ions of astrophysical interest; He-like Li + [27][28][29], Li atoms [30], Li-like B 2+ [31], C 3+ [32], Be-like B + [33] , C 2+ [34], B-like C + [35], C-like N + [11], N-like O + [36], F-like Ne + [37], neutral nitrogen [10] and oxygen [38][39][40][41], valence shell studies on B-like ions, N 2+ , O 3+ and F 4+ [26]. C -like Ar + [42], Mg-like Fe 14+ [43], As-like Se + [44][45][46] and Br-like Kr + [47][48][49]. All this experimental data has been compared with various modern theoretical methods.

Theoretical K-shell PI cross sections for the iso-nuclear C I-C IV ions assembled recently to model the Chandra x-ray absorption spectrum of the blazar Mkn 421 [6] were found to be in excellent agreement with the astrophysical observations. Additionally, K-shell PI cross sections calculations on neutral nitrogen showed excellent accord with high resolution measurements made at the Advanced Light Source radiation facility in Berkeley, California [10] as have similar cross section calculations on singly and doubly ionized atomic nitrogen when compared with measurements from the SOLEIL synchrotron facility in Saint-Aubin, France [11,50]. In fact, the majority of the high-resolution experimental studies from the third generation light sources have been shown to be in excellent agreement with detailed theoretical calculations performed using the state-of-the-art R-matrix method [51,52] and with other theoretical approaches.

The present experimental and theoretical work on the prototype Be-like and Li-like atomic nitrogen ions provides cross section data for the PI of x-rays in the vicinity of the K-edge, where strong n = 2 inner-shell resonance states are observed. This work follows our earlier successful work on K-shell investigations for singly and doubly ionized atomic nitrogen [11,50]. To our knowledge, there would appear to be limited experimental studies for resonance Auger energies reported to date on either Be-like or Li-like atomic nitrogen ions [24,53,54] for photon energies in the vicinity of the K-edge region. For Be-like atomic nitrogen, previous experimental and theoretical studies have been made in the valence region and in the near threshold region [26,43,55,56] where it was necessary to include both the ground state and metastable excited states in the theoretical work in order to achieve suitable agreement with experiment. We follow a similar prescription here in the vicinity of the K-shell energy region as the Be-like (N 3+ ) atomic nitrogen ions produced in the SOLEIL synchrotron radiation experiments are not purely in their ground state. Theoretical studies are an essential ingredient in order to determine the metastable constituents in the beam. In the case of Li-like atomic nitrogen ions one only needs to consider the case of the ground state as no metastable states are present in the parent beam.

K-shell PI when followed by Auger decay couples three or more ionization stages rather than two in the usual equations of ionization equilibrium [57]. The 1s → np photo-excitation processes involved in the interaction of a photon with the 1s 2 2s 2 1 S ground state of the Be-like nitrogen ion is;

hν + N 3+ (1s 2 2s 2 1 S) → N 3+ (1s2s 2 np[ 1 P • ]) N 4+ (1s 2 2s 2 S) + e -(k 2 ) or N 4+ (1s 2 np 2 P) + e -(k 2 ), n = 2, 3,
and where k 2 is the outgoing energy of the continuum electron with angular momentum . The strongest Auger decay channels being the spectator KLL channels where the Rydberg, np electron does not participate in the Auger decay.

Experimental studies on Be-like atomic nitrogen ions, in their ground state 1s 2 2s 2 1 S, are contaminated by the presence of metastable states. In the SOLEIL experiments, N 3+ ions are produced in the gas-phase from an electroncyclotron-resonance-ion-source (ECRIS) therefore metastable states 1s 2 2s2p 3 P • may be present in the parent ion beam. The 1s 2 2s2p 3 P

• metastable state, autoionization processes from the 1s → 2p photo-excitation process are;

hν + N 3+ (1s 2 2s2p 3 P • ) ↓ N 3+ (1s2s[ 1,3 S]2p 2 ( 3 P, 1 D, 1 S)] 3 S, 3 P, 3 D) N 4+ (1s 2 2s 2 S) + e -(k 2 ) or N 4+ (1s 2 2p 2 P) + e -(k 2 ).
For the case of Li-like ions, the strongest excitation processes in the interaction of a photon with the 1s 2 2s 2 S 1/2 ground state of the Li-like nitrogen ion is the 1s → 2p photo-excitation process;

hν + N 4+ (1s 2 2s 2 S 1/2 ) → N 4+ (1s[2s2p 1,3 P] 2 P • 1/2,3/2 ) ↓ N 5+ (1s 2 1 S 0 ) + e -(k 2 ).

Detailed experimental PI measurements have been performed

for the first time on these prototype Be-like and Li-like systems in the photon energy region of the K-edge.

In our previous publication on K-shell PI of N + ions [11] we high-lighted the limitations of central fields calculations using Harte-Slater of Dirac-Slater potentials [58][59][60][61][62]. We stress again that results determined from these methods used in spectral modelling should be treated with caution.

For Be-like atomic nitrogen ions, state-of-the-art ab initio calculations for Auger inner-shell processes were carried out by Petrini and de Araújo [57] and by Berrington and coworkers [63] using the R-matrix method [51] and followed a similar procedure to the work on K-shell studies for the Be-like B + ion [64]. Petrini and co-workers [65] noted that once the 1s-hole was created in the ions, by single PI, with simultaneous shake-up and shake-off processes, Auger decay populates directly excited states of the residual ions, which then produces UV lines. Similarly, for Li-like atomic nitrogen ions, Charro and co-workers [66] performed R-matrix calculations in LScoupling for PI cross sections and compared and contrasted their results with previous central field and R-matrix methods as only limited experimental data was available. However, no attempt was made in their calculations at determining resonance parameters. About a decade later this work was further extended by Garcia and co-workers [9], using the optical potential method within the Breit-Pauli R-matrix formalism [51,52,67,68]. PI from the ground state, along the nitrogen iso-nuclear sequence was investigated, in the photon energy region of the K-edge. We note that for K-shell PI, the cross section calculations of Garcia and co-workers [9] included both radiation and Auger damping, which cause the smearing of the K-edge. Due to the lack of experimental data being available at that time a comparison with previous theoretical results (of lower calibre) were only possible. Garcia and co-workers [9] pointed out these earlier central field calculations do not take account of the resonance features that dominate the cross sections near the K-edge. In the present study we compare our theoretical results from the multi-configuration Dirac Fock (MCDF) and the R-matrix with pseudo states methods (RMPS), an empirical fitting approximation [69] (derived from previous experimental measurements on resonance energies and Auger widths of the iso-electronic sequence), prior theoretical results [9,70,71] and current experimental measurements made at the SOLEIL synchrotron radiation facility.

In this paper detailed theoretical calculations and experimental measurements are presented for the K-shell single photon ionization cross sections of Be-like atomic nitrogen ions (410-415 eV and 460-460.4 eV) and Lilike atomic nitrogen ions, (420-426 eV). Our theoretical predictions from the MCDF and R-matrix methods enabled identification of the 1s → 2p resonances and their parameters, observed in the Be-like and Li-like nitrogen spectra and the 1s → 3p resonance around 460 eV in Be-like nitrogen. The present investigation provides absolute values (experimental and theoretical) for cross sections along with the n = 2 innershell resonance energies, natural line-widths and resonance strengths, for a photon colliding with the 1s 2 2s 2 1 S, and 1s 2 2s2p 3 P • states of the N 3+ ion. For Li-like ions, only the ground state 1s 2 2s 2 S 1/2 is present in the beam.

The layout of this paper is as follows. Section 2 presents the experimental procedure used. Section 3 gives an overview of the theoretical work. Section 4 compares and contrasts the results obtained from the experimental and theoretical methods. Finally in section 5 conclusions are drawn from the present investigation.

Experiment

Cross sections for PI of Be-like and Li-like atomic nitrogen ions were measured in the range where K-shell PI occurs. The experiment was performed at the multi-analysis ion apparatus (MAIA) set-up, permanently installed on branch A of the PLEIADES beam line [72,73] at SOLEIL, the French National Synchrotron Radiation Facility, located in Saint-Aubin, France. Details of the experimental setup were outlined in a previous publication on N + [11]. The N 3+ and N 4+ ions are produced in a permanent magnet ECRIS. Collimated N 3+ and N 4+ ion-beam currents up to 100 nA were extracted from the ion source after biasing the ion source by +2 kV and then selected by mass per charge ratio using a dipole magnet selector. The ion beam was placed on the same axis as the photon beam by using electrostatic deflectors and einzel lenses to focus the beam. After the interaction region between the photon and the ion beams, another dipole magnet separates the primary beam and the beam of ions which have gained one (or several) charge(s) in the interaction, the so-called photoions. The primary ions are collected in a Faraday cup and the photon-ions are detected by multi-channel plates detector.

The photon current is measured by a calibrated photodiode.

Ions with the same charge as the photo-ions can also be produced by collisions between the primary ions and the residual gas or stripping on the walls in the interaction region. This background signal is subtracted by chopping the photon beam, collecting the data with and without photons for 20 s accumulation time.

For the absolute measurements of the PI cross sections, a -1000 V bias is applied on the 50 cm long interaction region and the data are collected with 30 meV photon energy steps. The overlap of the two beams and the density distributions of the interacted particles is determined in three dimensions by using two sets of three scanning slits. The cross sections obtained have an estimated systematic uncertainty of 15%. In another spectroscopy mode, no bias is applied to the interaction region allowing the photon and ion beams to interact over about 1 m and to scan the photon energy with a finer step. In this mode, only relative cross sections can be measured. They are later normalized on the cross sections determined in the absolute mode assuming the area under the resonances to be the same. Table 1 gives typical experimental parameters used to evaluate absolute cross sections for N 3+ ions at a photon energy of 412.5 eV. The energy and band width of the photon beam are calibrated separately using a gas cell and N 2 (1s → π * g v = 0) PI lines, located at 400.87 eV [74] and Ar 2p 3/2

-1 4s at 244.39 eV [75]. The photon energy, once corrected for Doppler shift, has an uncertainty of approximately 30 meV. Outstanding possibilities in terms of spectral resolution and flux at the N 2 (1s -1 ) K-edge have been discussed recently by Miron and co-workers [76,77].

Theory

SCUNC: Li-like and Be-like nitrogen

In the framework of the screening constant by unit nuclear charge (SCUNC) formalism [69,78,79], the total energy of the core-excited states is expressed in the form given by,

E N n ; 2S+1 L π = -Z 2 1 N 2 + 1 n 2 1 -β N n ; 2S+1 L π ; Z 2 , (1)
where E N n ; 2S+1 L π is in Rydberg units. In this equation, the principal quantum numbers N and n are respectively for the inner and the outer electron of the He-like iso-electronic series. The β-parameters are screening constants by unit nuclear charge expanded in inverse powers of Z and are given by the expression,

β N n ; 2S+1 L π = q k=1 f k 1 Z k (2)
where f k N n ; 2S+1 L π are parameters to be evaluated empirically from a previous experimental measurements.

Similarly, one may get the Auger widths in Rydbergs (1 Rydberg = 13.605 698 eV) from the formula

(Ry) = Z 2 1 - f 1 Z × Z Z 0 - f 1 Z 3 × (Z -Z 0 ) Z 2 0 - f 1 Z 4 × (Z -Z 0 ) Z 3 0 2 . (3)
The experiment measurements of Müller and co-workers on Be-like carbon and Li-like carbon and boron [31,32,34] were used to determine all the appropriate empirical parameters. For Be-like carbon, we note the labelling of the 1s2s2p 2 3 D and 1s2s2p 2 3 P states of Müller and co-workers [34] should be reversed as pointed out in our recent calculations on the carbon iso-nuclear sequence [6] used to model the x-ray spectra of the bright Blazar Mkn 421 observed by the Chandra satellite.

MCDF: Li-like and Be-like nitrogen

MCDF calculations were performed based on a full intermediate-coupling regime in a j j-basis using the code developed by Bruneau [80]. Photo-excitation cross sections have been carried out for both Li-like and Be-like atomic nitrogen ions in the region of their respective K-edges.

Only electric dipole transitions have been computed using the Babushkin gauge. For Be-like atomic nitrogen ion the following initial configurations have been considered: 1s 2 2s 2 , 1s 2 2s 2p, and 1s 2 2p 2 . Where the bar over the orbital is to indicate that they are different for the initial and final states.

In order to describe the correlation and relaxation effects, multiple orbitals with the same quantum number have been used. Then, the following final configurations have been considered: 1s2s 2 2p, 1s2s2p 2 , 1s2p 3 , 1s2s 2 3p, 1s2s2p3p, and 1s2p 2 3p. Such notation means that radial functions for n = 2 orbitals are not the same for initial and final configurations. The wavefunctions have been calculated minimizing the following energy functional:

E = α (2J α + 1)E α 2 α (2J α + 1) + β (2J β + 1)E β 2 β (2J β + 1) (4)
where α and β run over all the initial and final states, respectively. Synthetic spectrum has been constructed as a weighted sum of photo-excitation cross sections from both the ground state level 1s 2 2s 2 ( 1 S 0 ) and the metastable state levels 1s 2 2s2p( 3 P • 0,1,2 ). Each electric dipole transition has been dressed by a Lorentzian profile, assuming a full width at half maximum (FWHM) equal respectively to 66 meV and 80 meV for the 1s-2p and 1s-3p transitions in Be-like atomic nitrogen. Such FWHM values have been deduced from the MDCF calculations that were performed separately. A mixture of 60% ground and 40% metastable states has been used to fit the experimental results. In order to compare directly with the Be-like experimental data, the synthetic spectrum has been convoluted with a Gaussian profile to simulate the experimental resolution. A similar procedure has been used for the Li-like atomic nitrogen ion, except that (i) the following configurations have been considered : 1s 2 2s, 1s2s 2p, and 1s2s3p, and (ii) a Lorentzian FWHM equal to 45 meV and 60 meV has been used respectively for the 1s-2p and 1s-3p transitions in Li-like atomic nitrogen. It should be pointed out that for the Li-like atomic nitrogen ion, only the ground state contribution has been retained to compare with experimental data.

R-matrix: Be-like nitrogen

The R-matrix method [51,52,67,68], using a version of the codes implemented on parallel architectures [44,48,81] determined the necessary cross sections. For Be-like ions both the initial 1 S ground state and the 3 P • metastable states were required. Cross section calculations were carried out in LS-coupling with 390-levels retained in the close-coupling expansion using the RMPS. The Hartree-Fock 1s, 2s and 2p tabulated orbitals of Clementi and Roetti [82] were used with n = 3 physical and n = 4 pseudo orbitals of the residual N 4+ ion. The n = 4 pseudo-orbitals were determined by energy optimization on the ground state of the N 4+ ion, with the atomic-structure code CIV3 [83]. The n = 4 pseudoorbitals are used to account for core relaxation and electron correlation effects, in the multi-configuration interaction target wavefunctions. The N 4+ residual 390 ion states used multiconfiguration interaction target wavefunctions. The nonrelativistic R-matrix method determined the energies of the N 3+ bound states and all the appropriate cross sections. We determined PI cross sections for the 1s 2 2s 2 1 S ground state and the 1s 2 2s2p 3 P • metastable state.

For the electron-ion collision work we allowed threeelectron promotions out of selected base configurations of N 3+ . The collision work was carried out with 20 continuum functions and a boundary radius of 8.2 Bohr radii. From our RMPS calculations, the 1 S ground state gave a bound state ionization potential of 5.69195 Rydbergs compared to the experimental value of 5.69420 Rydbergs. In the case of the 3 P • metastable state, the ionization potential from the RMPS calculations gave 5.08202 Rydbergs, compared to the experimental value of 5.08200 Rydbergs. A discrepancy respectively of approximately 31 meV for the ground state and 0.3 meV for the metastable state. It is seen that both RMPS theoretical results are in excellent agreement with the experimental values from the NIST tabulations [START_REF] Kramida | NIST Atomic Spectra Database (version 5)[END_REF].

For the 1 S ground state and the 3 P • metastable state, the outer region electron-ion collision problem was solved by selecting an appropriately fine energy mesh of 2 × 10 -7 Rydbergs (≈2.72 µeV) in order to delineate all the resonance features in the cross sections. Radiation and Auger damping were also included in the R-matrix calculations.

For a direct comparison with the SOLEIL experimental measurements (performed at the various energy resolutions), the R-matrix results were convoluted with an appropriate Gaussian function of FWHM and an admixture of 60% ground state and 40% metastable state was used to simulate experiment. The peaks found in the theoretical PI cross section spectrum were fitted to Fano profiles for overlapping resonances [START_REF] Fano | [END_REF][86][87][88][89][90][91] instead of the energy derivative of the eigenphase sum method [92][93][94].

R-matrix: Li-like nitrogen

For this Li-like system intermediate-coupling PI cross section calculations were performed using the semi-relativistic Breit-Pauli approximation which allows for relativistic effects to be included in a similar manner to our previous work on Lilike, boron [31] and carbon [32] ions. As in our previous work, radiation-damping [67] effects were also included within the confines of the R-matrix approach [51,52] for completeness [51,67,98]. An appropriate number of N 5+ residual ion states (19 LS, 31 LSJ levels) were included in our intermediate-coupling calculations. The n = 4 basis set of N 5+ orbitals obtained from the atomic-structure code CIV3 [83] were used to represent the wavefunctions. PI cross section calculations were then performed in intermediate coupling for the 1s 2 2s 2 S 1/2 initial state of the N 4+ ion in order to incorporate relativistic effects via the semi-relativistic Breit-Pauli approximation.

For cross section calculations He-like LS states were retained: 1s 2 1 S, 1sns 1,3 S, 1snp 1,3 P • , 1snd 1,3 D, and 1snf 1,3 F • , n 4, of the N 5+ ion core giving rise to 31 LSJ states in the intermediate close-coupling expansions for the J = 1/2 initial scattering symmetry of the Li-like N 4+ ion. The n = 4 pseudo states are included in an attempt to account for core relaxation, electron correlations effects and the infinite number of states (bound and continuum) left out by the truncation of the close-coupling expansion in our work. For the structure calculations of the residual N 5+ ion, all n = 3 physical orbitals and n = 4 correlation orbitals were included in the multi-configuration interaction target wavefunctions expansions used to describe the states.

The Hartree-Fock 1s and 2s orbitals of Clementi and Roetti [82] together with the n = 3 orbitals were determined by energy optimization on the appropriate spectroscopic state using the atomic-structure code CIV3 [83]. The n = 4 correlation (pseudo) orbitals were determined by energy optimization on the ground state of this ion. All the states of the N 5+ ion were then represented by using multiconfiguration interaction wavefunctions. The Breit-Pauli Rmatrix approach was used to calculate the energies of the N 4+ (LSJ) bound states and the subsequent PI cross sections. A minor shift (<0.1%) of the theoretical energies for the N 5+ ion core states to experimental values [START_REF] Kramida | NIST Atomic Spectra Database (version 5)[END_REF] was made so that they would be in agreement with available experimental thresholds. Both double and triple promotion models for the scattering wavefunctions were investigated. The accuracy of the bound initial state wavefunction is more difficult to assess. Earlier LS-coupling calculations of Charro and co-workers The MCDF (solid green line) and R-matrix (solid red line) calculations shown are convolution with a Gaussian profile of 111 meV FWHM and an appropriate weighting of the ground and metastable states (see text for details) to simulate the measurements. For the metastable 3 P • state, the MCDF calculations have been shifted up by +1.46 eV in order to match experiment. Table 2 gives the designation of the resonances 1 -3 and their parameters.

[66] gave a value of 7.1909 Rydbergs, for the 1s 2 2s 2 S 1/2 bound state. From our R-matrix calculation we obtain values of 7.19301 Rydbergs (double electron promotions), and 7.19332 Rydbergs (triple electron promotions) compared to a value of 7.19479 Rydbergs from the NIST tabulation [START_REF] Kramida | NIST Atomic Spectra Database (version 5)[END_REF].

A discrepancy of 24 meV and 20 meV respectively with experiment. PI cross sections out of the Li-like nitrogen ion N 4+ (1s 2 2s 2 S 1/2 ) ground state were then obtained for total angular momentum scattering symmetries of J = 1/2 and J = 3/2, odd parity, that contribute to the total. The R-matrix method [51,52,67] was used to determine all the PI cross sections for the initial ground state in LS and intermediate coupling. In previous R-matrix work on Li-like boron [31] and Li-like carbon [32] ions, two and three-electron promotions scattering models were both employed yielding similar results. A similar tendency is found here when compared with the high resolution SOLEIL experimental measurements. We illustrate only the R-matrix results from the triple-promotion scattering model with the SOLEIL experimental measurements.

The electron-ion collision work was carried out with twenty continuum functions and a boundary radius of 6.8 Bohr radii. For the 2 S 1/2 initial state the outer region electron-ion collision problem was solved using an appropriate fine energy mesh of 2.0 × 10 -7 Rydbergs (≈2.72 µeV) to delineate the resonance features in the cross sections.

For Li-like atomic nitrogen, the QB technique (applicable to atomic and molecular complexes) of Berrington and co-workers [92][93][94] was used to determine the resonance parameters and averaging was performed over final total angular momentum J values. All the resonance parameters for the Li-like atomic nitrogen ion are presented in table 3.

Finally, in order to compare directly with experiment, the theoretical cross section was convoluted with a Gaussian function of appropriate FWHM (125 meV) to simulate the energy resolution of the measurements. The experimental and theoretical results for this Li-like atomic nitrogen ion are presented in figure 5.

Results and discussion

Figures 1-4 compare our experimental cross sections (solid circles) with theoretical predictions made from the MCDF (green curve) and R-matrix (red curve) methods for Be-like atomic nitrogen (N 3+ ). Similarly in figure 5 we present those for Li-like atomic nitrogen (N 4+ ) ions. For the Be-like atomic nitrogen (N 3+ ) ion, the experimental cross section has been recorded with different band widths, ranging from 38 to 133 meV, presented in figures 1-4 respectively. Each cross section is obtained from the weighted mean of several sweeps (from three to eight following increasing resolution). The assignment, resonance excitation energy, Auger width and strengths of all the n = 2 observed lines are summarized in tables 2 and 3. For the determination of the experimental widths, each individual sweep has been fitted separately by Voigt profiles to avoid any possible shift in the energy delivered by the monochromator. Then the final width of the Lorentzian component was obtained from the weighted mean of the individual Lorentzian width determined from each fit. For the N 3+ ion, splitting of the J components of the initial state [START_REF] Kramida | NIST Atomic Spectra Database (version 5)[END_REF] was also taken into account, assuming a statistical distribution of the levels. The experimental and theoretical oscillator strengths were obtained from the area under the lines. The MCDF (solid green line) and R-matrix (solid red line) calculations shown are convolution with a Gaussian profile of 56 meV FWHM and an appropriate weighting of the ground and metastable states (see text for details) to simulate the measurements. For the metastable 3 P • state, the MCDF calculations have been shifted up by +1.46 eV in order to match experiment. Table 2 gives the designation of the resonances 1 -3 and their parameters. The MCDF (solid green line) and R-matrix (solid red line) calculations shown are convolution with a Gaussian profile of 38 meV FWHM and an appropriate weighting of the ground state and metastable state (see text for details) to simulate the measurements. For the metastable 3 P • state, the MCDF calculations have been shifted up by +1.46 eV in order to match experiment. Table 2 gives the designation of the resonances 1 -2 and their parameters.

For Be-like atomic nitrogen ions, three peaks were observed in the experimental spectrum in the photon energy range 412-415 eV investigated. Figure 1, shows the experimental and theoretical results obtained at a resolution of 111 meV, figure 2 illustrates the results at the higher resolution of 56 meV. The double peak around 412 eV is more visible at the higher resolution of 38 meV, see figure 3.

Due to the presence of metastable states in the beam of ions for N 3+ , in order to make a true comparison of theory with experiment, the theoretical cross sections were convoluted using a Gaussian profile function of the appropriate width and an admixture of 40% metastable and 60% ground state ions used to simulate experiment. From our results illustrated in figures 1-4 it is seen over the entire photon energy region 3 gives the resonances 1 -2 and their parameters.

investigated, excellent agreement is achieved with the present R-matrix calculation, both on the absolute cross sections scale and for resonance energies positions. The MCDF calculations show less favourable agreement, as the double peak resonances due to the 1s 2 2s2p 3 P • metastable state lies over 1.46 eV below the experimental locations. In figures 1-3, for Be-like nitrogen, we have shifted the MCDF cross section calculations for the 1s 2 2s2p 3 P • metastable up by 1.46 eV in order to match the experimental measurements. Table 2 presents the resonance parameters for all three resonances found in the Be-like atomic nitrogen experimental measurements. We note that our present RMPS estimates for the position of all three resonances are in excellent agreement with experiment.

Table 2. Be-like nitrogen (N 3+ ), present experimental and theoretical results for the resonance energies E (res) ph (eV), natural line-widths (meV) and resonance strengths σ PI (in Mb eV), for the dominant core photo-excited n = 2 states of the N 3+ ion, in the photon energy region 410 to 415 eV compared with previous investigations. The experimental error in the calibrated photon energy is estimated to be ± 30 meV for the resonance energies. Estimates for the resonance energies, of Be-like nitrogen made using the screening constant by unit nuclear charge (SCUNC) empirical fitting approach [69] show satisfactory agreement with the more sophisticated theoretical methods and experiment. In the case of the Auger widths, apart from the first 1s2s2p 2 [ 4 P] 3 P resonance we see they are all in respectable agreement with experiment. We note that the more sophisticated theoretical methods such as R-matrix and the saddle point method consistently give values in agreement with each other and in general with experiment. We point out that in recent K-shell measurements for Be-like boron (e.g., B + [33]) the Auger width for the 1s2s2p 2 [ 4 P] 3 P gave a value of ∼10 meV, consistent with theory. So we suspect that even higher resolution than the present work performed at SOLEIL would be required to fully resolve the double peak structure in the N 3+ spectrum at about 412 eV.

Resonance

Figure 4 shows the SOLEIL data taken in the region of the 1s 2 2s 2 1 S → 1s2s 2 3p 1 P • resonance located around 460 eV at an energy resolution of 133 meV. Due to the limited experimental data taken in this region (as illustrated in figure 4), it was not possible to fit the experimental data to extract a reliable Auger width. We find an experimental value for the energy of this resonance to be 460.280 ± 0.04 eV, a strength of 6.30 ± 2.6 which compares favourably with theoretical predictions from the RMPS value of 460.107 eV with a resonance strength of 6.59. Furthermore, the RMPS provides an Auger width of 72 meV for this 1s → 3p resonance. For the 1s → 3p resonance a position of 460.019 ± 0.045 eV with an Auger width of 87 meV was determined from the empirical fitting SCUNC approximation [69]. The optical potential R-matrix method ( [9]) for this same resonance gave a value of 462.373 eV with an Auger width of 68 meV. The present MCDF work gave values of 460.189 eV for the position and 80 meV for the Auger width. No other values appear to be available in the literature. We note that numerical values of cross sections from previous R-matrix work [9] are available for only the ground state for this Be-like ion and as such similar comparisons cannot be made in figures 1-4.

Table 3. Li-like nitrogen (N 4+ ), present experimental and theoretical results for the resonance energies E (res) ph (eV), natural line-widths (meV) and resonance strengths σ PI (in Mb eV), for the 1s 2 2s 2 S → 1s[2s2p 1,3 P] 2 P • core-excited states of the N 4+ ion, in the photon energy region 420 to 426 eV compared with previous investigations. The intermediate-coupling results have been averaged over the fine structure levels to compare with experiment and with other theoretical methods. The error in the calibrated photon energy is estimated to be ± 30 meV for the resonance energies and the experimental resolution was 125 meV determined from multi-function Voigt fits to the measurements. k Complex scaled multi-reference configuration interaction method (CMR-CI) [102]. l Saddle-point method with R-matrix (SPM-RM). [103]. m Screening constant by unit nuclear charge (SCUNC) approximation [69,78,79].

Resonance

Table 3 presents the resonance parameters for the two resonances observed in the Li-like atomic nitrogen experimental measurements illustrated in figure 5. We note that only the first narrow peak in the Li-like experimental spectrum located at 421.472 ± 0.03 eV was able to be properly fitted, giving a Auger line width of 11 ± 8 meV. Both the sophisticated R-matrix and Saddle point methods are in agreement with each other and the experimental value for the location of this resonance. Theoretical predictions from R-matrix calculations and other methods for the Auger width all lie within the experimental error estimate for this resonance. The second peak in the Li-like atomic nitrogen photoionization cross section, located at 425.449 ± 0.03 eV, had limited experimental data taken on the lower energy side of the peak and due to the rather large noise level (large scatter in the experimental cross section data) made it impossible to fit. The noise level was simply just too large to do a proper fit of this peak. For the natural line width of this 1s[2s2p 1 P] 2 P • resonance, R-matrix calculations give a value of 42 meV, present MCDF estimates of 45 meV, and the Saddle point method 42 meV. From previous R-matrix work [9] we have extracted resonance positions and widths. For the first resonance we find an energy of 420.612 eV and for the second resonance an energy of 424.823 eV, a discrepancy of about 0.9 eV and 0.6 eV lower than the present experiment, with values of 6 meV and 53 meV for the respective widths.

It is also seen that the present R-matrix calculations yield a resonance strength of 6.5 Mb eV, MCDF estimates a value of 9 Mb eV and both theoretical values are in suitable agreement with the experimental measurement of 7.3 ± 2.5. For the positions of both resonances, the theoretical values from the RMPS calculations are in more favourable agreement with the SOLEIL synchrotron measurements than either the earlier EBIT [24], the laser produced plasmas (LPP) [53] or those determined from electron spectroscopy in ion-atom collisions [54] measurements. Table 3 indicates that estimates for the resonance energies and Auger widths of Li-like nitrogen made using the SCUNC empirical fitting approach [69,78,79] are in satisfactory agreement with the more sophisticated theoretical methods and experiment.

Conclusions

K-shell photoionization for Be-like and Li-like atomic nitrogen ions, respectively N 3+ and N 4+ , has been performed using state-of-the-art experimental and theoretical methods in the vicinity of their respective K-edges. To our knowledge this would appear to be the first time high-resolution spectroscopy has been performed (at a photon energy resolution of respectively, 38 meV, 56 meV, 111 meV, 133 meV FWHM for Be-like and 125 meV FWHM for Li-like atomic nitrogen ions). The measurements at the SOLEIL synchrotron radiation facility, in Saint-Aubin, France, cover the photon energy ranges 410-415 eV and 460-460.4 eV for Be-like and 420-426 eV for Li-like atomic nitrogen ions. The strong peaks found in the respective PI cross sections in the photon energy regions studied are identified as the 1s → 2p and 1s → 3p transitions in the Be-like and 1s → 2p transitions in the Li-like K-shell spectrum that are assigned spectroscopically. All the n = 2 resonance parameters have been tabulated in table 2 for Be-like and in table 3 for Li-like atomic nitrogen ions. For the observed peaks, respectable agreement is seen with the present theoretical and experimental results both on the photon-energy scale and the absolute cross section scale. Some differences are highlighted and discussed.

The strength of the present study is the high resolution of the spectra along with theoretical predictions made using state-of-the-art MCDF and R-matrix methods. The present results have been compared with high-resolution experimental measurements made at the SOLEIL synchrotron radiation facility and with other theoretical methods so would be suitable to be incorporated into astrophysical modelling codes like CLOUDY [16,17], XSTAR [104] and AtomDB [105] used to numerically simulate the thermal and ionization structure of ionized astrophysical nebulae.

Introduction

Satellites Chandra and XMM-Newton currently provide a wealth of x-ray spectra of astronomical objects; however, a serious lack of high-quality atomic data impedes the interpretation of these spectra [1][2][3][4][5]. Recent studies on carbon and its ions have shown that high-quality data is required to model the observations in the x-ray spectrum of the bright blazar Mkn 421 observed by the Chandra LETG+HRC-S [6]. Spectroscopy in the soft x-ray region (5-45 Å) including K-shell transitions of C, N, O, Ne, S and Si, in neutral, singly or doubly ionized states and L-shell transitions of Fe and Ni, provides a valuable tool for probing the extreme environments in active galactic nuclei (AGNs), x-ray binary systems, cataclysmic variable stars (CVs) and Wolf-Rayet stars [7] as well as the interstellar media (ISM) [8]. The latter, 7 Authors to whom any correspondence should be addressed. recent work, for example, demonstrated that x-ray spectra from XMM-Newton can be used to characterize ISM, provided accurate atomic oxygen K-edge cross sections are available. Analogous results concerning the chemical composition of the ISM are to be expected with the availability of accurate data on neutral [9,10] and singly ionized atomic nitrogen K-edge cross sections. The globular cluster x-ray source CXO J033831.8-352604 in NGC 1399 has also recently been found to show strong emission lines of [O III] and [N II] in its optical spectrum in addition to ultraluminous x-ray emission with a soft x-ray spectrum [11,12]. This lack of available x-ray data has motivated not only the present authors to perform K-shell photoionization (PI) studies on such ions but also other groups who use the same or similar experimental techniques. K-shell PI cross section results have been obtained on a variety of ions of astrophysical interest: He-like Li + [13,14], Li-like B 2+ [15], C 3+ [16], Be-like B + [17] , C 2+ [18], B-like C + [19], N-like O + [20], F-like Ne + [21], along with valence shell studies on Mg-like Fe 14+ [22]. The majority of this experimental data has been benchmarked with state-of-the-art theoretical methods.

This work on this proto-type C-like ion provides benchmark values for cross sections on photoabsorption of x-rays in the vicinity of the K-edge, where strong n = 2 innershell resonance states of singly ionized atomic nitrogen are observed. No experiments have been reported to date on singly ionized atomic nitrogen in the K-edge photon energy region, and previous experimental studies have been restricted to the valence shell region [23].

Promotion of a K-shell electron in C-like nitrogen (N + ) ions to an outer np-valence shell (1s → np) from the ground state produces states that can autoionize, forming a N 2+ ion and an outgoing free electron. One of the strongest excitation processes in the interaction of a photon with the 1s 2 2s 2 2p 2 3 P ground-state of the C-like nitrogen ion is the 1s → 2p photoexcitation process:

hν + N + (1s 2 2s 2 2p 2 3 P) → N + (1s2s 2 2p 3 3 S o , 3 P o , 3 D o ) ↓ N 2+ (1s 2 2s 2 2p 2 P o ) + e -.
Experimental studies of this singly ionized atomic nitrogen ion in its ground state 1s 2 2s 2 2p 2 3 P are further hampered by the presence of metastable states as the N + ions are produced in the gas phase using an electron-cyclotron-resonance-ionsource (ECRIS). The metastable states 1s 2 2s 2 2p 2 1 D, 1 S and 1s 2 2s2p 3 5 S o are present in the ion beam. In the case of the 1s 2 2s2p 3 5 S o metastable state, auto-ionization processes occurring by the 1s → 2p photo-excitation process are hν + N + (1s 2 2s2p 3 5 S o ) ↓ N + [1s2s[ 1,3 S]2p 4 ( 3 P, 1 D, 1 S)] 5 P ↓ N 2+ (1s 2 2s2p 2 4 P) + e -.

Similarly, auto-ionization processes may also occur for the 1s 2 2s 2 2p 2 1 D, 1 S metastable states interacting with photons. To our knowledge this would appear to be the first time experimental measurements have been performed on this proto-type C-like system in the photon energy region of the K-edge.

Theoretical PI cross-section calculations for inner-shell processes in this C-like ion have been performed by Reilman and Manson [24] using the Hartree-Slater wavefunctions of Herman and Skillman [25] and by Verner and co-workers [26] using Dirac-Slater potentials ( [27,28]) within a central field approximation. PI cross sections obtained using these methods may be suitably accurate at very high photon energies but often give poor results near thresholds where configuration mixing is strong and resonance structure prevails, as is the case for the processes investigated here in the vicinity of the K-edge threshold. Central field methods neglect resonance features (observed in experimental measurements) as degenerate subshells of equivalent electrons rather than individual energy levels are considered. Therefore, any results obtained from these approaches in spectral modelling should be treated with due caution.

In this paper, we present detailed measurements of the single-and double-PI cross sections in the 398-406 eV region (the only one where peaks were observed) from the entire 388-430 eV photon energy range explored. Our multi-configurational Dirac-Fock (MCDF) and R-matrix calculations enabled the identification and characterization of the strong 1s → 2p resonances observed in the spectra. The present investigation provides absolute values (experimental and theoretical) for PI cross sections, n = 2 inner-shell resonance energies and linewidths occurring in the interaction of a photon with the 1s 2 2s 2 2p 2 3 P , 1 D , 1 S and 1s 2 2s2p 3 5 S o states of the N + ion.

The layout of this paper is as follows. Section 2 details the experimental procedure used. Section 3 presents a brief outline of the theoretical work. Section 4 presents a discussion of the results obtained from both the experimental and theoretical methods. Finally, in section 5 conclusions are drawn from the present investigation.

Experiment

Ion production

The present measurements were made using the new MAIA (Multi-Analysis Ion Apparatus) set-up, figure 1, permanently installed on branch A of the PLEIADES beam line [29] at SOLEIL, the French National Synchrotron Radiation Facility, located in Saint-Aubin, France. It is a merged beam setup, similar to the one originally designed by Peart et al [30] for measuring electron impact cross sections. A simplified scheme of the set-up is shown in figure 1.

The N + ions are produced in a permanent magnet ECRIS especially designed for this set-up at the CEA (Commissariat á l'Energie Atomique et aux Energies Alternatives) in Grenoble and previously tested at LURE (Laboratoire pour l'Utilisation de Rayonnement Electromagnétique) in Orsay [31]. To produce a nitrogen plasma, molecular nitrogen gas is injected into the plasma chamber and heated by a 12.6 GHz microwave. A power of a few watts is sufficient for an optimum production of the N + ions. The ions are extracted by applying a 2 kV bias on the source and selected in mass/charge ratio by a dipole magnet. The selected ions are deflected by a 45 • spherical electrostatic deflector (ED1) to be merged with the photon beam inside a 50 cm long interaction region (IR). Two sets of slits (S2 and S3) allow the matching of the size of the ion beam to the size of the photon beam. After interaction, the charge of the ions is analysed by a second dipole magnet. The parent ions are collected in a Faraday cup (FC4), and the so-called photoions which have gained one (or several) charge(s) are counted using channel plates.

Excitation source

The photon beam is monochromatized synchrotron radiation from the PLEIADES beam line. Two undulators with 256 and 80 mm period deliver photons in the 10-100 eV and 100-1000 eV energy ranges, respectively, with all types of polarizations above 55 eV. The light is monochromatized by a plane grating monochromator with no entrance slit. High spectral purity is obtained by a combination of a quasiperiodic design for the undulators and the use of a varied groove depth for the plane grating. Three varied line spacing gratings are available for the adjustment of the resolution. An ultimate resolving power of approximately 10 5 is achievable at 50 eV. After monochromatization, the light is distributed and focalized to three different branches. On branch A, used for this experiment, the photon beam is parallel, with a typical spot size of 2 mm × 2 mm. The photon flux is measured using a calibrated silicon photodiode (PD). The photon energy is determined using a double-ionization chamber of Samson type [32]. For this work, we used the 1s → π * transitions in the N 2 gas [33] and 2p → 3d transitions in Ar gas [34] for calibration purposes. The photon energy was corrected for Doppler shift resulting from the velocity of the N + ions. The estimated accuracy of the photon energy determination is 40 meV.

Experimental procedure

The merged-beam set-up allows a determination of the absolute PI cross sections. At a given photon energy, the cross sections σ PI are obtained from

σ PI = Se 2 ηνq I J L 0 dz x yF (z) , ( 1
)
where S is the counting rate of photoions measured with the channel plates and η is the efficiency of the photodiode. A chopper, placed at the exit of the photon beam line, allows us to subtract from the photoion signal, the noise produced by collisional ionization processes, charge stripping on the slits or autoionizing decay of metastable excited states of N + ions produced in the ECRIS. Here, q is the charge of the target ions, ν is the velocity of the ions in the interaction region determined from the accelerating potential applied to the ECRIS and I is the current produced by the photons on the calibrated photodiode. The efficiency η of the photodiode was calibrated in the 10-1000 eV energy range at the Physikalisch-Technische Bundesanstalt (PTB) beam line at BESSY in Berlin; e is the charge of the electron; J is the current of incident ions measured in FC4; is the efficiency of the microchannel plates determined by comparing the counting rate produced by a low intensity ion beam and the current induced by the same beam in FC4; x yF (z) is an effective beam area (z is the propagation axis of the two beams) and F(z) is a two-dimensional form factor determined using three sets of xy scanners placed at each end and in the middle of the interaction region. Each scanner is a 0.2 mm width slit moved across the ion and photon beams. The length L of the interaction region is fixed by applying a 200 V bias on a 50 cm long tube placed in the interaction region, resulting in a different velocity for the photoions produced inside and outside the tube. F(z) is defined by

F xy ≈ i xy j xy i xy j xy ( 2
)
where i xy =i(x,y) x y and j xy =j (x,y) x y are the ion and photon currents, respectively, passing through the slits and measured with FC4 for the ion beam and the photodiode for the photon beam, and x and y are the step sizes used to scan the slits, typically 0.2 mm. Typical values of the parameters involved in equation ( 1) measured at a photon energy of 400 eV are given in table 1. The contribution of N 2+ 2 molecular ions in the incident beam was subtracted. It was measured after the beam time using 14 N 15 N isotopic gas to be of the order of 3%.

The accuracy of the measured cross-sections is determined by statistical fluctuations on the photoion and noise counting rates and a systematic contribution resulting from the measurement of the different parameters in equation (1). The latter is estimated to be 15% and is dominated by the uncertainty on the determination of the photon flux, the form factor and detector efficiency.

To record the single-(double)-PI spectra, the field in the dipole magnet was adjusted to detect N 2+ ions (N 3+ ions) with the channel plates as the photon energy was scanned. Two modes have been used. One with no voltage applied on the interaction tube, allowing a better statistic since the whole interaction length of the beams is used. In this mode, only relative cross sections are obtained. In the second mode, the voltage is applied to the tube to define the interaction length L (50 cm), which allows the determination of the cross sections in absolute value.

Theory

Multi-configurational Dirac-Fork calculations

We have performed MCDF calculations based on full intermediate coupling in a jj basis using the code developed by Bruneau [35]. The photoexcitation cross sections involving the 1s→np transitions have been considered for all levels of the N + ground configuration, namely 1s 2 2s 2 2p 2 3 P 0,1,2 , 1 D 2 and 1 S 0 . The metastable 1s 2 2s2p 3 5 S o 2 level has also been considered. The calculations were limited to principal quantum number up to 4.

Only electric dipole transitions have been computed using Babushkin and Coulomb gauges, respectively. The oscillator strengths obtained in the two gauges differ by less than 10%. Using the MCDF code we have calculated the average autoionization rate av (meV) of the 1s2s 2 2p 3 configuration (with a 1s vacancy) to be equal to 96 meV (which corresponds to a lifetime τ of 3.43 fs from the uncertainty principle E t = h/2). From the MCDF calculations synthetic spectra are constructed as a sum of Lorentzian profiles using, for all lines, the av value as the full width at half maximum (FWHM). In order to compare directly with the experimental measurements made at 300 and 60 meV, the MCDF results were convoluted with Gaussian functions of FWHM of 300 and 60 meV, respectively, to simulate the measurements. A nonstatistical distribution of the ground and metastable states gave best agreement with experiment by weighting the contribution of the 1s 2 2s 2 2p 2 3 P, 1 D, 1 S and 1s 2 2s2p 3 5 S o states by (0.54, 0.11, 0.03, 0.32), respectively.

R-matrix

The R-matrix method [36][37][38], with an efficient parallel implementation of the codes [39], was used to determine all PI cross sections, for both the initial 3 P ground state and the 1 S, 1 D and 5 S o metastable states in LS-coupling using 390 levels in the close-coupling expansion. The Hartree-Fock 1s, 2s and 2p tabulated orbitals of Clementi and Roetti [40] were used together with the n = 3 orbitals of the N 2+ ion determined by energy optimization on the appropriate holeshell state using the atomic structure code CIV3 [41]. The n = 3 correlation (pseudo)orbitals are included to account for core relaxation and additional correlation effects in the multiconfiguration interaction wavefunctions. All N 2+ residual ion states were then represented by using multi-configuration interaction wavefunctions. The non-relativistic R-matrix approach was used to calculate the energies of the N + bound states and the subsequent PI cross sections. PI cross sections out of the 1s 2 2s 2 2p 2 3 P ground state and the 1 S, 1 D along with the 1s 2 2s2p 3 5 S o metastable states were then obtained for all total angular momentum scattering symmetries that contribute.

All of our PI cross sections were determined in LScoupling with the parallel version [39] of the R-matrix programs [36][37][38] and 390 levels of the N 2+ residual ion included in our close-coupling calculations. Due to the presence of meta stable states in the beam, PI cross-section calculations were performed for both the 1s 2 2s 2 2p 2 3 P ground state and the 1s 2 2s 2 2p 2 1 S, 1 D and 1s 2 2s2p 3 5 S o metastable states of the N + ion. Here again, in order to compare directly with the experimental measurements made at 300 and 60 meV, the R-matrix results were convoluted with Gaussian functions of FWHM of 300 and 60 meV, respectively, to simulate the measurements. A non-statistical distribution of the ground and metastable states gave best agreement with experiment by weighting the contribution of the 1s 2 2s 2 2p 2 3 P, 1 D, 1 S and 1s 2 2s2p 3 5 S o states by (0.54, 0.11, 0.03, 0.32), respectively.

The scattering wavefunctions were generated by allowing all possible two-electron promotions out of select base configurations of N + into the orbital set employed. Scattering calculations were performed with 20 continuum functions and a boundary radius of 6.2 Bohr radii. For both the 3 P ground state and the 1 S, 1 D, 5 S o metastable states, the outer region electron-ion collision problem was solved (in the resonance region below and between all thresholds) using a suitably chosen fine energy mesh of 2×10 -7 Rydbergs (≈ 2.72 μeV) to fully resolve all of resonance structure in PI cross sections. Radiation and Auger damping were also included in the cross-section calculations. Finally, the total spectrum was obtained assuming a weighting of the 1s 2 2s 2 2p 2 3 P, 1 D, 1 S and 1s 2 2s2p 3 5 S o states by (0.54, 0.11, 0.03, 0.32), respectively, in order to compare directly with the PI cross-section measurements obtained using the ion beam produced from the ECRIS at the SOLEIL radiation facility.

The multi-channel R-matrix QB technique (applicable to atomic and molecular complexes) of Berrington and co-workers [42][43][44] was used to determine the resonance parameters. The resonance width may be determined from the inverse of the energy derivative of the eigenphase sum δ at the position of the resonance energy E r via

= 2 dδ dE -1 E=E r = 2[δ ] -1 E=E r . ( 3
)
The results for the resonance parameters determined from the QB method are presented in table 2.

Results and discussion

In the entire 388-430 eV photon energy range we explored, only three intense structures were observed around 400 eV. Figure 2 shows our experimental and theoretical (MCDF and R-matrix) results over the photon energy range 398 eV-406 eV. The experimental results were taken with a photon energy resolution of 300 meV. The black triangles show the (res) ph (eV), and the natural linewidths (meV) for the core photoexcited n = 2 states of N + ions in the photon energy region 399-403 eV. The experimental uncertainty is ± 40 meV for the resonance energies. The earlier MCDF calculations of Chen and co-workers [45] measurements of the single-PI cross section obtained in the absolute mode. The yellow circles have been obtained in the relative mode with a smaller energy step. They have been put on the absolute scale keeping the same area under the absolute and relative measurements. The open blue triangles give the double PI cross section determined in the relative mode. To place them on the absolute scale it was assumed that the length of the interaction region was the same while recording the relative single-and double-PI cross sections.

In order to compare directly with the experimental measurements both our MCDF and R-matrix results were convoluted with a Gaussian function of FWHM of 300 meV to simulate the measurements. Furthermore, we found that a nonstatistical distribution of the ground and metastable states gave best agreement with experiment by weighting the contribution of the 1s 2 2s 2 2p 2 3 P, 1 D, 1 S and 1s 2 2s2p 3 5 S o states by (0.54, 0.11, 0.03, 0.32), respectively. A statistical weighting showed greater disparity between theory and experiment. Figure 3 shows the same experimental and theoretical (MCDF and R-matrix) single-PI cross section results over the photon energy range 399-403 eV but at the higher resolution of 60 meV. Voigt profiles were used to fit the peaks in the experimental data, to extract the linewidths, assuming a Gaussian instrumental contribution of 60 meV for each peak. The experimental data, obtained in the relative mode, were normalized keeping the same total area under the experimental single-PI cross section measured with 60 and 300 meV band pass (yellow circles on figures 2 and 3). The error bars give the statistical uncertainty. shown in figure 2 by the open blue triangles. Here again, theory has been convoluted with a Gaussian of 60 meV FWHM and the weighting of the states used as before. Both MCDF and R-matrix calculations are in good qualitative agreement with the experimental results. The position of the lines is better described by the R-matrix calculations, from matching the calculated and experimental ionization thresholds. The R-matrix calculations include Auger and radiation damping missing from the MCDF calculations and give line intensity in closer agreement with experiment. The R-matrix calculations indicate that radiation damping can contribute up to 30%.

In table 2, we present the MCDF theoretical values for the resonance energies and linewidths from our work which were obtained by fitting the various peaks found in the cross sections to Fano profiles for overlapping resonances. This enabled the extraction of the resonance energies and natural linewidths for the various states. The assignment of the peaks in the PI cross sections (due to 1s → 2p photoexcitation) is based on our R-matrix theoretical work, since it was not possible to extract experimental values for each individual state as the lines are a blend of different states. The first strong peak at about 399.85 eV is primarily a blend of resonance A, 1s2s 2 2p 3 3 D o , resulting from photoexcitation of the ground 1s 2 2s 2 2p 2 3 P state along with that from resonance B, 1s2s2p 4 5 P from the 1s 2 2s2p 3 5 S o metastable state. This is clearly seen in the higher resolution experimental data taken at 60 meV (figure 3) as opposed to the measurements made at the lower resolution of 300 meV (figure 2). The valley between the first and the second peak is filled with the 1s2s 2 2p 3 1 P o resonance C resulting from photo-excitation of the 1s 2 2s 2 2p 2 1 S metastable. The next strong peak at about 400.63 eV is a blend of the 1s2s 2 2p 3 3 S o resonance D, resulting from photo-excitation from the ground 1s 2 2s 2 2p Experimental linewidths were extracted from the ionyield measurements in figure 3 by fitting Voigt profiles to the peaks assuming a Gaussian instrumental contribution of 60 meV for each peak. The first resonance peak in the experimental data at 399.844 ± 0.04 eV has a linewidth of 138 ± 41 meV, the second peak located at 400.633 ± 0.04 eV has a linewidth of 86 ± 10 meV and the third peak at 401.346 ± 0.04 has a linewidth of 143 ± 21 meV. The linewidth gives only an upper limit of the natural width since several lines can contribute to the structures. According to the R-matrix calculations, only peak 3 in the experimental data is pure, and following MCDF calculations, only peak 2, which may be due to the S → P dipole selection rules. Table 2 presents the theoretical results for the resonance positions and widths from our present investigations. For comparison purposes we have also included the results from the earlier MCDF calculations of Chen and co-workers [45]. From the R-matrix results we see that the average autoionization rate for the 1s2s 2 2p 3 configuration (with a 1s vacancy) is 112 meV compared to a value of 96 meV determined from our present MCDF calculations.

In table 2, the results from our present theoretical predictions from both the MCDF and R-matrix methods overall indicate that they are in suitable agreement with each other. Furthermore, it is seen that the R-matrix results give closer agreement with experiment for the resonance energies of all peaks, deviating at most by 45 meV (cf the average for A+B), which is just outside the experimental error of 40 meV). All other resonance energies from the R-matrix work are within the 40 meV experimental uncertainty; however, greater disparity is seen between experiment and the MCDF results. The use of a much larger basis in the R-matrix work along with coupling to the continuum could possibly improve the agreement with experiment but vastly increases the computational complexity of the problem. As seen from table 2, discrepancies occur in the position and autoionization linewidth of the resonances from our work when compared to the early MCDF calculations of Chen et al [45]. This was also highlighted by Garcia and coworkers [9], where outstanding discrepancies were found with that MCDF work, in particular wavelengths and A-coefficients for this C-like ion, which those authors believed to be due to numerical error by Chen et al [45].

Conclusions

Photoionization of C-like nitrogen ions, N + , has been investigated using state-of-the-art experimental and theoretical methods. High-resolution spectroscopy was performed with E/ E = 7000 at the SOLEIL synchrotron radiation facility, in Saint-Aubin, France, covering the energy range 388-430 eV where several strong peaks in the cross sections were found in the energy region around 400 eV. For these observed peaks, suitable agreement is found between the present theoretical and experimental results both on the photon-energy scale and on the PI cross-section scale for this prototype C-like system. The strength of this study is in its excellent experimental resolving power coupled with theoretical predictions using the MCDF and R-matrix method. Given that the present results have been benchmarked with high-resolution experimental data and with state-of-the-art theoretical methods they would be suitable to be included in astrophysical modelling codes such as CLOUDY [46,47] and XSTAR [48]. The theoretical R-matrix cross sections for the individual states are available by contacting one of the authors, B M McLaughlin, b.mclaughlin@qub.ac.uk.

Introduction

X-ray spectra from XMM-Newton may be utilized to characterize the interstellar medium, if accurate atomic K-edge cross sections are available [1][2][3][4][5][6][7]. Single and multiply ionization stages of C, N, O, Ne and Fe have been observed in the ionized outflow in NGC 4051 measured with XMM-Newton (x-ray multi-mirror mission-Newton) [8] in the soft x-ray region and low ionized stages of C, N and O have also been used in modelling x-ray emission from OB supergiants [9]. Radiative/photo recombination of singly and doubly charged nitrogen ions also play an important role in the chemistry of the atmosphere of Titan [10]. Detailed photoionization (PI) models of the brightest knot of star formation in the blue compact dwarf galaxy Mrk 209 required abundances for ions of oxygen and nitrogen [11]. The XMM-Newton x-ray spectra of WR 1 is rich in nitrogen ions [12] and PI cross section data and abundances for carbon, nitrogen, and oxygen in their various stages of ionization are essential for PI models applied to the plasma modelling in a variety of planetary nebulae [13]. In the present study we focus our attention on obtaining detail spectra on the doubly ionized nitrogen ion N 2+ (N III) in the vicinity of its K-edge.

PI cross sections used for the modelling of astrophysical phenomena have mainly been provided by theoretical methods, due to limited experimental data being available. Major effort has gone into improving the quality of calculated data using state-of-the-art theoretical methods. Recent advances in the determination of atomic parameters for modelling K lines in cosmically abundant elements have been reviewed by Quinet and co-workers [4].

Absolute experimental K-shell PI cross section results have been obtained by various groups on a variety of atoms and ions of astrophysical interest; He-like Li + [14-16], Li-like B 2+ [17], C 3+ [18], N 4+ [6], Be-like B

+ [19], C 2+ [20], N 3+ [6], B-like C + [21], C-like N + [5], N-like O + [22], F-like Ne + [ 23
], neutral nitrogen [24] and oxygen [25][26][27][28].

Recent studies on K-shell PI cross sections calculations for neutral nitrogen and oxygen showed excellent accord with high-resolution measurements made at the advanced light source (ALS) radiation facility [24,28] as have similar cross section calculations on singly and multiply ionized stages of atomic nitrogen compared with high-resolution measurements at the SOLEIL synchrotron facility [5,6]. The majority of the high-resolution experimental data from third generation light sources have been shown to be in excellent agreement with the state-of-the-art R-matrix method [29,30] and with other modern theoretical approaches.

The present investigation for this proto-type B-like atomic nitrogen ion gives accurate values of PI cross sections produced by x-rays in the vicinity of the K-edge, where strong n = 2 inner-shell resonance states of N 2+ are observed. This work complements our previous studies on K-shell PI of singly and multiply ionized atomic nitrogen ions [5,6] in the vicinity of the K-edge. Previous experimental studies on B-like atomic nitrogen have been performed only in the valence shell region [31,32]. To date no experimental studies for the K-edge region have been reported in the literature. For the B-like ions N 2+ , O 3+ and F 4+ , absolute experimental cross section measurements in the near threshold region were made by Bizau and co-workers [31,32]. Close-coupling calculations performed by Li-Guo and Xin-Xiao [33] based on the Rmatrix formalism [34] obtained excellent agreement with the experimental work of Bizau and co-workers [32]. In this near threshold region it was necessary to include both the ground state and metastable excited states in the theoretical work in order to achieve suitable agreement with experiment.

We follow a similar approach here for the K-shell energy region. N 2+ ions produced in the SOLEIL synchrotron radiation experiments are not purely in their ground state. K-shell PI contributes to the ionization balance in a more complicated way than outer shell PI. In fact K-shell PI when followed by Auger decay couples three or more ionization stages instead of two in the usual equations of ionization equilibrium [35].

The 1s → 2p photo-excitation process on the 1s 2 2s 2 2p 2 P o ground state of B-like nitrogen ion is, hν + N 2+ (1s 2 2s 2 2p 2 P o ) → N 2+ (1s2s 2 2p 2 [ 3 P, 1 D, 1 S] 2 S, 2 P, 2 D) which can decay to N 3+ (1s 2 2s 2 1 S) + e -(k 2 ), or N 3+ (1s 2 2s np 1

P • ) + e -(k 2 ),
where k 2 is the outgoing energy of the continuum electron with angular momentum . Experimental studies on this doubly ionized atomic nitrogen ion, in its ground state 1s 2 2s 2 2p 2 P o , are also hampered by metastable states present in the parent ion beam. In the present experimental studies performed at the SOLEIL radiation facility, N 2+ ions are produced in the gas-phase using an electron-cyclotron-resonance-ion-source (ECRIS) so the metastable state 1s 2 2s2p 2 4 P can be present in the parent ion beam.

For the 1s 2 2s2p 2 4 P metastable state, autoionization processes occurring by the 1s → 2p photo-excitation process are mainly;

hν + N 2+ (1s 2 2s2p 2 4 P) ↓ N 2+ [1s2s[ 1,3 S]2p 3 ( 4 S o , 2 D o , 2 P o )] 4 S o , 4 P o , 4 D o ↓ N 3+ (1s 2 2s np 3 P o ) + e -(k 2 ).
In the vicinity of the K-edge our current investigations appear to be the first time this B-like system has been studied experimentally for such an energy region.

State-of-the-art ab initio calculations for Auger innershell processes were first carried out on this B-like system by Petrini and de Araújo [35] using the R-matrix method [29] and followed a similar procedure to the work on K-shell studies for the Be-like B + ion [36]. Stoica and co-workers [37] noted that once the 1s-hole was created in the ions, by single PI, with simultaneous shake-up and shake-off processes, Auger decay populates directly excited states of the residual ions, which then produces UV lines. This work was further extended by Garcia and co-workers [38], using the optical potential method within the Breit-Pauli R-matrix formalism [29,30,34,39], for PI of the ground state only, along the nitrogen iso-nuclear sequence. Garcia and co-workers [38] pointed out that the earlier central field calculations [40][41][42] did not account for the strong autoionizing resonance features that dominate the cross sections near the K-edge.

In the present study we compare our theoretical cross section results from the multi-configuration Dirac-Fock (MCDF) and R-matrix methods with previous theoretical results [38,43,44] and with the current experimental measurements made at the SOLEIL synchrotron radiation facility. Detailed measurements of the K-shell single photon ionization cross sections for B-like nitrogen ions, were made in the 404-409 eV region (where strong peaks were observed) and in the photon energy range 439-442 eV. The results for resonance energies and Auger widths are compared with detailed theoretical predictions made using the MCDF [45], R-matrix [34] and the SCUNC empirical methods [46,47]. The theoretical predictions assist in the identification and characterization of the strong 1s → 2p and the weaker 1s → 3p resonances observed in the B-like nitrogen spectra. The current study gives absolute values (experimental and theoretical) for PI cross sections along with n = 2 and n = 3 inner-shell resonance energies, natural line widths and resonance strengths, for the situation of a photon interacting with the ground 1s 2 2s 2 2p 2 P o , and metastable 1s 2 2s2p 2 4 P states of the N 2+ ion.

In section 2 we briefly outline the experimental procedure used and section 3 presents the theoretical procedures used. Section 4 gives a discussion of our experimental and theoretical results. Finally in section 5 conclusions are drawn from the present study.

Experiment

Cross sections for PI of B-like atomic nitrogen ions were measured in the range where K-shell PI occurs. The experiment was performed at the multi-analysis ion apparatus set-up, permanently installed on branch A of the PLEIADES beam line [48,49] at SOLEIL, the French National Synchrotron Radiation Facility, located in Saint-Aubin, France. Further details of the experimental set-up were outlined in our previous publications on N + [5], N 3+ and N 4+ [6] therefore only a brief summary will be given here. The N 2+ ions are produced in a permanent magnet ECRIS. Collimated N 2+ ion-beam currents up to 160 nA were extracted from the ion source after biasing the ion source by +2 kV and then selected by mass per charge ratio using a dipole magnet selector. The ion beam was placed on the same axis as the photon beam by using electrostatic deflectors and einzel lenses to focus the beam. After the interaction region between the photon and the ion beams, another dipole magnet separates the primary beam and the beam of ions which have gained one (or several) charge(s) in the interaction, the so-called photo-ions. The primary ions are collected in a Faraday cup and the photon-ions are detected by micro-channel plates detector. The photon current is measured by a calibrated photodiode. Ions with the same charge as the photo-ions can also be produced by collisions between the primary ions and the residual gas or stripping on the walls in the interaction region. This background signal is subtracted by chopping the photon beam, collecting the data with and without photons for 20 s accumulation time.

For the absolute measurements of the PI cross sections, a -1000 V bias is applied on the 50 cm long interaction region and the data are collected with 30 meV photon energy steps. The overlap of the two beams and the density distributions of the interacted particles is determined in three dimensions by using three sets of scanning slits. The cross sections obtained have an estimated systematic uncertainty of 15%. In another spectroscopy mode, no bias is applied to the interaction region allowing the photon and ion beams to interact over about 1 m and to scan the photon energy with a finer step. In this mode, only relative cross sections can be measured. They are later normalized on the cross sections determined in the absolute mode assuming the area under the resonances to be the same. The energy and band width of the photon beam are calibrated separately using a gas cell and N 2 (1s → π * g v=0) PI lines, located at 400.87 eV [50] and Ar 2p -1 3/2 4s at 244.39 eV [51].

The photon energy, once corrected for Doppler shift, has an uncertainty of approximately 30 meV. The relative uncertainty on the band passes is of the order of 10%. Outstanding possibilities in terms of spectral resolution and flux at the N 2 (1 s -1 ) K-edge have been discussed recently by Miron and co-workers [52,53].

Theory

SCUNC: B-like nitrogen

In the framework of the Screening Constant by Unit Nuclear Charge (SCUNC) formalism [46,47], the total energy of the core-excited states is expressed in the form given by,

E N n ; 2S+1 L π = -Z 2 1 N 2 + 1 n 2 1 -β(N n ; 2S+1 L π ; Z) 2 (1)
where E(N n ; 2S+1 L π ) is in Rydberg units. In this equation, the principal quantum numbers N and n are respectively for the inner and the outer electron of the He-like iso-electronic series. The β-parameters are screening constants by unit nuclear charge expanded in inverse powers of Z and are given by the expression,

β(N n ; 2S+1 L π ) = k=1 f k 1 Z k (2)
where f k N n ; 2S+1 L π are parameters that are evaluated empirically from existing experimental measurements on resonance energies. Similarly one may get the Auger widths in Rydbergs (1 Rydberg = 13.605698 eV) from the formula

(Ry) = Z 2 1 - q f q 1 Z q 2 . ( 3
)
The ALS experimental measurements of Schlachter and coworkers on K-shell PI of B-like carbon ions [21] were used to determine the appropriate empirical parameters f q for the Auger widths.

MCDF: B-like nitrogen

MCDF calculations were performed based on a full intermediate coupling regime in a j j-basis using the code developed by Bruneau [45]. Photoexcitation cross sections have been carried out for B-like atomic nitrogen ions in the region of the K-edge. Only electric dipole transitions have been computed using the length form. For this B-like atomic nitrogen ion the following initial configurations were considered: 1s 2 2s 2 2p and 1s 2 2s 2p 2 . The bar over the orbital indicates that they are different for the initial and final states. In order to describe the correlation and relaxation effects, multiple orbitals with the same quantum number and the following final configurations were considered: 1s2s 2 2p 2 , 1s2s 2 2pnp, 1s2s2p 3 , 1s2s2p 2 np. Such notation means that radial functions with principal quantum number n = 1 or 2 are not the same for initial and final configurations. Radial functions with principal quantum number up to n = 6 were included in our calculations. Photoexcitation cross sections from the 1s 2 2s 2 2p and 1s 2 2s2p 2 configurations were calculated separately. The wavefunctions were calculated minimizing the following energy functional:

E = α (2J α + 1)E α 2 α (2J α + 1) + β (2J β + 1)E β 2 β (2J β + 1) (4)
where α and β run over all the initial and final states, respectively. Synthetic spectra were constructed as a weighted sum of the photoexcitation cross sections from the ground state levels 1s 2 2s 2 2p( 2 P 1/2,3/2 ) and the metastable state levels 1s 2 2s2p 2 ( 4 P • 1/2,3/2,5/2 ). In order to compare with the SOLEIL experiments the weights used were 80% of the 2 P • levels, and 20% for the 4 P. Each electric dipole transition has been dressed by a Lorenztian profile with a full width half maximum (FWHM) given by the autoionization width of the upper level. The autoionization widths were evaluated from our MCDF calculations and for the photo-excited configurations 1s2s 2 2p 2 and 1s2s2p 3 . The average Auger widths were found respectively to be 64 and 105 meV.

R-matrix: B-like nitrogen

The R-matrix method [34], implemented in a parallel version of the codes [54][55][56] was used to determine all the cross sections. The PI cross sections were carried out for the initial 2 P o ground state and the 4 P metastable states. All the PI cross section calculations were performed in LS-coupling with 390-levels of the residual ion retained in the close-coupling expansion. The 1s, 2s and 2p tabulated orbitals from the work of Clementi and Roetti [57] were used together with n = 3 physical and n = 4 pseudo-orbitals of the N 3+ residual ion. The n = 4 pseudo-orbitals were energy optimized on the appropriate 1s -1 hole-shell states [58], employing the atomic structure code CIV3 [59]. These n = 4 so-called pseudo-orbitals are incorporated into the scattering basis set to try and accommodate for core relaxation and electroncorrelation effects in the multi-reference-configuration interaction (MRCI) wavefunctions used to describe the atomic ion states. All the N 3+ residual ion states were then represented by using MRCI wave functions. The non-relativistic R-matrix approach was used to calculate the energies of the N 2+ bound states and the subsequent PI cross sections.

The R-matrix with pseudo-states method (RMPS) was used to determine all the cross sections (in LS -coupling) with 390 levels of the N 3+ residual ion included in the closecoupling calculations. Since metastable states are present in the parent ion beam, theoretical PI cross section calculations are required for both the 1s 2 2s 2 2p 2 P o ground state and the 1s 2 2s2p 2 4 P metastable states of the N 2+ ion for a proper comparison with experiment.

The scattering wavefunctions were generated by allowing two-electron promotions out of selected base configurations of N 2+ . Scattering calculations were performed with twenty continuum functions and a boundary radius of 9.4 Bohr radii. For the 2 P o ground state and the 4 P metastable states the electron-ion collision problem was solved with a fine energy mesh of 2×10 -7 Rydbergs (≈ 2.72 µeV) to delineate all the resonance features in the PI cross sections. Radiation and Auger damping were also included in the cross section calculations.

For a direct comparison with the SOLEIL results, the R-matrix cross section calculations were convoluted with a Gaussian function of appropriate width and an admixture of 80% ground and 20% metastable states used to best simulate experiment.

The peaks found in the theoretical PI cross section spectrum were fitted to Fano profiles for overlapping resonances [60][61][62] as opposed to the energy derivative of the eigenphase sum method [63][64][65]. The theoretical values for the natural linewidths determined from this procedure are presented in tables 1 and 2 and compared with results obtained from the high-resolution SOLEIL synchrotron measurements and with other theoretical methods.

R-matrix calculations were also performed using an n = 2 basis (1s, 2s and 2p) comparable to the work of Garcia and co-workers [38] for the 4 P metastable. The appropriate resonance parameters (for the 4 P metastable and the 2 P o ground state similar to the work Garcia and co-workers [38]) are included in tables 1 and 2 from these calculations for completeness.

Results and discussion

Over the two photon energy ranges investigated we found in the cross sections that the most intense resonance peaks were located in the photon energy region, 404-409 eV. All the experimental measurements are shown in figure 1, taken at the various spectral photon energy resolutions, ranging from 30 to 90 meV at FWHM. We note that since the 4 P metastable state of the N 2+ ion is present in the parent experimental beam, theory may be used to estimate its content. Figure 2 shows our experimental and theoretical (MCDF and R-matrix) results for the restrictive photon energy range of 404-409 eV. From our theoretical R-matrix studies on this system we find that an admixture of 80% ground state and 20% metastable state in the parent ion beam indicates satisfactory agreement between theory and experiment. This admixture was determined by comparing the area under the strong peaks between experiment and theory. For a direct comparison with the experimental work both the MCDF and R-matrix theoretical cross sections were convoluted with a Gaussian distribution function having a profile width at FWHM of 60 meV and the above admixture used to simulate the SOLEIL measurements and the results presented in figure 2. Figure 2 illustrates the experimental and theoretical (MCDF and R-matrix) single photonionization cross sections results over the photon energy range from 404-409 eV at the energy resolution of 60 meV. Tables 1 and2 gives the experimental and theoretical estimates for all of the resonance parameters of the peaks located in the photon energy range 404-409 eV. From table 2, one sees that the MCDF predictions for the 2 S resonance energy is at 410.1 eV, outside of the energy range investigated. The peaks found in the PI cross section spectra measured at SOLEIL for this B-like atomic nitrogen were fitted to Voigt functions. Voigt profiles were used to fit the peaks in the experimental data, in order to extract the line widths, with a Gaussian instrumental profile of 60 meV assumed for each peak. For the determination of the experimental widths, each individual sweep has been fitted separately by Voigt profiles to avoid any possible shift in the energy delivered by the monochromator. Then the final width of the Lorentzian component was obtained from the weighted mean of the individual Lorentzian width determined from each fit. For the N 2+ ion, splitting of the J components of the initial state [66] was also taken into account, assuming a statistical distribution of the levels. The experimental uncertainties in the tables are the total uncertainties.

The MCDF and R-matrix cross sections results indicate suitable agreement with experiment from matching the calculated and experimental ionization thresholds. Estimates for the resonance parameters, for the N 2+ doublet and quartet states, made using the SCUNC empirical fitting approach [46,47] can be seen to be in satisfactory agreement with the more sophisticated MCDF and R-matrix theoretical methods and experiment (cf tables 1 and 2).

For the peaks found in the cross sections in the photon energy range 404-409 eV, the resonance energies and linewidths (cf tables 1 and 2) all indicate suitable agreement between theory and the available experimental measurements. The MCDF (dashed line), R-matrix RMPS (solid line, red, 4 P, green, 2 P o ), and the optical potential (dash-dot line, 2 P o only) [38] calculations shown were obtained by convolution with a Gaussian distribution having a profile width at FWHM of 60 meV and a weighting of the ground and metastable states (see text for details) to simulate the measurements. Tables 1 and2 give the designation and the parameters of the resonances 1 -6 in this photon energy region.

be much weaker than predicted. Tentatively there appears some experimental indication of this weak peak around about 405.2 eV but higher statistics than that used in the present experiment would be required to resolve this feature. Similarly, missing from the present experimental measurements is the 4 P • peak (cf tables 1), which is observed weakly, in analogous ALS experiments on B-like carbon (C + ) by Schlachter and co-workers [21]. For the N 2+ ion our theoretical predictions indicate that the 1s 2 2s2p 2 4 P → 1s2s[ 3 S]2p 3 ( 2 P) 4 P o resonance energy is located at an energy of about 407.6 eV which is outside of the energy region scanned in the present experimental measurements. For B-like nitrogen, from tables 1 and 2, we note that peak 1 (1s2s[ 3 S]2p 3 4 D o ) and 5 (1s2s 2 2p 2 2 P) have comparable

Auger widths of about 50 meV. Peaks 4 (1s2s 2 2p 2 2 D) and 6 (1s2s 2 2p 2 2 S) also have similar widths but are about a factor of two larger being around 120 meV. On physical grounds we attribute this difference due to different branching ratios and decay mechanisms to the ground and excited states of the residual N 3+ ion. An analogy may be drawn with the decay The MCDF (dashed line, blue), R-matrix RMPS (solid line, red 4 P, green 2 P o ) and the optical potential (dash-dot line, 2 P o only) [38] calculations shown were obtained by convolution with a Gaussian distribution having a profile width at FWHM of 67 meV and a weighting of the ground and metastable states (see text for details) to simulate the measurements. mechanisms in higher charged B-like ions, from the recent saddle-point calculations of Sun and co-workers [67] where similar trends were found.

Due to dipole selection and Hund rules, angular momentum coupling consideration give two possible 4 S o states from the 1s2s[ 1,3 S]2p 3 ( 4 S o ) states as opposed to a single one from the 1s2s[ 3 S]2p 3 ( 2 D o ) 4 D o state. Highly correlated R-matrix [68] and saddle-point [69] calculations on B-like carbon confirm the earlier findings on the ordering of the quartet states by Schlachter and co-workers [21]. For the present B-like system our RMPS calculations give a similar ordering of the quartet states presented in table 1 as in C + [21,68,69]. Figure 2, illustrates all of the N 2+ doublet states predicted by theory, which are observed in this spectral range. Theoretical predictions from the MCDF and R-matrix methods show suitable agreement with experiment, apart from the location of the 2 S resonance in the MCDF case. For the doublet states, the R-matrix results indicate better agreement with experiment (for cross sections and resonance parameters cf tables 2) than other theoretical methods.

Figure 3 shows the photon energy range in the vicinity of the 1s 2 2s 2 2p 2 P o → 1s2s 2 2p[ 3 P]3p 2 D transition which occurs from the ground state of the N 2+ ion. The SOLEIL measurements in this energy region were taken with a spectral resolution of 67 meV. To compare directly with experiment the MCDF and R-matrix calculations were convoluted with a Gaussian distribution function having a profile width at FWHM of 67 meV and weighted 80% for the ground state and 20% for the metastable. From figure 3 it is seen that the resonance line intensities, from the present RMPS, Rmatrix and MCDF calculations compare more favourably with the SOLEIL measurements. The experimental energy of this resonance was found at 440.46 ± 0.04 eV with an Auger width of 90 ± 47 meV and a strength of 3.05 ± 0.83 Mb eV. For this same resonance, R-matrix predictions give an energy of 440.441 eV, a width of 90 meV and a strength of 2.79 Mb eV. MCDF predictions give an energy of 440.652 eV with a strength of 2.91 Mb eV, while SCUNC estimates give values of 440.580 eV and 73 meV respectively for the energy and the width. It is seen that the present theoretical estimates for this resonance (position and width) are in harmony with the experimental findings. On physical grounds we note that the RMPS approach provide additional electron-correlation which greatly improves the residual ion target structure, the ionization potential of the system, resonance energy positions and Auger widths found in the PI cross sections compared to other approaches as can be seen from the results presented in tables 1 and 2.

We note that XMM-Newton x-ray spectral observations of WR 1 stars indicate the NIV (N 3+ ) K-edge absorption edge feature is at about 460 eV (26.954 Å) [12]. For the K-edge a value of 459 eV (27.011 Å) was quoted by Daltabuit and Cox [70], obtained from the screening constant method of Slater [71,72], which tends to overestimate the threshold as indicated by Bethe and Salpeter [73]. We point out that similar wavelength λ differences ( λ ∼ 0.06 Å) occur between K-edge experiment, observation and theoretical estimates for atomic oxygen and its ions [28,74]. Modern methods such as the saddle-point with complex rotation [75] give a value of 457.445 eV (27.1036 Å), relative to the NIII (N 2+ ) ground state for this 1s2s 2 2p 3 P o hole state of NIV (N 3+ ) in close agreement with our R-matrix estimate of 457.441 eV (27.1038 Å). Similarly for the NIV (N 3+ ) 1s2s 2 2p 1 P o hole state, an estimate of 461.416 eV (27.870 Å) from the saddle-point method [76], is in close agreement with our R-matrix values of, 461.359 eV (26.874 Å, n = 3 basis), 461.422 eV (26.868 Å, n = 4 basis), and 461.317 eV (26.876 Å, n = 2 basis), from the earlier work of Garcia and co-workers [38].

Conclusions

K-shell photoionization of B-like nitrogen ions, N 2+ , has been investigated using state-of-the-art experimental and theoretical methods. High-resolution spectroscopy was able to be achieved with E/ E = 13,500 at the SOLEIL synchrotron radiation facility, in Saint-Aubin, France, covering the photon energy ranges; 404-409 eV and 439-442 eV. Several strong peaks are found in the cross sections in the energy regions, 404-409 eV and 439-442 eV which are identified as the 1s → 2p and 1s → 3p transitions in the N 2+ K-shell spectrum and assigned spectroscopically with their resonance parameters tabulated in tables 1 and 2. For the observed peaks, suitable agreement is found between the present theoretical and experimental results both on the photon-energy scale and on the absolute cross section scale for this prototype B-like system. The strength of the present study is the high resolution of the spectra along with theoretical predictions made using state-of-the-art MCDF, R-matrix with pseudo-states and empirical methods. The present results have been compared with high-resolution experimental measurements made at the SOLEIL synchrotron radiation facility and with other theoretical methods so would be suitable to be incorporated into astrophysical modelling codes like CLOUDY [77,78], XSTAR [79] and AtomDB [80] used to numerically simulate the thermal and ionization structure of ionized astrophysical nebulae.

  Photoionisation en Couche de Valence d'Ions Atomiques et de Petits Ions

Figure 1 :

 1 Figure 1: Variation des sections efficaces de photoionisations expérimentales et calculées en fonction de l'énergie des photons. Les mesures ont été réalisées avec une bande passante de 300 meV.

Figure 2 :

 2 Figure 2: Sections efficaces expérimental de photoionisation en couche K pour des ions N 2+ mesurées avec différentes bandes passantes dans la région d'énergie de photon 404 eV -409 eV. (a) Section absolue prise avec une bande passante de 90 meV FWHM, (b) Sections efficaces obtenues en mode relatif, normalisées sur la section absolue (a). Pour le scan (a), les barres d'erreur donnent l'incertitude totale, et l'incertitude statistique pour les spectres (b) et (c).

Figure 3 :

 3 Figure 3 : Variation des sections efficaces de photoionisation pour les ions N 3+ mesurés avec une bande passante de 111 meV, où les cercles pleins et les triangles représentent la photoionisation expérimentale. Les sections efficaces calculées avec les codes MCDF (ligne solide verte) et R-matrice (ligne rouge continue) ont été convoluées avec un profil Gaussien de 111 meV FWHM, après pondération appropriée par les populations de l'état fondamental et des états métastables pour simuler les mesures.

Figure 4 :

 4 Figure 4: Variation des sections efficaces de photoionisation en fonction de l'énergie des photons pour l'ion N 4+ mesurée avec une bande passante de 125 meV FWHM. Cercles pleins: sections efficaces absolues de photoionisation. Les barres d'erreur donnent l'incertitude totale des données expérimentales. R-matrice (ligne rouge continue), MCDF (ligne verte continue). Les calculs sont convolués avec un profil Gaussien de 125 meV FWHM pour simuler les mesures.

3 -

 3 Photoionisation des ions N2 + et CO2 + Les figures 7 et 8 montrent les résultats préliminaires obtenus sur les ions moléculaires N et CO ,. La figure 7 présente la section efficace mesurée pour le cation N2 + utilisant à la fois MAIA (panneau du bas) et MICRA (panneau supérieur). En raison du faible rapport signal sur bruit dans les deux expériences, les mesures n'ont pu être effectuées qu'en utilisant le rayonnement blanc émis par les onduleurs à l'ordre zéro du monochromateur, Les onduleurs des lignes PLEIADES et DESIRS ont le même nombre de périodes, délivrant une bande

Figure 7 :

 7 Figure 7: Variation de la section efficace de photoionisation en fonction de l'énergie des photons pour l'ion moléculaire N mesurée avec le piège FTICR (panneau supérieur) et le montage expérimental MAIA en faisceaux colinéaires (panneau inférieur).

Figure 8 : 4 -

 84 Figure 8: Variation de la section efficace de photoionisation en fonction de l'énergie des photons pour l'ion moléculaire CO2 + mesurée avec le piège à ions FTICR.

Finally

  , I would like to thank my wife "Huda" for her understanding and love during the past few years. Her support, motivation and encouragement were at the end what made this dissertation possible and to my lovely little girl" Shadn" who changed my vision for the life and gave it that unique taste. xxiv xxv

Figure 1

 1 Figure 1 Partitioning of configuration space into an internal and an external region by a

Figure 1 .

 1 Figure1. Scheme of the first merged beam setup. The ions produced in the source S were deflected by magnetic field M1 and collimated at C. In the biased cylinder I, the ions were merged with a VUV synchrotron radiation beam chopped by a rotating chopper wheel (L). The parent ions were collected in a Faraday cup C1 while the doubly charged ions produced in the interaction with the photons (the so-called photoions) were counted by the detector D. The ion beams were separated by a magnetic dipole (M2) and the kinetic energy of the photoions analyzed by an electrostatic deflector (E) fields. The photon flux was monitored by a calibrated diode (PD). The inset shows the collimator C and the biased cylinder on a larger scale[2].

Figure 4 :

 4 Figure 4: Schematic layout of the SOLEIL beamlines.

Figure 6 :

 6 Figure 6: Variation as a function of photon energy of flux measured on the MAIA branch for the different gratings and linear polarization.

10

 10

  Photon energy (eV)

Figure 7 :

 7 Figure 7: Scheme of the MAIA set-up on the PLEIADES beam line. The ionic optics used for the ion beam transport is represented in red. ECRIS: electron cyclotron resonance ion source, E: einzel lens, St: set of horizontal and vertical steerers, S: collimating slits, FC: Faraday cup, ED: electrostatic deflector, IR: interaction region, EQ: electrostatic quadrupole, SR: synchrotron radiation, PD: photodiode.

Figure 8 .

 8 Figure 8. : Scheme of the ECR ion source and the extraction optics installed on MAIA setup.

Figure 9 :

 9 Figure 9: Mass scan of ions extracted from the ECRIS recorded by scanning the current in the coils of MAGNET 1. The ion source was optimized for the production of N 2+ .

Figure 11 .

 11 Figure 11. Example of measured profiles for the photon beam (red) and ion beam (blue) in the horizontal plane (left) and in the vertical plane (right).

  Photon Flux 3.0 10 11 s -1 5%

3. 1

 1 Photoionization of carbon-like N + ion 3.1.1 Introduction For this ion, the single and double photoionization cross sections have been measured in the 398-406 eV regions. In parallel, Multi-Configurational Dirac-Fock (MCDF) and Rmatrix calculations have been realized for the identification and characterization of the resonances observed in the spectra.

Figure 1 :

 1 Figure 1: Variation of the experimental and calculated photoionization cross sections as a function of photon energy. The measurements were performed with 300 meV band pass.

Figure 2

 2 Figure 2 displays the same results obtained with a better energy resolution of 60 meV. To

Figure 2 :

 2 Figure 2: Variation of the experimental and calculated photoionization cross sections as a function of photon energy. The measurements were performed with 60 meV band pass.

Figure 3 :

 3 Figure 3: Experimental K -shell photoionization cross sections for N 2+ ions measured with various band passes over the photon energy region 404 eV -409 eV. (a) absolute cross section taken with a band pass of 90 meV FWHM, (b) Cross sections obtained in the relative mode then normalized to the scan (a). For scan (a), the error bars give the total uncertainty, and the statistical uncertainty for scans (b) and (c).

Figures 4 and 5

 5 Figures 4 and 5 compares our experimental cross section obtained with 60meV band width

Figure 4 :

 4 Figure 4: Photoionization cross sections for N 2+ ions measured with a 60 meV bandpass. Solid circles: total photoionization. The error bars give the statistical uncertainty on the experimental data. The MCDF (dashed line) and R-matrix (solid line, red, 4 P, green, 2 P o ) calculations shown were obtained by convolution with a Gaussian distribution having a profile width at FWHM of 60 meV and a weighting of the ground and metastable states (see text for details) to simulate the measurements. Tables 4 give the designation and the parameters of the resonances 1 -6 in this photon energy region.

Figure 5

 5 Figure 5 displays the variation of the measured and calculated absolute photoionization

Figure 5 :

 5 Figure 5: Photoionization cross sections for N 2+ ions measured with a 67 meV FWHM band pass. Solid circles: total cross sections. The error bars give the statistical uncertainty of the experimental data. The MCDF (dashed line, blue) and R-matrix (solid line, red 4 P, green 2 P o ) RMPS calculations shown were obtained by convolution with a Gaussian distribution having a profile width at FWHM of 67 meV and a weighting of the ground and metastable states (see text for details) to simulate the measurements.

Figure 7 :

 7 Figure 7: Photoionization cross sections for N 2+ ions measured with 275 meV FWHM band pass. Solid circles: total cross sections. The error bars give the uncertainty of the experimental data. R-matrix (solid line (green), 20% 4 P and 80% 2 P o ) RMPS calculations shown were obtained by convolution with a Gaussian distribution having a profile width of 275 meV FWHM. The relative cross section measurements have been normalized to the Rmatrix results at 457 eV.

Figure 8

 8 Figure 8 Variation of the photoionization cross sections for Be-like atomic nitrogen (N 3+ ) ions measured with a 111 meV band pass, where the solid circles represent the total photoionization measured in the relative mode. The error bars represent the statistical uncertainty; the absolute measurements are given by the open triangles. In this case, the error bars give the total uncertainty on the experimental data. The MCDF (solid green line)and R-matrix (solid red line) calculations shown were convoluted with a Gaussian profile of 111 meV FWHM and an appropriate weighting of the ground and metastable states to simulate the measurements. For the metastable 3 P 0 term, the MCDF calculations have been shifted up by +1.46 eV in order to match experiment.

Figure 9

 9 Figure 9 Variation of the photoionization cross sections for Be-like atomic nitrogen (N 3+ ) ions measured with a 56 meV band pass at the SOLEIL radiation facility. Solid circles: total photoionization. The error bars give the statistical uncertainty of the experimental data. The MCDF (solid green line) and R-matrix (solid red line) calculations shown were convoluted with a Gaussian profile of 56 meV FWHM and an appropriate weighting of the ground and metastable states to simulate the measurements. For the metastable 3 P 0 term, the MCDF calculations have been shifted up by +1.46 eV in order to match experiment.

Figure 10 :

 10 Figure 10: Variation of the photoionization cross sections for Be-like atomic nitrogen (N 3+ ) ions measured with a 38 meV band pass. Solid circles: Experiment. The error bars give the statistical uncertainty of the experimental data. The MCDF (solid green line) and R-matrix (solid red line) calculations shown were convoluted with a Gaussian function of 38 meV FWHM and an appropriate weighting of the ground state and metastable state (see text for details) to simulate the measurements. For the metastable 3 P 0 term, the MCDF calculations have been shifted up by +1.46 eV in order to match experimental measurements.

Figure 11 :

 11 Figure 11: Variation of the photoionization cross sections for Be-like atomic nitrogen (N 3+ ) ions measured with a 133 meV band pass for the 1s→3p resonance. Solid circles: Experiment. The error bars give the statistical uncertainty of the experimental data. The MCDF (solid green line) and R-matrix (solid red line) calculations shown were convoluted with a Gaussian function of 133 meV FWHM and an appropriate weighting of the ground and metastable states to simulate the measurements.

Figure 12 :

 12 Figure 12: Variation of photoionization cross sections as a function of photon energy for Li-like atomic nitrogen (N 4+ ) measured with a 125 meV FWHM band pass. The pink solid circles indicate the absolute measurements, the error bars giving the statistical uncertainty on the experimental data; the solid red line shows the results of the R-matrix intermediate coupling calculations; the solid green line indicates those of the MCDF calculations. The theoretical calculations shown were convoluted with a Gaussian profile of 125 meV FWHM to simulate the measurements.

  measurements have been compared to state-of-art theoretical methods, including SCUNC, MCDF and R matrix, performed in collaboration with several groups of theoreticians.Using the merged beam set up MAIA on PLEIADES beam line, a high resolving power up to 13500 has been achieved, allowing measurement of absolute excitation energy, natural widths and oscillator strength of the most intense 1s → np (n=2,3) photoexcitations. Such measurements are essential to improve the results of theoretical codes, as demonstrated here by the excellent agreement found between theories and experiment. The energy of the resonances is calculated in general within 300 meV for the SCUNC results and 200 meV for the MCDF and R matrix values. Concerning the oscillator strengths, our MCDF calculations predict values in agreement with the experimental values within their uncertainties, except for the N 4+ 1s(2s2p 3 P) 2 P resonance calculated higher by 20%. The width of the resonances was measured between 36 and 132 meV, corresponding to lifetime of the K vacancy in the highly excited states between 18 and 5 fs, respectively. In addition to provide a stringent test of state-of-the-art theoretical calculations, our measurements are important also for interpreting spectra recorded by the x-ray satellites from astrophysical sources. Some inconsistencies with measurements of ions in the same isoelectronic series were also found. Chapter 4: Photoionization Studies Using a FTICR Ion Trap 4.1 IntroductionWe have seen in the chapter on nitrogen ions that one problem encountered using the merged beam setup is the contribution of ions in metastable states, making more difficult the comparison with theoretical data as the relative population in the various states is most of the time unknown. We have investigated one way to avoid this problem, to trap the ions to allow them to radiatively decay to the ground state. In this chapter, I will present the results I have obtained on the photoionization of atomic ions (Xe + and Kr + ) and molecular ion (N2 + , CO2 + ) using two different techniques, the merged beam technique and a FTICR (Fourier Transform Ion Cyclotron Resonance) ion trap. FTICR traps are largely used in mass spectrometry, for example for the study of molecular structure. The technique is based upon the motion of charged particles in a combination of electric and magnetic

Figure 1

 1 Figure 1 displays a typical ICR cubic cell. The magnetic field is produced by permanent or

Figure 3 .

 3 Figure 3. Typical time sequence used in a FTICR ion trap experiment.

Figure 4 :

 4 Figure 4: MICRA setup.

Figure 5

 5 Figure 5 gives as an example of the temporal sequence used in the Xe experiment. In a first

Figure 5 :

 5 Figure 5: Schematic representation of the Xe + ions preparation and irradiation timing procedure. The target ions are first produced by electron impact ionization of Xe atoms in the gas pulse and are subsequently photoionized by a pulse of monochromatic synchrotron radiation, with a delay of more than 1 s [1].

Figure 6 :

 6 Figure 6: Variation of the photoionization cross section as a function of photon energy measured with the merged beam setup for the Kr + ion on the upper panel and for Xe + ion on the lower panel. The solid black curve gives the results of MCDF calculations for the direct photoionization.

Figure 7 :

 7 Figure7: Variation of the photoionization cross section for Kr + ion near the threshold recorded with a mean bandpass of 30 meV. The long black vertical lines above the spectrum gives the position of the np 4 (n=4) ionization thresholds for the ions in the np5 2 P3/2 ground level and np5 2 P1/2 metastable level, and the short lines gives the position of np 4 1 D2 md Rydberg series.

Figure 8 :

 8 Figure8: Variation of photoionization cross section for Xe + ion near the threshold recorded with a mean bandpass of 25 meV. The long black vertical lines above the spectrum gives the position of the np 4 (n=5) ionization thresholds for the ions in the np5 2 P3/2 ground level and np5 2 P1/2 metastable level, and the short lines gives the position of np 4 1 D2 md Rydberg series.

Figure 10 :

 10 Figure 10: Recent measurements performed at ALS for the absolute single photoionization cross section of Kr + ions as a function of the photon energy compared to R-matrix results (red curve). The ALS measurements are at a nominal energy resolution of 7.5 meV and normalized to the ASTRID/SOLEIL measurements at 26.5025 eV [11].

Figure 11 :

 11 Figure 11: Comparison of the photoionization cross section of Kr + ion as a function of

Figure 12 :

 12 Figure 12: Comparison of the variation of the photoionization cross section of Xe + ion as a function of photon energy using the ASTRID merged beam set up (top) and FTICR set up middle panel. The bottom panel is the pure 2 P1/2 metastable state after subtracting the top and the middle panels.

  Photon Energy (eV)

dication [ 16 ]

 16 , indicated as the vertical green bars on the figure. This result has been obtained in the very last hour of a beam time on DESIRS beam line. They must be reproduced during our next beam time to improve the scattering of the data.

Figure 14 : 2 +

 142 Figure 14: Variation of the photoionization cross section as a function of photon energy for CO2 + molecular ion measured with the FT-ICR ion trap.

Figure 1

 1 Figure 1 shows a schematic of the ECRIS used on MAIA and a new three extraction

Figure 1 :

 1 Figure 1: Scheme of the ECR ion source used on the MAIA setup at SOLIEL and a new three electrodes extraction system.

  velocity and the z-components of the velocity, β=v/c, γ= 1/(1-β 2 ), where c is the speed of light, and M is the mass of the ion. The components of the normalized emittance in the x and y directions are defined by: In our set up, the divergence is limited by the size of the optics[3]. Several softwares, most of them solving the equation of motion for a group of ions in three dimensional space (Poisson's equations) and taking into account the presence of electromagnetic fields and a growing number of side effects, are available now, like CPO, AXCEL, LORENTZ3D, IGUN and Transport. In this work I have used the code IGUN to simulate the extraction of ion, then ECRopt code, this last one avoiding to solve Poisson's equations in the case of unipotential lenses. 5.1.2 Presentation of the Codes 5.1.2.1 IGUN IGUN is a program which simulates the plasma sheath for the extraction of positive ions. IGUN was originally written to calculate electron trajectories in electrostatic and magnetostatic fields with rectangular or cylindrical coordinates. Poisson's equation is solved by finite difference equations using boundary conditions defined by specifying the type and position of the boundary. Electric fields are determined by differentiating the potential distribution. The electron trajectory equations are fully relativistic and account for all possible electric and magnetic field components. Space charge forces are realized through appropriate deposition of charge on one cycle followed by another solution of Poisson's equation which is in turn followed by another cycle of trajectory calculations [4].

Figure 2 :

 2 Figure 2: Two extraction electrodes system, plasma electrode Ep (in blue) and Extraction electrode Eext (in red). The gap is set at their minimal value (18 mm). The dashed curve gives the intensity of the magnetic field along the z axis.

Figures 3 and 4 Figure 3 :

 43 Figures 3 and 4 show the variation of the ellipse emittance in π*cm*mrad and the

Figure 4 :

 4 Figure 4: The ellipse emittance and divergence (Maximum Angle) as a function of the gap between the Ep and EExt, for the Ar 5+ ion in the plane Z=95mm. The bias on the plasma electrode is fixed at 4KV.

Figures 5

 5 Figures 5 and 6 shows the trajectories of the Ar 5+ ions for the optimum gap distance at

Figure 5 :

 5 Figure 5: Ions trajectories for the optimum gap= 40 mm. The potential of the plasma

Figure 6 :

 6 Figure 6: Ion trajectories for the optimum gap= 45 mm. The potential of the plasma electrode is fixed at 4KV.

Figure 7 :

 7 Figure 7: Simulation of the trajectories of Ar 5 + ions extracted (at optimum gap) from the plasma at 2KV inside the einzel lens E1 and the dipole magnet. The paths in the horizontal plane are shown in red, in the vertical plane in blue. The black rectangles give the geometrical dimensions of the einzel lens E1 and magnet M1.

Figure 8 :Figure 9 :

 89 Figure 8: Simulation of the trajectories of Ar 5+ ions extracted (at optimum gap) from the plasma at 4KV inside the einzel lens E1 and the dipole magnet M1. The paths in the horizontal plane are shown in red, in the vertical plane in blue.

Figure 10 :Figure 11 :

 1011 Figure 10: Emittance of the Ar 5+ ion beam at the position of S2: in red is the emittance in the horizontal plane and in blue in the vertical plane. The Ar 5+ ions were extracted at 4KV.

Figure 12 :

 12 Figure 12: Emittance of the ions beam measured after the slit two, S2 in the vertical plane.

Figure 13

 13 Figure13shows this new extraction system. It is composed of the plasma electrode Ep , the

Figure 12 :

 12 Figure 12: Three extraction electrodes system. The gaps g1 and g2 between each electrode can be adjusted.

  Ions propagation axis (mm)Magnetic field strength (gauss) Radius (mm)

  to 17 show the variation of the maximum angle (Divergence), in mrad, as a function of the voltage applied on the intermediate electrode, with g1+g2 constant, at 2KV and 4KV voltages applied on the plasma electrode. The minimum divergences for 2KV and 4KV extraction voltage are found at 62.4 mrad and 42.5 mrad, respectively, for the same value of g1 which is equal to 25 mm. After moving the extraction electrode while fixing the interspacing g1, the best values of g2 value and divergence for 2KV and 4KV, respectively, are 6.96 mrad at g2=5mm and 6.97 mrad at g2= 25mm. The ion trajectories are shown on figures 18 and 19 for 2 KV and 4 KV extractions, respectively, for optimum gaps g1 and g2.

Figure 14 :

 14 Figure 14: Divergence (Maximum Angle) as a function of the voltage applied to Eext for different gaps g1, for Ar 5+ ion in the plane Z=75 mm. The bias of the plasma electrode is fixed at 2KV.

Figure 15 :

 15 Figure 15: Divergence (Maximum Angle) as a function of gap g2, for Ar 5+ ion in the plane Z=75 mm. The bias of the plasma electrode is fixed at 2KV voltage.

Figure 16 :

 16 Figure 16: Divergence (Maximum Angle) as a function a function of the voltage applied to Eext for different gaps g1, for Ar 5+ ion in the plane Z=75 mm. The bias of the plasma electrode is fixed at 4KV voltage.

Figure 17 :

 17 Figure 17: Divergence (Maximum Angle) as a function of gap g2, for Ar 5+ ion in the plane Z=75 mm. The bias of the plasma electrode is fixed at 4KV voltage.

Figure 18 :

 18 Figure 18: Trajectories of the ion beam of Ar 5+ at the optimum g1= 25 mm and g2=5 mm. The bias of the plasma electrode is fixed at 2KV.

Figure 19 :

 19 Figure 19: Trajectories of the ion beam of Ar 5+ at the optimum g1= 25 mm and g2=25 mm. The bias of the plasma electrode is fixed at 4KV.

Figure 20 :

 20 Figure 20: Ar 5+ beam trajectories inside the einzel lens E1 and the dipole magnet M1, in red in the horizontal plane, in blue in the vertical plane. The Ar 5+ ions were extracted at 2KV. Ions Propagation axis (m)

Figure 21 :

 21 Figure 21: Emittance of Ar 5+ ion beam at the position of slit S2: red in the horizontal plane, blue in the vertical plane. The Ar 5+ ions were extracted at 2KV.

Figure 22 :Figure 23 :

 2223 Figure 22: Ar 5+ beam profiles trajectories inside the einzel lens E1 and the dipole magnet M1, the ion trajectories in the horizontal plane is in red color and the vertical plane is in blue color. The Ar 5+ ions were extracted at 4KV. Ions Propagation axis (m)

Figure 1 .

 1 Photon energy (eV)

Figure 2 .

 2 Photon energy (eV)

Figure 3 .

 3 Photon energy (eV)

  Photon energy (eV)

Figure 4 .Figure 5 .

 45 Figure 4. Photoionization cross sections for Be-like atomic nitrogen (N 3+ ) ions measured with a 133 meV band pass at the SOLEIL radiation facility for the 1s → 3p resonance. Solid circles: total photoionization. The error bars give the statistical uncertainty of the experimental data. The MCDF (solid green line) and R-matrix (solid red line) calculations shown are convolution with a Gaussian profile of 133 meV FWHM and an appropriate weighting of the ground and metastable states (see text for details) to simulate the measurements.

Figure 1 .

 1 Figure 1. Scheme of the MAIA set-up on the PLEIADES beam line. The ionic optics used for the ion beam transport are represented in red. ECRIS: electron cyclotron ion source, E: einzel lens, St: set of horizontal and vertical steerers, S: collimating slits, FC: Faraday cup, ED: electrostatic deflector, IR: interaction region, EQ: electrostatic quadrupole, SR: synchrotron radiation, PD: photodiode.

Figure 1 .

 1 Photon energy (eV)

Figure 2 .

 2 Photon energy (eV)

Figure 3 .

 3 Photon energy (eV)

Table of Contents

 of

1 Chapter 1: Introduction

Table 1 :

 1 Some technical parameters for SOLEIL Synchrotron[11].

		Unit	Value
		GeV	2.75
	Energy spread σE	MeV	2.79
	Bunch length σz	mm	10.5

Table 1 :

 1 Configuration, level and energy relative to the ground state N

	Configuration Level Energy (eV)
	N + 2s 2 2p 2	3 P	0
	N + 2s 2 2p 2	3 P	0.00604
	N + 2s 2 2p 2	3 P	0.01622
	N + 2s 2 2p 2	1 D	1.89897
	N + 2s 2 2p 2	1 S	4.05289
	N + 2s2p 3	5 S	5.80055
	N 2+ 2s 2 2p ( 2 P ) limit 29.60128

+ (2s 2 2p 2 3 P ) of all the initial states contributing in this work

[1]

.

Table 2 :

 2 Resonance energies E

( )

Table 3 :

 3 Configurations, terms and energy relative to ground state N 2+ (2s 2 2p 2 2 P

	Configuration	Level	Energy (eV)
	N 2+ 2s 2 2p	2 P /	0
	N 2+ 2s 2 2p	2 P /	0.02162
	N 2+ 2s2p 2	4 P /	7.09030
	N 2+ 2s2p 2	4 P /	7.09770
	N 2+ 2s2p 2	4 P /	7.10775
	N 3+ 2s 2	( 1 S / ) limit	47.4453

/ ) energy

[1]

.

Table 4 :

 4 Comparison of the present experimental and theoretical results for the resonance energies E

(

)

  .65  0.04. Using the energy of the 1s 2 2s2p 3 P° metastable state of C 2+ at -

	987.9612 eV [22], we obtain from Eq. (5.b)
	f ( 3 D) = 2.5090  0.0006
	f ( 3 P) = 2.5128  0.0006
	f ( 3 S) = 2.5910  0.0006

2 3 S) states respectively at (in eV) 292.80  0.03, 292.98  0.03 and 296

Table 6 :

 6 Experimental and theoretical results for the resonance energies, natural line widths and the average photoionization cross sections for the K-shell photoexcitation within the photon energy range between 420 eV to 426 eV.

	Resonance			SOLIEL/Others	R-matrix	MCDF/Others
	(Label)			(Experiment a )	(Theory)	(Theory)
	1s 2 2s 2 S → 1s[2s2p 3 P] 2 P 0	(	)	421.472 ± 0.03 a	421.448 e	421.390 f
	(1)			421.521 ± 0.05 b	420.890 j	420.940 g
				421.228 ± 0.05 c		421.321 h
				421.120 ± 0.07 d		421.572 i
						421.169 k
						421.390 l
						421.842 m
	1s 2 2s 2 S →1s[2s2p 1 P] 2 P 0 (2)	Г σ PI		11 ± 8 47.3 ± 7.4	4 e 6 j	5 g 4 h 6 k
						6 l
						5 m
		(	)	425.449 ± 0.03 a	60 a	425.654 f
				425.624 ± 0.40 c	425.606 e	426.020 g
				424.890 ± 0.15 d	425.650 j	425.421 h
						425.770 i
						425.530 l
				___		425.666 m
		Г			42 e	16 g
					53 j	42 h
						43 l
						44 m
		σ PI			6.5 e	___

a SOLEIL, experimental work

[29]

. b EBIT measurements

[36]

. c Laser produced plasmas (LLP) measurements.

[37]

d Electron spectroscopy in ion-atom collisions measurements.

[38]

e R-matrix, intermediate-coupling (31 levels)

[29]

, level averaged. f MCDF method

[29]

, c MCDF

[39,40]

. g Saddle-point with complex-rotation method (SPM-CR)

[41]

. h intermediate-coupling, semi-relativistic method

[42]

. i R-matrix, intermediate coupling, results, level averaged

[6]

.

Table 2 :

 2 Energy, quantum defect and assignment of Rydberg series obtained using equation before for the Kr + ion.

	Energy (eV)	δ	Assignment
	20.195	0.16 2 P1/2 → 1 D2 6d
	20.594	0.25 2 P1/2 → 1 D2 7d
	20.876	0.28 2 P1/2 → 1 D2 8d
	21.062	0.34 2 P1/2 → 1 D2 9d
	21.210	0.30 2 P1/2 → 1 D2 10d
	21.425	0.29 2 P3/2 → 1 D2 6d
	21.886	0.29 2 P3/2 → 1 D2 7d
	22.177	0.30 2 P3/2 → 1 D2 8d
	22.375	0.31 2 P3/2 → 1 D2 9d
	22.510	0.35 2 P3/2 → 1 D2 10d
	22.610	0.41 2 P3/2 → 1 D2 11d
	22.683	0.51 2 P3/2 → 1 D2 12d
	22.747	0.49 2 P3/2 → 1 D2 13d

Table 3 :

 3 Energy, quantum defect and assignment of Rydberg series obtained using equation before for the Xe + ion.

Table 1 :

 1 Summary of the present simulations for the Ar 5+ ions extraction from ECRIS at 2 KV and 4 KV and transportation through the first dipole magnet M1.

	Two electrodes	Three electrode
	system	systems
	2KV	4KV	2KV	4KV

Table 1 .

 1 Experimental parameters used for evaluating the absolute cross section for N 3+ ions measured at a photon energy of 412.5 eV.

	Signal	340 Hz
	Noise	250 Hz
	Velocity Photon flux Ion current	2.0 × 10 5 ms -1 3.0 × 10 11 s -1 100 nA
	Detector efficiency 0.36
	Form factor	59

Table 1 .

 1 Typical values for the experimental parameters involved in evaluating the absolute cross section measured at a photon energy of 400 eV.

	S	30 Hz
	Noise	17 Hz
	ν	1.6 10 5 m s -1
	Photon flux 4.1 10 11 s -1
	J	38 nA
		0.62
	F xy	30

Table 2 .

 2 Resonance energies E

  are included for comparison purposes.

	Resonance		SOLEIL	R-matrix	MCDF
	(label)		(experiment † )	(theory)	(theory)
	1s 2 2s 2 2p 2 3 P → 1s2s 2 2p 3 3 D o E (res) ph A	-	399.706 a	399.894 b 394.665 c
			-	124 a	78 b
					207 c
	1s 2 2s2p 3 5 S o → 1s2s2p 4 5 P B	E (res) ph	-	399.891 a	400.834 b 401.568 c
			-	6 2 a	104 b
					43 c
	Average A + B	E (res) ph	399.844 ± 0.04 †	399.799 d	400.364 e
	1s 2 2s 2 2p 2 1 S → 1s2s 2 2p 3 1 P o C	E (res) ph	138 ± 41 † -	93 d 400.159 a	91 e 400.583 b 400.146 c
			-	115 a	106 b
					164 c
	1s 2 2s 2 2p 2 3 P → 1s2s 2 2p 3 3 S o D	E (res) ph	-	400.579 a	399.999 b 396.13 c
			-	7 8 a	64 b
					122 c
	1s 2 2s 2 2p 2 1 D → 1s2s 2 2p 3 1 D o E (res) ph E	-	400.681 a	400.031 b 397.859 c
			-	105 a	108 b
					170 c
	Average D + E	E (res) ph	400.633 ± 0.04 †	400.630 d	400.015 e
	1s 2 2s 2 2p 2 3 P → 1s2s 2 2p 3 3 P o F	E (res) ph	86 ± 10 † 401.346 ± 0.04 †	92 d 401.347 a	86 e 402.342 b 396.430 c
			143 ± 21 †	121 a	110 b 200 c
	1s 2 2s 2 2p 2 1 D → 1s2s 2 2p 3 1 P o E (res) ph G	-	402.320 a	402.503 b 399.619 c
			-	132 a	107 b
					164

c † SOLEIL, experimental lines are not pure, see text for discussion. a R-matrix LS-coupling, present work. b MCDF, present work. c MCDF, Chen and co-workers [45]. d Average R-matrix, present work. e Average MCDF, present work.

  The measured double-PI cross section is only 4% of the single PI; these data are Photoionization cross sections measured with a 300 meV band pass. Open circles: single photoionization. The absolute measurements (solid black triangles); single photoionization have been obtained with a larger energy step. Open blue triangles: double photoionization. The error bars give the total uncertainty of the experimental data. The MCDF (dashed line) and R-matrix (solid line) calculations shown were obtained by convolution with a Gaussian profile of 300 meV FWHM and a weighting of the states (see the text for details) to simulate the measurements.

		30
	Cross Section (Mb)	5 10 15 20 25	N	+				EXPT (SOLEIL) Double Absolute MCDF R-matrix ΔE=300 meV
		0
		398		399	400	401	402	403	404	405	406
						Photon energy (eV)
	Figure 2.

  Comparison of the single-photoionization cross section (yellow circles) measured with 60 meV band pass (points with error bar giving the statistical uncertainty). The MCDF (dashed line) and R-matrix (solid line) calculations shown were obtained by convolution with a Gaussian profile of 60 meV FWMH and a weighting of the states (see the text for details) to simulate the measurements. Table2gives designation of resonances A-G, and the 1s2s 2 2p 3 1 D o resonance E, caused by photo-excitation of the 1s 2 2s 2 2p 2 1 D metastable state. The third peak at about 401.35 eV is due to resonance F, 1s2s 2 2p 3 3 P o , resulting from photo-excitation of the 1s 2 2s 2 2p 2 3 P ground state. Finally, a broad resonance peak G, at about 402.3 eV (barely visible in the experimental data, figure2), resulting from the photoexcitation of the 1s 2 2s 2 2p 2 1 D metastable is assigned to the 1s2s 2 2p 3 1 P o resonance state. These assignments are summarized in the first column of table 2.

		50
		40	2		N	+		R-matrix EXPT (SOLEIL) MCDF
	Cross Section (Mb)	10 20 30	1	3	4	+	5	6	ΔE=60 meV 7
		0
		399	399.5	400	400.5	401	401.5	402	402.5	403
					Photon energy (eV)
	Figure 3.
	2 3 P

Table 1 .

 1 B-like atomic nitrogen ions, quartet core-excited states. Comparison of the present experimental and theoretical results for the resonance energies E (res) ph (eV), natural linewidths (meV) and resonance strengths σ PI (in Mb eV), for the dominant core photoexcited n = 2 states of the N 2+ ion, in the photon energy region 404 to 409 eV with previous investigations.

	Resonance (Label)	SOLEIL (Experiment a )	R-matrix MCDF/Others (Theory) (Theory)
	1s2s[ 3 S]2p 3 4 D • E (res) ph 1	404.794 ± 0.03 a 404.776 b 404.784 c 404.623 f 405.630 d 404.930 e
		36 ± 19	62 b 68 f	31 d 56 e
	σ PI	3.21 ± 0.74 2.53 b	-
	1s2s[ 3 S]2p 3 4 S • E (res) ph 2	-	4 0 5 . 2 3 4 b 405.780 c 405.553 f 404.354 d 405.510 e
		-	1 5 b	-
			16 f	12 d 11 e
	σ PI	-	2 . 1 9 b	-
	1s2s[ 3 S]2p 3 4 P o E (res) ph 3	-	4 0 7 . 5 8 4 b 408.543 c 408.273 f 408.411 d 407.580 e
		-	4 8 b	-
			51 f	24 d 15 e
	σ PI	-	1 . 3 5 b	-

a SOLEIL, experimental work. b R-matrix, n = 4 basis, LS-coupling, present work. c MCDF, present work d MCDF, Chen and co-workers [43, 44]. e Screening Constant by Unit Nuclear Charge (SCUNC) approximation, present work. f R-matrix, n = 2 basis, intermediate coupling, level averaged.

Table 2 .

 2 B-like atomic nitrogen ions, doublet core-excited states. Comparison of the present experimental and theoretical results for the resonance energies E (res) ph (eV), natural linewidths (meV) and resonance strengths σ PI (in Mb eV), for the dominant core photoexcited n = 2 states of the N 2+ ion, in the photon energy region 404 to 409 eV with previous investigations.

	Resonance (Label)	SOLEIL (Experiment a )	R-matrix MCDF/Others (Theory) (Theory)
	1s2s 2 2p 2 2 D E (res) ph 4	405.814 ± 0.03 a 405.703 b 405.890 c 405.965 f 406.128 d
			-	4 0 5 . 9 8 0 d
			-
		122 ± 19	122 b 109 f	55 d 123 e
	σ PI	10.1 ± 1.9	10.44 b	-
	1s2s 2 2p 2 2 P E (res) ph 5	406.547 ± 0.03 a 406.656 b 406.380 c 406.387 f 406.404 d 406.561 e
		58 ± 7	62 b 43 f	25 d 66 e
	σ PI	16.80 ± 2.7	19.32 b	-
	1s2s 2 2p 2 2 S E (res) ph 6	408.376 ± 0.03 a 408.344 b 410.085 c 408.297 f 408.087 d 408.414 e
				-
		120 ± 60	106 b 94 f	25 d 132 e
	σ PI	1.18 ± 0.47 1.87 b	-

a SOLEIL, experimental work. b R-matrix, n = 4 basis, LS-coupling, present work. c MCDF, present work d MCDF, Chen and co-workers.

[43, 44]

e Screening Constant by Unit Nuclear Charge (SCUNC) approximation, present work. f R-matrix, n = 2 basis, intermediate coupling, level averaged.

'Max Angle (mrad) (10mm)' 'Max Angle (mrad) (12.5mm)' 'Max Angle (mrad) (15mm)' 'Max Angle (mrad) (17.5mm)' 'Max Angle (mrad) (20mm)' 'Max Angle (mrad) (22.5mm)' 'Max Angle (mrad) (25mm)'

©

IOP Publishing Ltd Printed in the UK & the USA

Present address: Laboratoire Structures, Propriétés et Modélisation des Solides (SPMS) UMR CNRS 8580, Ecole Centrale Paris, 1, Grande Voie des Vignes, F-92295 Châtenay-Malabry, France.

Author to whom any correspondence should be addressed.

Acknowledgments

The experimental measurements were performed on the PLEIADES beamline [72, 73], at the SOLEIL Synchrotron radiation facility in Saint-Aubin, France. The authors would like to thank the SOLEIL staff and, in particular those of the PLEIADES beam line for their helpful assistance. B M McLaughlin acknowledges support from the US National Science Foundation through a grant to ITAMP at the Harvard-Smithsonian Center for Astrophysics, the RTRA network Triangle de le Physique and a visiting research fellowship from Queen's University Belfast. I Sakho acknowledges the hospitality of the Université Paris-Sud and support from the University Assane Seck of Ziguinchor during a recent visit. We thank Dr John C Raymond and Dr Randall K Smith at the Harvard-Smithsonian Center for Astrophysics for discussions on the astrophysical applications and Dr Javier Garcia for numerical values of the cross sections [9]. We gratefully acknowledge Dr Wayne C Stolte for assistance in fitting and interpreting the SOLEIL experimental measurements with the WinXAS software. The computational work was carried out at the National Energy Research Scientific Computing Center in Oakland, CA, USA, the Kraken XT5 facility at the National Institute for Computational Science (NICS) in Knoxville, TN, USA and at the High Performance Computing Center Stuttgart (HLRS) of the University of Stuttgart. We thank Stefan Andersson from Cray Research for his advice and assistance with the implementation of the parallel R-matrix codes on the Cray-XE6 at HLRS. The Kraken XT5 facility is a resource of the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575.

Acknowledgments

Data collection was performed on the PLEIADES beam line, at the SOLEIL Synchrotron Radiation Facility in Saint-Aubin, France. The authors would like to thank the staff of SOLEIL and, in particular, the staff of the PLEIADES beam line for their helpful assistance. The experimental research work described here has been supported by Triangle de le Physique contract 2007-001T. BMM acknowledges support by the US National Science Foundation through a grant to ITAMP at the Harvard-Smithsonian Center for Astrophysics. The computational work was carried out at the National Energy Research Scientific Computing Center in Oakland, CA, USA, and on the Tera-grid at the National Institute for Computational Sciences (NICS) in Knoxville, TN, USA, which is supported in part by the US National Science Foundation.

Acknowledgments

The experimental measurements were performed on the PLEIADES beamline [48, 49], at the SOLEIL Synchrotron radiation facility in Saint-Aubin, France. The authors would like to thank the SOLEIL staff and, in particular those of the PLEIADES beam line for their helpful assistance, C Miron, Xiao-Jing Liu, E Robert and C Nicolas. B M McLaughlin acknowledges support from the US National Science Foundation through a grant to ITAMP at the Harvard-Smithsonian Center for Astrophysics, the RTRA network Triangle de le Physique and a visiting research fellowship from Queen's University Belfast. I Sakho acknowledges the hospitality of the Université Paris-Sud and support from the University Assane Seck of Ziguinchor during a recent visit. We thank John C Raymond and Randall K Smith at the Harvard-Smithsonian Center for Astrophysics for discussions on the astrophysical applications and Javier Garcia for numerical values of the cross sections [38]. The computational work was carried out at the National Energy Research Scientific Computing Center in Oakland, CA, USA, the Kraken XT5 facility at the National Institute for Computational Science (NICS) in Knoxville, TN, USA and at the High Performance Computing Center Stuttgart (HLRS) of the University of Stuttgart. We thank Stefan Andersson from Cray Research for his advice and assistance with the implementation and optimization of the parallel R-matrix codes on the Cray-XE6 at HLRS. The Kraken XT5 facility is a resource of the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575.

This linearly polarized electric field can be separated into two counter-rotating components,

Under the action of this oscillating electric field the ions speed up and their radius increases linearly with time. The rate of power absorbed during the excitation time texcite is given by:

The integration between t=0 to t=texcite gives the total energy absorbed by the ion:

and the relation between the radius in the presence of the excitation electric field and the excitation time is:

The strength of this equation is that post-excitation ion cyclotron orbital radius is independent of the mass to charge ratio. Thus all the ions can be excited to the same ICR orbital radius.

Let's note that, by increasing the intensity E0 of the oscillating electric field, it is possible to selectively eject some ions from the trap and to store only one selected ionic species.

The excited ions passing by the walls of the cell used as antenna induce an oscillating current which is amplified and stored. The magnitude of this signal is proportional to the total charge and the orbital radius, and it is independent of the magnetic field strength. By

= + ( -)

The equations are solved by the integration corresponding to the position z at certain time, t. In this case the equation for x and y coordinates can be written as,

Where = / is the charge to mass ratio, Ex, Ey and Ez are the components of the electric field along the x, y and z, Bx, By and Bz are the magnetic field components along the x, y and z.

The calculation of the behavior of beams in the electrostatic lens usually requires solving the Laplace equations, using heavy codes. An alternative method is obtained using a limited axial distribution of the real potential for fixed geometry. In the general case of unipotential lens ("einzel" in German), there is no exact formula describing such an axial distribution. Nevertheless, it is possible to obtain an approximate expression for the distribution in cylindrical geometry between two tubular electrodes of different diameters which closely reflects the actual distribution. The electric fields inside the einzel lens can be easily expressed with Taylor expansions:

The expression of the components of the electric field in three axis are:

Perspectives

The measurements of photoionization cross sections in multi-charged ions are important tools in atomic physics and plasma physics, and laboratory data are needed, as an example, for the interpretation of the spectra sent by X-ray laboratories like XMM Newton and Chandra. Our data on nitrogen ions allowed to improve the code used for the modelization of astrophysical and laboratory plasmas. This work will be continued by the K-shell photoionization study on the Oxygen iso-nuclear series. A proposal has already been accepted to do these experiments with the MAIA setup at the PLEIADES beamline at SOLEIL. In addition, the proposal to improve the photoionization cross sections obtained on CO and N molecular ions using MAFIA (Mobile Actual Field Ion Analyzer) at the DESIRS beamline also has been accepted by the last program committee.

The simulation of the MAIA set up till the interaction region will be also continued in order to increase the ions current in the interaction region. For the ion trap experiment, the simulation of extraction of the ions from MAFIA will be performed to study the feasibility.

It will allow to improve greatly the efficiency of the photoions detection, which has been the main limitation up to now for the FTICR ion trap.

Appendices