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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 Fe IV


Conf. Term tex2html_wrap_inline1202 tex2html_wrap_inline1410 tex2html_wrap_inline1200 Conf. Term tex2html_wrap_inline1202 tex2html_wrap_inline1410 tex2html_wrap_inline1200
tex2html_wrap_inline1420 tex2html_wrap_inline1160 -3.984 -3.961 -4.028 tex2html_wrap_inline1430 tex2html_wrap_inline1432 -2.229 -2.214 -2.266
tex2html_wrap_inline1420 tex2html_wrap_inline1442 -3.665 -3.640 -3.734 tex2html_wrap_inline1450 tex2html_wrap_inline1452 -2.184 -2.189 -2.237
tex2html_wrap_inline1420 tex2html_wrap_inline1462 -3.660 -3.585 -3.706 tex2html_wrap_inline1450 tex2html_wrap_inline1472 -2.115 -2.135 -2.184
tex2html_wrap_inline1420 tex2html_wrap_inline1482 -3.613 -3.563 -3.674 tex2html_wrap_inline1490 tex2html_wrap_inline1492 -2.005 -2.035 -2.090
tex2html_wrap_inline1420 tex2html_wrap_inline1502 -3.485 -3.426 -3.548 tex2html_wrap_inline1510 tex2html_wrap_inline1472 -1.971 -2.024 -2.075
tex2html_wrap_inline1520 tex2html_wrap_inline1522 -2.941 -2.774 -2.857 tex2html_wrap_inline1530 tex2html_wrap_inline1532 -1.967 -2.035 -2.065
tex2html_wrap_inline1520 tex2html_wrap_inline1482 -2.795 -2.685 -2.767 tex2html_wrap_inline1490 tex2html_wrap_inline1552 -1.972 -2.006 -2.058
tex2html_wrap_inline1560 tex2html_wrap_inline1462 -2.648 -2.531 -2.616 tex2html_wrap_inline1510 tex2html_wrap_inline1572 -1.936 -1.997 -2.048
tex2html_wrap_inline1580 tex2html_wrap_inline1582 -2.618 -2.531 -2.620 tex2html_wrap_inline1530 tex2html_wrap_inline1452 -1.932 -1.987 -2.038
tex2html_wrap_inline1600 tex2html_wrap_inline1502 -2.613 -2.516 -2.605 tex2html_wrap_inline1490 tex2html_wrap_inline1532 -1.949 -1.988 -2.040
tex2html_wrap_inline1620 tex2html_wrap_inline1442 -2.580 -2.488 -2.578 tex2html_wrap_inline1530 tex2html_wrap_inline1472 -1.910 -1.970 -2.026
tex2html_wrap_inline1640 tex2html_wrap_inline1482 -2.523 -2.427 -2.519 tex2html_wrap_inline1650 tex2html_wrap_inline1492 -1.906 -1.956 -2.010
tex2html_wrap_inline1430 tex2html_wrap_inline1662 -2.275 -2.256 -2.303 tex2html_wrap_inline1650 tex2html_wrap_inline1452 -1.902 -1.959 -2.012
tex2html_wrap_inline1430 tex2html_wrap_inline1682 -2.267 -2.250 -2.296 tex2html_wrap_inline1650 tex2html_wrap_inline1532 -1.877 -1.929 -1.982
tex2html_wrap_inline1700 tex2html_wrap_inline1462 -2.280 -2.194 -2.292 tex2html_wrap_inline1710 tex2html_wrap_inline1472 -1.819 -1.900 -1.959
tex2html_wrap_inline1720 tex2html_wrap_inline1502 -2.270 -2.198 -2.293 tex2html_wrap_inline1710 tex2html_wrap_inline1572 -1.816 -1.895 -1.947
tex2html_wrap_inline1430 tex2html_wrap_inline1572 -2.226 -2.224 -2.274 tex2html_wrap_inline1710 tex2html_wrap_inline1452 -1.806 -1.882 -1.939
Table 3: Comparison of the calculated energies for Fe IV, tex2html_wrap_inline1202, with the previous OP results, tex2html_wrap_inline1404, by Sawey & Berrington (1992), and observed energies, tex2html_wrap_inline1200, from Sugar & Corliss (1985)


The calculations begin with the energies of 746 LS terms of Fe IV corresponding to all possible bound states with principal quantum number tex2html_wrap_inline1156. In Table 3 (click here) we compare the computed energies for these terms with those calculated by SB, and experimental values from Sugar & Corliss (1985). The energies obtained in the present work, especially those for the low lying states, agree typically within 2% with the experimental values. This level of agreement, for most terms, is better than the results of SB.

4.2. Oscillator strengths

Dipole oscillator strengths (f-values) for 34,635 transitions among the calculated states of Fe IV 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).

In the absence of experimental f-values it is difficult to ascertain the overall accuracy of the data. However, a 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 (click here) we plot tex2html_wrap_inline1768 vs. tex2html_wrap_inline1770. We have included all the symmetries since each exhibits roughly the same dispersion. The dispersion between length and velocity values with gf's greater than unity < 13% for the quartets, and < 22% for the sextets. This, added to the good agreement between the calculated energies and those observed experimentally, suggests that the overall uncertainty for such transitions should be 20% or less. However, weaker transitions are likely to have greater uncertainties.

Figure 1: log gfV plotted against log gfL for transitions between calculated LS terms

Table 4 (click here) presents a comparison of the present gf-values with those of SB and Fawcett (1989) for a small sample of transitions. Fawcett's work is based on a semi-empirical adjustment of Slater parameters in the relativistic Hartree-Fock code by Cowan (1981) to minimize the differences between calculated and observed wavelengths. Very good agreement between the present results and Fawcett's values is observed for most transitions.


Configuration Transition PresentSBFawcett
tex2html_wrap_inline1786 tex2html_wrap_inline1788 6.11 5.92 6.11
tex2html_wrap_inline1790 10.2 9.83 8.49
tex2html_wrap_inline1792 13.8 13.3 13.9
tex2html_wrap_inline1794 3.77 3.64 2.20
tex2html_wrap_inline1796 7.02 6.67 6.57
tex2html_wrap_inline1798 9.17 8.79 8.75
tex2html_wrap_inline1800 tex2html_wrap_inline1802 12.7 11.5 12.0
tex2html_wrap_inline1804 17.5 16.6 16.9
tex2html_wrap_inline1806 tex2html_wrap_inline1808 0.11 0.08 0.10
tex2html_wrap_inline1810 tex2html_wrap_inline1812 0.11 0.08 0.10
tex2html_wrap_inline1814 tex2html_wrap_inline1816 1.90 2.24 1.94
Table 4: Comparison of calculated gf values in LS coupling for Fe IV with the OP calculations by Sawey & Berrington (1992), and the semi-empirical results by Fawcett (1989) using Cowan's code


4.3. Photoionization cross sections

Photoionization cross sections were calculated for all 746 bound states of Fe IV considered here. These cross sections include detailed autoionization resonances resulting from the coupling to states dominated by tex2html_wrap_inline1820, and tex2html_wrap_inline1822 configurations in the core ion. Figure 2 (click here)a shows the photoionization cross section of the tex2html_wrap_inline1824 ground state of Fe IV. In the same figure we have plotted the results of Reilman & Manson (1979) and those of SB. One interesting feature in the present cross section is the huge resonance, more than 1 Ry wide and two orders of magnitude higher than the background, just above the ionization threshold. Such a feature should have a large effect on the ionization rate and the opacity of Fe IV; thus a careful and detailed study of this resonance is worthwhile.

Figure 2: Photoionization cross section (tex2html_wrap_inline1826 (Mb)) of the ground state tex2html_wrap_inline1828 of Fe IV as a function of photon energy (Rydbergs). a) the cross section obtained with the present 31CC expansion (solid curve); b) the cross section excluding the tex2html_wrap_inline1830 configuration; c) the cross section with the tex2html_wrap_inline1832 target terms of Fe V included explicitly (the Rydberg series tex2html_wrap_inline1834 for n=3 to 10 is marked). The dashed curve shows the results of Sawey & Berrington (1992) and the filled dots, those of Reilman & Manson (1979)

The first thing to investigate is what electron configuration of the Fe IV system is responsible for the resonance, and whether this is possibly a pseudo-resonance that sometimes arise in close coupling calculations owing to inconsistencies between the two summations in Eq. (1). Pseudo-resonances can arise if the first summation involving all channels coupled to the target terms does not explicitly include the parent configurations of some (N+1)-electron correlation configurations included in the second summation. Such configurations then do not have corresponding thresholds for the Rydberg series of resonances in the target expansion (first summation in Eq. (1)). These manifest themselves as large pseudo-resonances, which in a sense represent the entirety of resonance series belonging to the missing thresholds. In order to rule out this possibility, the ground state cross section was calculated several times with different subsets of the (N+1)-electron correlation configurations list given in Table 2 (click here).

It appears that the configuration tex2html_wrap_inline1830 gives rise to the particular resonance. Figure 2 (click here)b shows the cross section obtained when this configuration is excluded. The tex2html_wrap_inline1830 configuration of Fe IV corresponds directly to the tex2html_wrap_inline1832 correlation configuration in the expansion for the Fe V core ion. Thus, this resonance appear to be real. In addition, the term energies for Fe IV calculated with an accurate CI expansion that includes the tex2html_wrap_inline1830 configuration, using the code SUPERSTRUCTURE (Eissner 1991), indicates that an autoionizing tex2html_wrap_inline1850 state is indeed expected at about 4.58 Ry above the ground state. That the energy of this state agrees well with the position of the resonance adds weight to our identification.

As this resonance arises from the tex2html_wrap_inline1830 configuration in Fe IV, one might expect that explicit inclusion in the calculation of the thresholds due to the tex2html_wrap_inline1832 parent configuration in the Fe V target would change the shape of the resonance and even break it into a series of narrow Rydberg resonances. The list of term energies for Fe V in the present target expansion reveals that above these 31 terms the next higher terms coupled to the ground tex2html_wrap_inline1160 state of Fe IV are tex2html_wrap_inline1858. Therefore, the ground state cross section was re-calculated including these states; the result is shown in Fig. 2 (click here)c. A number of narrow resonances converging on to the new thresholds are present; however, the large resonance under investigation remains unchanged.

The more extended calculation also allows for a better identification of the origin of the resonance, which seems to belong to the tex2html_wrap_inline1834 Rydberg series, as indicated in Fig. 2 (click here)c, for n= 3 to 10. An alternative series could be the tex2html_wrap_inline1864; however, the percentage channel contribution with the tex2html_wrap_inline1866 parent is smaller.

Thus the nature of the large resonance in the ground state cross section seems to be understood, and its identification as the autoionizing equivalent electron state tex2html_wrap_inline1850 explains in large part the broadness of the feature. Nevertheless, one should be aware that the position of the resonance may be uncertain in the absence of experimental data for the tex2html_wrap_inline1832 thresholds in Fe V. Also, the position of the resonance in the present calculation relies entirely on the accuracy of the tex2html_wrap_inline1832 wavefunctions, which is rather difficult to assess.

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 approximation used by Reilman & Manson (1979). The close-coupling calculation by SB included only the 16 states of the target ion dominated by the tex2html_wrap_inline1874 configuration. This means that only contributions from the tex2html_wrap_inline1876 ground state of the Fe V core ion were included in the photoionization cross sections of states with multiplicity (2S+1)=6. Therefore, all coupling effects for the cross sections of these states and, in particular, of the the ground state of Fe IV, were also missing.

4.3.1. Partial photoionization cross sections

In constructing non-LTE spectral models of astrophysical objects it may be important to determine accurately the populations of excited levels of the residual ion following photoionization. This requires not only total photoionization cross section but also partial cross sections into the excited states of product ion. Therefore, we have obtained the partial cross sections for photoionization of the states of Fe IV into at least the lowest few (particularly the metastable) terms of Fe V. These partial photoionization cross sections are also necessary in the calculation of unified electron-ion recombination rate coefficients (Nahar & Pradhan 1995). Fig. 3 (click here) presents these partial cross sections for photoionization of some excited states of Fe IV into the ground state of Fe V.

Figure 3: Partial photoionization cross sections of some excited states (tex2html_wrap_inline1882, panel a); (tex2html_wrap_inline1884, panel b); (tex2html_wrap_inline1886, panel c)) of Fe IV into the ground state of Fe V

4.3.2. Photoexcitation-of-core resonances

One interesting feature observed in certain photoionization cross sections ia the so-called photoexcitation-of-core (PEC) resonances that result from strong dipole transitions between the ground state and opposite parity states within the target ion (Yu & Seaton 1987; see also Bautista & Pradhan 1995, for Fe I). The PEC features are prominent in the photoionization of excited bound states along a Rydberg series, as the outer electron is weakly bound and photo-excitation takes place within the ion core - the PEC process is thus the inverse of the di-electronic recombination process with the outer electron as a "spectator" (Nahar & Pradhan 1995). Such PEC resonances are seen in Fig. 4 (click here) which displays the photoionization cross sections of Fe IV bound states in the Rydberg series tex2html_wrap_inline1890 with n = 5-9. At the Fe V target thresholds tex2html_wrap_inline1894, the incident photon energies equal those of the strong dipole transitions from the ground state tex2html_wrap_inline1896 and large PEC autoionizing resonances are formed, enhancing the effective cross section up to several orders of magnitude above the background. The prominent peaks shown in Fig. 4 (click here) correspond to these dipole transition energies tex2html_wrap_inline1898.

Figure 4: Photoionization of bound states in the tex2html_wrap_inline1900 Rydberg series showing PEC resonances

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