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4 Results

The results for the energy levels, oscillator strengths, and total and partial photoionization cross sections are described in the following sections.

4.1 Energy levels of Ni II

The calculations begin with the energies of 836 LS terms of Ni II corresponding to all possible bound states with principal quantum number $n\leq 10$.Table 2 shows some comparisons between the computed energies and experimental values from Sugar & Corliss (1985). The energies obtained in the present work agree with the experimental values typically within 2% throughout the entire data set with the only exception of the equivalent electron states of the form $\rm 3d^7 4s^2$. For these states, the calculated energies differ from experiment by about 6%. It is also found that near the Ni II ionization limit, where the density of states is large, there are numerous mismatches between the identification of states in the Sugar & Corliss compilation and the present results. The present level identifications are based on both percentage channel contributions and quantum defects and they do seem to be secure. Thus, more experimental work on the detection and proper identification of highly excited state is needed.

Table 2: Comparison between calculated energies for Ni II, $E_{\rm cal}$, and observed energies, $E_{\rm obs}$, from Sugar & Corliss (1985)

Conf. & Term & $E_{\rm cal}$\space & $E_{\r...
 ...\rm 3d^8 (^1D) 4d $&$\rm ^2P $& $-0$.305995 & $-0$.296769 \\ \hline\end{tabular}

4.2 Oscillator strengths

Dipole oscillator strengths (f-values) for $23\,738$ transitions among the calculated states of Ni II were obtained in LS coupling. This set includes transitions for which the lower state lies below the first ionization threshold and the upper state lies above. These transitions can be important in opacity calculations because they contribute to the total photo-absorption, but do not appear as resonances in the photoionization cross sections (strictly speaking, the upper bound state does autoionize if departure from LS coupling is considered and fine structure continua are explicitly allowed).

Comparison of length and velocity oscillator strengths provides a systematic consistency check on the accuracy of the wavefunctions and, therefore, on the reliability of the f-values. In Fig. 1 we plot $\log(gf_V)$ vs. $\log(gf_L)$. We have included all the symmetries since each exhibits roughly the same dispersion. The dispersion between length and velocity values is $\sim 9\%$ for gf-values greater than unity and $\sim 12\%$ for gf-values greater than 0.1.

\includegraphics [width=8cm,height=6cm,clip]{ds1715f1.eps}\end{figure} Figure 1: log gfV plotted against log gfL for transitions between calculated LS terms
The first experimental determination of Ni II f-values in the VUV was recently reported by Fedchak & Lawler (1999). While this work awaits publication all determinations of Ni II abundance from absorption lines in the diffuse ISM (e.g. Morton 1991; Zsargó & Federman 1998) have been based on theoretical data by Kurucz & Peytremann (1975) and Kurucz & Bell (1995).

Table 3 presents a comparison of the present f-values with the experimental data from Fedchak & Lawler (1999), the recommended values by Fuhr et al. (1988), and those from semiempirical computations of Kurucz & Bell (1995). Notice the recommended data by Fuhr et al. is mostly based on the values of Kurucz & Peytremann adjusted such as the radiative lifetime of the levels agreed with a few experimental measurements. Table 4 compares radiative lifetimes obtained from the present data with those measured by Fedchak & Lawler and calculated using the Kurucz & Bell data.

Table 3: Comparison of calculated f-values in LS coupling for Ni II with experimental data from Fedchak & Lawler (1999; FL), recommended values by Fuhr et al. (1988), and theoretical data from Kurucz & Bell (1995; KB)

Configuration & Transition &Present& FL& Fuhr et...
 ...226\\  & $\rm ^2F-^2D^o$\space & 0.183& --- & 0.166& 0.166\\ \hline\end{tabular}

Table 4: Comparison of calculated radiative lifetimes (in 10-9 s) of Ni II states with experimental data from Fedchak & Lawler (1999; FL) and theoretical values from Kurucz & Bell (1995)

Configuration & Multiplet &Present&FL&KB \\ \hlin...
 ...& 2.313 \\  & $\rm z\ ^2D^o$\space & 1.951 & 2.00 & 1.564 \\ \hline\end{tabular}

The tables show good agreement between the present data and the experimental values of Fedchak & Lawler but, there are significant differences with respect to the f-values of Kurucz & Bell and Fuhr et al. Particular discrepancies exist for transitions that involve the $\rm 3d^9~^2D$ ground state of Ni II. For these the Kurucz & Bell f-values seem to be overestimated by factors of two to four. These transitions are also the ones commonly observed from spectra of the ISM; furthermore, the gas phase abundance of nickel in the ISM maybe higher than previously estimated.

The good agreement between length and velocity f-values and the agreement for both absolute f-values and level lifetimes with experimental determinations suggests that the overall uncertainty for such transitions should be near 10%. However, weaker transitions are likely to have greater uncertainties. For weak transitions relativistic effects, not included here, may be important. Similarly, algebraic splitting of the present f-values in LS coupling into fine structure f-values would lead to large errors for relativistic effects are quite important at the fine structure level in Ni II.

4.3 Photoionization cross sections

Photoionization cross sections were calculated for all bound states. These cross sections include detailed autoionization resonances. Figure 2 shows the photoionization cross section of the $(\rm 3d^9~^2D)$ ground state of Ni II. In the same figure we have plotted the results of Verner et al. (1993) and Reilman & Manson (1979), both using central field type approximations. One interesting feature in the present cross section is the packs of large resonances between 1.75 and 2 Ryd that rise several orders of magnitude above the background. These resonances result from the coupling of the ground state ($\rm 3d^8~^1D$) with the excited state of the form ($\rm 3d^7\ 4s$) of the Ni III target.

\includegraphics [width=7cm]{ds1715f2.eps}\end{figure} Figure 2: Photoionization cross section ($\sigma$ (Mb)) of the ground state $\rm 3d^9 (^2D)$ of Ni II as a function of photon energy (Rydbergs). The dotted curve shows the results of Verner et al. (1993) and the filled squares, those of Reilman & Manson (1979)
The failure of other authors to obtain resonance structures in the cross sections is due to the absence of the relevant electron correlations in those calculations. This is always the case for the central field approximations.

Figure 3 shows the photoionization cross sections for a few excited states of Ni II.

\includegraphics [width=18cm,height=18cm,clip]{ds1715f3.eps}\end{figure} Figure 3: Photoionization cross section ($\sigma$ (Mb)) of excited states of Ni II as a function of photon energy (Rydbergs)


\includegraphics [width=18cm,height=13cm,clip]{ds1715f4.eps}
\end{center}\end{figure} Figure 4: Partial photoionization cross sections of the Ni II ground state going into the lowest six states of Ni III

In addition to the total photoionization cross sections we obtained also partial cross sections for photoionization going into each of the states of the target ion. These cross sections are needed for the computation of recombination rates (Nahar & Pradhan 1995) and in constructing non-LTE spectral models where it may be important to determine accurately the populations of excited levels of the residual ion following photoionization. Figure 4 presents these partial cross sections for the Ni II ground state going into into the lowest six levels of Ni III.

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