The ionization equilibrium of Fe from a theoretical viewpoint

In this respect, our concern is that an appreciable departure from LTE (Saha) ionization equilibrium (i.e. so-called overionization) may be more or less predicted because of the imbalance between the photoionization rate R (aJ), determined mainly by the mean radiation field (J), and the photo-recombination rate R* (aB), determined by the Planck function (B) corresponding to the local electron temperature. Namely, since the radiation from hot deeper layers contributes mostly to J, the inequality of J > B turns out to hold in the optically thin

Figure 32.2. The photo-recombination-to-photoionization ratio (R*/R) with depth at three Fe i ionization edges for models with various metallicities (Teff = 6,000 K and log g = 4.0).

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Figure 32.3. The neutral (Fe i) and singly-ionized (Fe ii) fractions relative to the total amount of Fe atoms with depth, for the Solar-metallicity log g = 4.0 models of Teff = 4,500, 5,000, 5,500, and 6,000 K.

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Figure 32.3. The neutral (Fe i) and singly-ionized (Fe ii) fractions relative to the total amount of Fe atoms with depth, for the Solar-metallicity log g = 4.0 models of Teff = 4,500, 5,000, 5,500, and 6,000 K.

atmospheric layer, especially for the T-sensitive UV wavelength region within which the important Fe i ionization edges are located. In effect, this reduces the Fe i/Fe ii ratio compared with that expected from the LTE Saha equation since (Fe i/Fe ii)nlte < (Fe i/Fe ii)lte.

There are several key points requiring attention in discussing how and when the NLTE effect caused by this overionization becomes quantitatively significant.

(1) First is the metallicity which affects the opacity; i.e. the radiation field tends to be ther-malized as the metallicity becomes higher. This situation is depicted in Figure 32.2, which shows that the departure of R*/R from unity (overionization) becomes progressively insignificant with increasing [M/H], especially at the important UV ionization edge of ground or low-excited Fe i levels (X ~ 1,800 A).

(2) Second is the population fraction of each ionization stage relative to the total number of Fe atoms, which determines the significance of the overionization effect on the abundance determination. As a general rule, influences on abundances derived from lines of the dominant stage are negligible, while those for the trace species are sensitively affected. In the case of unevolved dwarfs, most Fe atoms are singly ionized and only a small fraction remains neutral in the atmospheres of G stars and stars in earlier stages, whereas neutral Fe becomes predominant in K stars (Figure 32.3). In the former case, therefore, Fe i lines appreciably suffer the NLTE overionization effect while Fe 11 lines do not, and vice versa for the latter case. (3) Third, one should pay attention to the excitation potential of the levels under question. That is, the populations which suffer appreciable overionization are generally those of lower excitation levels, and highly excited levels often do not conform to it because of a comparatively stronger coupling to the continuum (i.e. the situation is closer to the condition of LTE). Some empirically derived tendency, e.g. that found by Ruland et al. (1980) for Fe i and Ti i in their analysis of Pollux, may be interpreted in this connection.

Accordingly, in the case of the F-G stars (most Fe are in the Fe ii state) commonly used for galactic chemical-evolution studies, the fact that a lower metallicity favors an appreciable NLTE effect causes the following trend which is theoretically predicted.

• The LTE abundances from Fe i lines tend to be underestimated while those from Fe ii lines are hardly affected.

• Quantitatively, however, this effect becomes progressively more appreciable as we go into the very-metal-poor regime of halo stars; while it is almost negligible for disk stars (-1 < [Fe/H]), not to mention for metal-rich stars with [Fe/H] > 0 (Thevenin & Idiart 1999, Figure 9).

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