Characterization of anomalous Zeeman patterns in complex atomic spectra

The modeling of complex atomic spectra is a difficult task, due to the huge number of levels and lines involved. In the presence of a magnetic field, the computation becomes even more difficult. The anomalous Zeeman pattern is a superposition of many absorption or emission profiles with different Zeeman relative strengths, shifts, widths, asymmetries and sharpnesses. We propose a statistical approach to study the effect of a magnetic field on the broadening of spectral lines and transition arrays in atomic spectra. In this model, the sigma and pi profiles are described using the moments of the Zeeman components, which depend on quantum numbers and Land\'{e} factors. A graphical calculation of these moments, together with a statistical modeling of Zeeman profiles as expansions in terms of Hermite polynomials are presented. It is shown that the procedure is more efficient, in terms of convergence and validity range, than the Taylor-series expansion in powers of the magnetic field which was suggested in the past. Finally, a simple approximate method to estimate the contribution of a magnetic field to the width of transition arrays is proposed. It relies on our recently published recursive technique for the numbering of LS-terms of an arbitrary configuration.Comment: submitted to Physical Review

THÉORIE DES GRAPHES EN SPECTROSCOPIE ATOMIQUE

Le traitement graphique substitué par Jucys et El-Baz au traitement algébrique initial des méthodes de Racah pour l'étude des propriétés angulaires des fonctions d'onde atomiques, a été amélioré et développé. Des graphes représentant des fonctions antisymétriques ou numérotées ont été définis, aussi bien dans le cas d'une fonction à N électrons équivalents que dans celui d'une fonction à groupes d'électrons non équivalents. Les graphes des opérateurs tensoriels doubles du type w(Kk)i(nl, n' l') ont été également définis, ce qui a amené à reconsidérer d'une manière un peu différente de celle d'El-Baz les graphes des opérateurs tensoriels irréductibles et les règles d'intégration graphique. L'ensemble de la méthode peut être présenté comme un tout cohérent, incluant les graphes de Jucys qui apparaissent comme le résultat de l'intégration des graphes bra, ket, et opérateurs. La théorie de la seconde quantification a été utilisée pour justifier de manière plus fondamentale les définitions graphiques adoptées. Nous donnons une représentation des opérateurs annihilation-création et de leurs relations d'anticommutation. A partir de celle-ci, les règles de couplage habituelles conduisent à la forme graphique de l'opérateur W(Kk)(nl, n' l'). Plusieurs relations tensorielles entre opérateurs annihilation-création ont été retrouvées à titre d'exemple. Une représentation graphique des coefficients de parenté fractionnelle est proposée. L'utilisation pratique de la méthode dans les calculs d'éléments de matrice concrets est grandement facilitée par la simplification des différentes étapes du calcul et en particulier de l'intégration. D'autre part, un programme Fortran, utilisant cette méthode graphique, a été mis au point sur l'UNIVAC 1108 de la Faculté des Sciences d'Orsay : il permet d'établir automatiquement les formules littérales correspondant aux éléments de matrice donnés sous leur forme mathématique habituelle.The graphic method for the study of the angular properties of atomic wave functions, substituted by Jucys and El-Baz for the original algebraic treatment by the methods of Racah, has been developped and improved. The graphs representing antisymmetrical or numbered functions have been defined for the case of N equivalent electrons as well as for groups of non-equivalent electrons. The graphs of the double irreducible tensor operators of the type w(Kk)i(nl, n' l') have been also defined in a way slightly different of the one of El-Baz, as have the graphical rules for integration. The method is presented in unified way, including Jucys' graphs which appear as the result of the integration of the graphs : bra, ket and operator. The theory of second quantization has been used to justify the graphical definitions in a more fundamental manner. A graphic repsesentation of the annihilation and creation operators, and of the anticommutation rules is given. Using this representation the usual coupling rules give the graphic form of the operator W(Kk)(nl, n' l'). For example, several tensorial relations between annihilation and creation operators have been found. A graphic representation of fractional parentage coefficients is proposed. The practical use of the method in actual matrix element calculations is greatly improved by the simplification of the different steps of the calculation, particularly the integration. Furthermore a Fortran program, based on the graphic method, which gives automatically the literal formulas that correspond to matrix elements given in their usual mathematical form, has been written for the UNIVAC 1108 of the Faculty of Science in Orsay

CONTRIBUTION A L'EMPLOI DES GRAPHES DE YUTSIS ET EL-BAZ EN SPECTROSCOPIE ATOMIQUE

Le travail qui a été exposé est la continuation de ceux qui ont été décrits par Yutsis et coll. [1] et par El-Baz et coll. [2] [3] [4]. L'ensemble de la méthode, y compris les modifications apportées par l'auteur, sera prochainement publié au Journal de Physique.The work which has been described is a continuation of that described by Yutsis et al [1] and by El-Baz et al [2] [3] [4]. The whole method as well as the alterations introduced by the author will be published soon in the Journal de Physique

Étude des structures hyperfines des raies d'arc de 169Tm

Using a Hypeac spectrometer, hyperfine structures of 70 lines of the Tm I spectrum have been measured. Energies, hyperfine splittings and J - values are deduced for 27 levels and 58 lines are classified. The hyperfine splitting of the ground-state is measured accurately : 8W = — 49.5 mK. A theoretical study permits a prediction of the energy and the hyperfine splitting of a level lying in the 4f13 6s 7s configuration.A l'aide d'un spectromètre Hypeac on a déterminé les structures hyperfines de 70 raies d'arc du thulium 169. On en a déduit les énergies, les écarts hyperfins et les valeurs de J de 27 niveaux, ce qui a permis de classer 58 raies. L'écart hyperfin du fondamental a été déterminé avec précision : δW = — 49,5 mK. Une étude théorique a permis de prévoir l'énergie et l'écart hyperfin d'un niveau de la configuration 4f13 6s 7s

Quantitative Evaluation of the Fetal Cerebellar Vermis Using the Median View on Two-Dimensional Ultrasound

BACKGROUND: Evaluation of the cerebellum and vermis is one of the integral parts of the fetal cranial anomaly screening. OBJECTIVES: The aim of this study was to create a nomogram for fetal vermis measurements between 17 and 30 gestational weeks. PATIENTS AND METHODS: This prospective study was conducted on 171 volunteer pregnant women between March 2013 and December 2014. Measurements of the fetal cerebellar vermis diameters in the sagittal plane were performed by two-dimensional transabdominal ultrasonography. RESULTS: Optimal median planes were obtained in 117 of the cases. Vermian diameters as a function of gestational age were expressed by regression equations and the correlation coefficients were found to be highly statistically significant (P < 0.001). The normal mean (± standard deviation) for each gestational week was also defined. CONCLUSION: This study presents the normal range of the two-dimensional fetal vermian measurements between 17 and 30 gestational weeks. In the absence of a three-dimensional ultrasonography, two-dimensional ultrasonography could also be used confidently with more time and effort