Astron. Astrophys. 333, 219-230 (1998)

2. Atomic model

The Grotrian diagram of the Mg I model is shown in Fig. 1. We include all levels n up to n and which results in a total of 83 Mg I terms. The model is completed by the doublet ground state of Mg II. All energies were taken from the compilation of Martin & Zalubas (1980) except for those terms with which were obtained using the polarisation formula of Chang & Noyes (1983). Fine structure splitting has been neglected. This model is nearly the same as that used by Carlsson et al. (1992).

Fig. 1. Mg I Grotrian diagram including all permitted and a few important intercombination radiative bound-bound transitions considered in this paper. The atomic model is complete for n 9,

2.1. Radiative transition data

The bound-bound transition probabilities for allowed radiative transitions with n and were taken from the Opacity Project (Butler et al., 1993). For the remaining transitions we use Coulomb approximation results according to Bates & Damgaard (1949). The accuracy of the transition probabilities is estimated to be quite high, with errors 10%. We only consider a few important intercombination transitions. The oscillator strength of 3s  S  - 3p    was taken to be the mean value of that of Wiese et al. (1969) and the recent calculation of Moccia & Spizzo (1988). 3p    - 3d  D , 4d  D, 5d  D, 6d  D were taken from Kurucz & Peytremann (1975).

The photoionization cross-sections of the lowest three levels, namely 3s  S, 3p    and 3p    were fitted using an exponential power law to the theoretical calculations of Butler et al. (1993). The resulting photoionization cross-sections of 3s  S, 3p    and 3p    at threshold are 2.5 MB, 17.5 MB and 150 MB respectively; the theoretical cross-sections are displayed in Fig. 2a-c. We thus ignore the explicit influence of the resonances. Consequently, the corresponding photoionization rates obtained by a power law approximation are systematically smaller by values of only 16.5%, 7.4%, and 17.8%, respectively, at all optical depths throughout the atmosphere. This introduces no significant errors in both departure coefficients and line profiles mostly because the wavelength region blueward of 2078Å  is dominated by opacity from Al I. For longer wavelengths, the simple fits provide a reasonable approximation to the cross-sections.

Fig. 2a-c. Mg I photoionization cross-sections for the three lowest terms (Butler et al. 1993). The edges are near a 1620 Å (3s  S,); b 2514 Å (3p  ); and c 3757 Å (3p  ).

The quantum defect formulae of Peach (1967) are used to determine the photoionization cross-sections for n and . For all other levels the hydrogenic approximation is used.

2.2. Collisional transition data

2.2.1. Electron collisions

The cross-sections of the allowed transitions for electron impact excitation were computed following van Regemorter's (1962) formula. The mean gaunt factor g in the formula was set equal to 0.2 if the principal quantum number n changes in the transition and to 0.7 if it does not. When van Regemorter's formula is compared with the electron impact excitation (Seaton 1962), it turns out that the latter is systematically smaller. Mashonkina (1996) has shown that for Mg I differences up to a factor of 100 do occur, however, with a line-to-line scatter of similar amplitude. Our comparatively high values for allowed electron collisions may therefore represent an upper limit towards level thermalization. Any smaller collision rates will change the departure coefficients towards stronger non-LTE effects. The cross-sections of the forbidden transitions for electron impact excitation were computed from the semi-empirical formula of Allen (1973), in which the collision strength is set equal to 1 for all transitions. Here, Mashonkina (1996) shows that calculations using the Born approximation may lead to significantly greater collision cross-sections, in particular for transitions between highly excited levels with small energy separation. Our choice therefore may imply that our forbidden electron collision rates are systematically low. We note that there is no simple argument that could assist in the proper choice of the electron collision rates, and we return to this problem when discussing the infrared emission lines. The cross-sections for electron impact ionization were calculated using the formula of Seaton (1962) for energy levels up to n 9 and , since their photoionization cross-sections at threshold were available. For the few high value levels a semi-empirical formula (Drawin 1969) was employed.

2.2.2. Collisions with hydrogen atoms

Excitation and ionization by inelastic collisions with heavy particles are often considered to be unimportant compared with electron collisions since they have much smaller cross-sections (Omont 1977, Petitjean & Gounand 1984, Carlsson et al. 1992, Caccin et al. 1993). However, taking the larger number density of neutral hydrogen in the solar photosphere into account, the collision rates, could be of interest in cases where the statistical equilibrium depends sensitively on collisions. Thus some investigators consider this kind of collision to be very important (e.g.  Steenbock 1985, Lemke & Holweger 1987) since the ratio of number densities in cool stars may easily exceed . Baumüller & Gehren (1996) in their analysis of the solar aluminium line formation encountered a similar case. They applied the hydrogen collision formula as derived by Drawin (1969) but allowing for a scaling factor   that varied in an almost step-like fashion at some excitation energy . We carefully investigated the influence of hydrogen collisions in our atomic model by fitting all available Mg I lines. As for the Al I atom we again had to modify Drawin's formula by a scaling factor. However, for Mg I the factor varies exponentially with upper level excitation energy (in eV),

This was determined in a fully empirical manner, recomputing the complete non-LTE line formation with statistical equilibrium equations including the differing hydrogen collision rates, and it enabled us to fit lines of different excitation energies. The notion of hydrogen collision cross-sections decreasing systematically with excitation energy is also in rough agreement with Kaulakys' (1985, 1986) prediction for Rydberg transitions. Although Lemke & Holweger (1992) were unable to reproduce the infrared Mg I emission lines using the Holweger-Müller model atmosphere they point out that the emission lines may require hydrogen collision cross-sections that depend on the excitation energy of the lower level of a transition. This trend seems to compensate the very rough approximation of the atomic collisions in a similar way in both Al I and Mg I. The consequences of this particular form of the hydrogen collision rates will be discussed in Sect.  5.

© European Southern Observatory (ESO) 1998

Online publication: April 15, 1998
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