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3. Results

In this section we present relativistic BP results for the maxwellian averaged collision strength, tex2html_wrap_inline1152, with radiation damping included. The effects of relativity and radiation damping on the maxwellian averaged collision strengths are examined.

The maxwellian averaged collision strength, or the effective collision strength, is given by:
where tex2html_wrap_inline1154tex2html_wrap_inline1156 is the collision strength for excitation from level i to level j, averaged over a maxwellian distribution of outgoing electron energies tex2html_wrap_inline1162tex2html_wrap_inline1164 above the excitation threshold of the level j, at temperature T. This slowly varying function of temperature can then be used to obtain the rate coefficient, tex2html_wrap_inline1170, for electron impact excitation,
where tex2html_wrap_inline1172 is the energy difference between levels i and j and tex2html_wrap_inline1178 is the statistical weight of level i.

Table 1: Labeling of energy levels included in the calculation and the comparison of the observed and calculated energies (in rydbergs) in Fe XXII

Table 2: Comparison of the effective collision strengths tex2html_wrap_inline1272 obtained from the relativistic Breit Pauli R-matrix method with the 45 level target expansion (45 BP) and with the 15 level expansion (15 BP), and by the term coupling method with the 8 term (15 level) target expansion (TCC) for Fe XXII. All three sets of the results include radiation damping effect

The tex2html_wrap_inline1278 values for the 105 transitions between the 15 n=2 levels (see Table 1 (click here)), the 450 transitions from these 15 n=2 levels to the 30 n=3 levels and the 435 transitions between these 30 n=3 levels are tabulated for the same 32 z-scaled temperatures as those for the B-like ions in Paper III. We also calculated a set of results for the 105 transitions between the 15 n=2 levels using a 15-level BP target expansion (15BP). In order to show the physical effects mentioned above, we compare in Table 2 (click here) the tex2html_wrap_inline1292 results for 2 temperatures, T = 441 000 and 2 205 000 K, obtained with the 45-level target expansion (45BP), the 15BP, and the TCC calculations with the 8-term target (that corresponds to the same 15 levels as in 15BP) for transitions from the ground level tex2html_wrap_inline1296 to the fourteen excited n=2 levels. The two sets of BP results for excitation to the low-lying levels are larger than those with the TCC (those for the lower temperature enhanced more), indicating the effect of the more extensive resonances due to relativistic level splitting in the BP calculations. For the intercombination transitions to the tex2html_wrap_inline1300 levels (indices 3, 4 and 5 in the tables), the results with the BP are considerably larger than those with the TCC, obviously due to the relativistic intermediate coupling. For transitions to the high-lying n=2 levels, namely the tex2html_wrap_inline1304 levels (indices 11 to 15 in the tables), the values for 45BP are generally higher than the 15BP, especially for the higher temperature T = 2 205 000 K, indicating the effect of resonances due to the n=3 levels.

Figure 1: Comparison of the present effective collision strengths by the relativistic Breit-Pauli R-matrix method with the 45-level target (the solid lines) and the 15-level target (the dash-dotted lines), by the term-coupling method (the dashed lines) and by the relativistic distorted-wave method (the dotted lines) for the transitions from the ground level tex2html_wrap_inline1310 to a) tex2html_wrap_inline1312, b) tex2html_wrap_inline1314, c) tex2html_wrap_inline1316 and d) tex2html_wrap_inline1318

To further illustrate these effects, in Fig. 1 (click here) we compare the tex2html_wrap_inline1320 values from the 45BP, 15BP, TCC and RDW (Zhang & Sampson 1994a) calculations for transitions from the ground level to the tex2html_wrap_inline1322, and the three tex2html_wrap_inline1324 levels. It is seen that the values do not differ very much between the 15BP, the TCC and the RDW for the higher temperatures. The 45BP results are mostly higher than the three other sets at these temperatures, indicating the effect of the n=3 resonances. For lower temperatures, the two sets of BP results are the same. However, they differ significantly from the TCC and the RDW results, especially for the intercombination transitions, showing that the relativistic effects affect the low-energy resonances. Obviously the RDW results are too low due to absence of resonance contributions. The TCC results are also low as explained by Fig. 1 (click here) of Zhang & Pradhan (1995a). From that figure we see that the TCC method takes relativistic effects into account only when the electron energies are greater than the target thresholds coupled to the upper level of the transition through intermediate mixing, while the BP method, in contrast, includes the relativistic effects ab initio for the entire energy region. Here it is again shown that the TCC method is not an adequate approximation for the relativistic effects in highly charged ions.

To study the radiation damping effect on the rate coefficients, we have also calculated the tex2html_wrap_inline1328 values with this effect turned off. By examining the two sets of results, it is found that, although radiation damping affects collision strengths near a few target thresholds for some transitions, especially for the forbidden and the intercombination transitions, the effect on the rate coefficients is minimal for most transitions, only reducing the results by 1% or 2%, except for three transitions, tex2html_wrap_inline1330, tex2html_wrap_inline1332 and tex2html_wrap_inline1334. For these three transitions, not only is the radiation damping effect on the collision strengths large, but its contribution to the rate coefficient is enhanced since the scattered electron energies, which enter the exponential term in Eq. (1), are small near the next target threshold where radiation damping is most pronounced. In Fig. 2 (click here) comparisons of the BP tex2html_wrap_inline1336 values with and without radiation damping are made for four transitions, including the above three transitions and the transition tex2html_wrap_inline1338. For the former three transitions the radiation damping effect reduces the tex2html_wrap_inline1340 values by about 8 - 20% at low temperatures, but for the latter transition the reduction is just about 2% (although its effect on the collision strength can be seen in Fig. 2 (click here) of Zhang & Pradhan 1995a).

The tex2html_wrap_inline1344 values for the 105 tex2html_wrap_inline1346 transitions, with n=2, are tabulated in Table 3 at 32 z-scaled temperatures (T in tex2html_wrap_inline1354 K, so that actual T values are obtained by multiplying by tex2html_wrap_inline1358 = 441). Results for the 450 n=2-3 transitions and for the 435 tex2html_wrap_inline1362 transitions with n=3 are tabulated in Table 4 and Table 5, respectively. Tables 3, 4 and 5 are not reproduced here, but are available in electronic form at the CDS, or via ftp from the authors at

Figure 2: Radiation damping effects on the effective collision strengths for the transitions a) tex2html_wrap_inline1366, b) tex2html_wrap_inline1368, c) tex2html_wrap_inline1370 and d) tex2html_wrap_inline1372. The solid lines indicate the results with radiation damping and the dashed line those without radiation damping

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