Issue 
A&A
Volume 577, May 2015



Article Number  A113  
Number of page(s)  9  
Section  Atomic, molecular, and nuclear data  
DOI  https://doi.org/10.1051/00046361/201525914  
Published online  13 May 2015 
Research Note
Effective collision strengths between Mg i and electrons
^{1} Institut d’Astronomie et d’Astrophysique, Université Libre de Bruxelles, CP 226, Boulevard du Triomphe, 1050 Brussels, Belgium
email: tmerle@ulb.ac.be
^{2} Université de Nice SophiaAntipolis, Laboratoire Lagrange, UMR 7293 CNRS, OCA, CS34229, 06304 Nice Cedex 4, France
^{3} Department of Physics and Astronomy, Drake University, Des Moines, IA 50311, USA
Received: 17 February 2015
Accepted: 13 March 2015
The treatment of the inelastic collisions with electrons and hydrogen atoms are the main source of uncertainties in nonLocal Thermodynamic Equilibrium (LTE) spectral line computations. We report, in this research note, quantum mechanical data for 369 collisional transitions of Mg i with electrons for temperatures comprised between 500 and 20 000 K. We give the quantum mechanical data in terms of effective collision strengths, more practical for nonLTE studies.
Key words: atomic data / atomic processes / line: formation / line: profiles
© ESO, 2015
1. Introduction
Atomic data are the main ingredient for a good nonlocal thermodynamic equilibrium (nonLTE) description of the interaction of a chemical species with matter and radiation in atmospheres of latetype stars. Huge efforts have been made over the years regarding experimental and theoretical data (see e.g. Henry 1993). Semiclassical or empirical formulae are widely used because they provide rough estimates of collision rates based on oscillator strength for radiatively permitted transitions while no formulae exist to correctly treat the forbidden transitions. The most used of such formulae is the impact parameter method (IPM, Seaton 1962a,b) which implies computing collision crosssections for weak and strong coupling. A simpler way (but also not as precise) is to use the semiempirical formulation from van Regemorter (1962) which is an approximation of the IPM using experimental data available at that time. A more recent version that takes into account recent experimental data give a formulation with a dependance on the change of the principal quantum number (Fisher et al. 1996) but this version is not used very often in the literature.
Energy levels of Mg i.
Fig. 1
Grotrian diagram of Mg i with mean energy levels labeled (top) and with the collision transitions (bottom). Theoretical collision transitions are shown with black lines whereas the calculated ones are shown with grey lines. 
Magnesium is an αelement produced by type II supernovæ of which is important to study its enrichment relative to iron in different stellar populations. The numerous optical lines of Mg i, including the green b triplet, allow the determination of its abundance in many kinds of stars which is why a good description of its atomic data and especially the inelastic collisions are needed. Zatsarinny et al. (2009) lead a very detailed study of angledifferential crosssections for electron scattering from neutral magnesium using a Bspline Rmatrix (BSR) method. However, only results for five collision crosssections with the ground state were published in this paper, whereas many collision crosssections were computed. In the present research note, we decided to publish the other data in the form of effective collision strengths useful for nonLTE applications and which concern 369 transitions between excited states.
2. Previous calculations
In Zatsarinny et al. (2009), 37 target states were included in the scattering calculations. The configurations, terms, energy levels, and statistical weights g are given in Table 1. The Grotrian diagram of the energy levels are given in the top panel of Fig. 1. The double excited level 3p^{2}^{1}S located above the ionization energy is also included owing to big mixing of this configuration to the ground state. This improves the convergence of the total scattering function. The quality of the model atom description can be assessed from the theoretical oscillator strengths they calculated compared with the NIST^{1} counterparts. The differences are less than 3%. Accurate oscillator strengths are important for subsequent calculations of collision crosssections.
The Bspline Rmatrix code (Zatsarinny 2006) was used for the scattering calculations. Angledifferential crosssections for electronimpact excitation are obtained and described in detail in Zatsarinny et al. (2009). In their Fig. 8, they show angleintegrated crosssections for five transitions with the ground state but crosssections for 369 transitions were actually obtained and not published. Such crosssections are extremely important for nonLTE studies. With 37 levels included, there are a total of 666 collision transition possibilities. We present here all the transitions between the first 17 levels up to 3s5p ^{1}P°, and all effective collision strengths for excitation of these levels to the more excited level indicated in Table 1. There are 369 overall (see bottom panel of Fig. 1). Though the crosssection for transitions between highlyexcited levels can also be extracted from the BSR calculation, they cannot be considered reliable because of slow partial wave convergence for these levels. In nonLTE calculations, we hardly need detailed crosssections, but rather thermal average quantities such as effective collision strengths.
3. Effective collision strengths
The collision rate from state i to state j with electrons can be written as (1)where n_{e} is the electron density, g_{i} the statistical weight of the lower level, E_{ij} the transition energy, k the Boltzmann constant, T the temperature of the medium and the constant (2)where a_{0} is the Bohr radius, is the Rydberg unit of energy, and m_{e} the mass of electron. Moreover, we use the effective collision strenght defined as (3)where x = E/kT and x_{ij} = E_{ij}/kT are the kinetic energy after excitation and the energy of the transition in unit of kT, σ_{ij} is the collision crosssection expressed in unit of , and is dimensionless and symmetric with respect to the transition (i.e. ). The resulting effective collision strengths are given in Table 2 for nine temperatures between 500 and 20 000 K. The level indexes in Table 2 refer to the levels in Table 1.
These data will be useful for nonLTE studies of magnesium in stellar atmospheres of cool stars since they come from reliable quantum mechanical calculations. We should mention that such efforts toward calculating and collecting quantum mechanical data for inelastic collisions is extremely important so as to derive reliable nonLTE abundance corrections. Special efforts are also currently underway for computing inelastic collisions with hydrogen atoms (Barklem et al. 2011; Belyaev et al. 2014).
Effective collision strengths.
Available at: http://www.nist.gov/pml/data/asd.cfm
Acknowledgments
T.M. is shortterm foreign postdoctoral fellow from FNRS (FRFC convention, Ref: PDR T.0198.13).
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All Tables
All Figures
Fig. 1
Grotrian diagram of Mg i with mean energy levels labeled (top) and with the collision transitions (bottom). Theoretical collision transitions are shown with black lines whereas the calculated ones are shown with grey lines. 

In the text 
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