2 Transition probabilities

To calculate the (3d⁸+3d⁷4s) $\rightarrow$ 3d⁷4p transitions in Co II properly, it is necessary to take several neighbouring configurations into account: we used (3d⁸+3d⁷4s+3d⁶4s²+3d⁷4d+3d⁷5s) for the even and (3d⁷4p+3d⁶4s4p+3d⁵4s²4p+3d⁷5p+3d⁷4f) for the odd system. Interactions with other (far-lying) configurations are taken into account by means of so-called effective operators. The overall mean deviations are 39 cm^-1 for the even system and 19 cm^-1 for the odd system. The mean deviations for the individual lower configurations are 1.4, 7.1 and 15 cm^-1 for 3d⁸, 3d⁷4s and 3d⁷4p, respectively.

2.1 E1 results

In Table 2 the log(gf) values for the (3d⁸+3d⁷4s) $\rightarrow$ 3d⁷4p electric dipole (E1) transitions are given. This system is selected by cutting off energy values higher than 120000 cm^-1 of both the even and the odd system in the final printing procedure; only log(gf) values larger than -3 are included. In this paper a sample focusing on the resonance lines is given, as these UV lines are of most interest in ISM absorption observations; as explained in a footnote to the abstract, the complete table can be obtained at CDS.

The first column of this table shows the wavelength obtained from the energy differences between the experimental level values. Wavelengths below 2000 Å are given as vacuum wavelengths and above 2000 Å as air wavelengths. The second column gives the log(gf) values followed by the J-value, energy value and the name of the lower (even) level. The first character of the level name designates the configuration number: for the even levels "1" refers to 3d⁸ and "2" to 3d⁷4s; for the odd levels "1" refers to 3d⁷4p. An "*" after the energy value indicates that the level is known, in which case the experimental level value is given. When unknown, the calculated energy value is given and used to approximate the wavelength. Full results including weaker lines and lines involving higher lying levels can be found on Internet.

To have an indication of the accuracy, we compare our results with data from three different experiments as well as with the calculations from the well known database of Kurucz (Kurucz 1993).

• First of all, we compare in Table 3 Kurucz's and our log(gf) values with the results of a recent "state of the art" experiment to determine absolute oscillator strengths (Mullman et al. 1997). It can be seen that the present results are always close to the experimental ones, and 16 out of 28 can be considered "equal" to within the uncertainties specified.
• In Table 4, we make a comparison with the intensity numbers obtained by the recent FTS experiment (Pickering et al. 1997). The intensity numbers given by Pickering et al. are signal-to-noise ratios of the lines of the FT spectra. These values are useful in the comparison of one line to another; however, they do not show absolute intensities. All transitions considered are based on the lowest even term, the 3d⁸ (³₂F). The order of our log(gf) values completely agrees with the order of experimental intensity numbers, and confirms the present approach.
• In the region 2300 to 2800 Å a number of 3d⁷4s-3d⁷4p transition probabilities can be compared with measured oscillator strengths of the 28 strongest lines in the multiplets UV7, UV8 and UV9 published in 1985 (Salih et al. 1985). With respect to earlier work of Kurucz (Kurucz & Peytremann 1975), Salih et al. observe good agreement for strong lines but less so for intercombination ($\Delta S=1$) lines. At this point we would like to emphasize that the agreement with the current data of Kurucz's database (Kurucz 1993, semi-empirical Fe-group) is much better. The present method is not inherently different, except that use of orthogonal operators stabilizes the fit and allows the introduction of higher order effects into the model. As a result the eigenvector compositions should be more reliable, which is most obvious in intercombination lines. From Table  5 it seems that our calculated log(gf) values agree somewhat better for $\Delta S=0$ also.

In general, the current data turn out to prevail in all three cases.

 

Table 2: Calculated log(gf) values for the (3d⁸+3d⁷4s) - 3d⁷4p transition array of Co II

 

Table 2: continued

 

Table 2: continued

 

Table 3: Comparison between theory and experiment^a of log(gf)-values for the (3d⁸+3d⁷4s) $\rightarrow$ 3d⁷4p transitions involving the lowest even multiplets

 

Table 4: Comparison between experimental^a intensity numbers and theoretical log(gf)-values for the 3d⁸ - 3d⁷4p transitions involving the ground ³₂F term

 

Table 5: Comparison between theory and experiment^a of log(gf)-values for the 3d⁷4s - 3d⁷4p transitions involving the lower ^3,5F multiplets

  

Table 6: Values for the electric quadrupole transition integrals in Co II calculated by means of MCDF including core polarization

 

Table 7: Calculated A-values for the (3d⁸+3d⁷4s) - (3d⁸+3d⁷4s) M1 and E2 transition arrays of Co II; the notation x(y) means $x \times 10^y$

2.2 Forbidden lines

Transition probabilities of forbidden, i.e. magnetic dipole (M1) or electric quadrupole (E2), transitions are only given for energy levels below 50000 cm^-1 in view of their astrophysical relevance.

The forbidden transitions observed in the infra-red at 18.8 $\mu$m, 10.52 $\mu$m and 1.547 $\mu$m by Jennings et al. (1993), are all M1 transitions. In their interpretation, Jennings et al. used a calculation of Nussbaumer & Storey (1988), with A-values 1.08, 2.23 and 2.89 $\ 10^{-2}$ s^-1. These values agree with ours at the percent level (in Table 7, we give 1.09, 2.24 and 2.81 $\ 10^{-2}$ s^-1), which is not very surprising as it concerns here strong transitions based on relatively pure levels; moreover, no radial transition integrals are needed for M1 transitions. For the (a³F-b³F) E2 transitions given by Nussbaumer & Storey the discrepancies are larger, our values being roughly 20% lower. For levels that are less pure, the present method is expected to be especially effective. Radiative data for infrared lines arising from forbidden transitions are needed to study the debris of Type II supernova explosions like SN 1987 A (Li et al. 1993).

Similar to the E1 case, the radial part of the E2 transitions is calculated from relativistic wavefunctions. In Table 6 the radial integrals for the electric quadrupole transitions are given in the form of a symmetric matrix. For E2-transitions within the 3d⁷4d configuration, there are two non-zero contributions, one for the 3d-3d and one for the 4d-4d transition. For this case, there are two rows in the table, the upper for the 3d-3d integral and the lower for the 4d-4d transition integral.

The A-values for the forbidden lines are restricted to the magnetic dipole (M1) and electric quadrupole (E2) transitions within the 3d⁸+3d⁷4s configurations, from levels with an energy of less than 50000 cm^-1 above the ground and with A-values larger than 10^-3 s^-1. The level with the lower J-value is given first in the designation of the transition. A specimen of the table available at CDS is included in the paper as Table 7. We selected data for this sample in three wavelength regions: the first includes the observed infra-red transitions, the second illustrates a region where M1 and E2 transitions occur simultaneously and the last gives ultra-violet transitions involving the lowest configuration 3d⁸. As the sample is necessarily incomplete, only the full CDS table should be used to calculate radiative lifetimes.