3 Comparison with previous calculations

3.1 3s²3p³ - 3s3p⁴ transitions

Previous calculations of collision strengths for the 3s²3p³$\to$3s3p⁴ electric dipole transitions in FeXII by F77 and TH88 are available for a detailed comparison with the present results.

F77 adopted a distorted wave approach coupled with a limited target model including 4 configurations (3s²3p³, 3s3p⁴, 3s²3p²3d and 3p⁵), and computed collision strengths at only one value of energy (6.6Ry). Due to the approximation used in the scattering process and to limitations in computational resources, resonance effects and the variation of $\Omega$ with E could not be explored. In Table 4 we compare the two sets of data by choosing a cut-off of $\pm\,50\%$ of the present results. Values in F77 which do not agree within $50\%$ with our collision strengths have been denoted with the superscript ^a. Out of 35 transitions, 6 ($17\%$) do not satisfy the $\pm\,50\%$ criterion and have not been included in the analysis. It should be noted that of these 6 transitions only 1 has $\Omega\,\gt\,0.1$ in our calculation. For the remaining 29 transitions we find an average difference of $15\%$ from our results.

TH88 adopted a more comprehensive target representation, also including correlation orbitals, and used the more sophisticated R-matrix approach in their scattering calculation. Details of their computation have been extensively discussed in relation to ours in BMS. We compare the two sets of collision strengths in Table 4 at the lowest (6.6Ry) and highest (30Ry) energies for which they tabulate values. At 6.6Ry 10 transitions ($29\%$) do not satisfy the $\pm\,50\%$ criterion and we find an average difference of $18\%$ from our data for the remaining values. It is important to note that all the $\Omega$'s excluded are $<\,0.1$ in our computation. At 30Ry we find an average difference of $16\%$ from our data, after exclusion of 9 transitions ($26\%$), of which only 1 has $\Omega\,\gt\,0.1$ in our set. A comparison over the entire energy range tabulated by TH88 reveals that the two sets of $\Omega$'s have the same trend as a function of E.

We stress the point that all the collision strength values which show the most severe disagreement between the different sets of calculations, up to more than an order of magnitude in some cases, are "weak'' transitions, i.e. transitions with $\Omega\,\le\,0.1$. For these transitions the strong mixing of levels and consequent cancellation effects in the matrix elements, which depend on the particular choice of the target model, are more likely to introduce big discrepancies in the collision strengths, as shown by the above analysis. This shows once more the importance of a proper choice of target representation in a scattering calculation, i.e. one in which full account is taken of interaction effects between configurations of a parity complex.

As is the case for the forbidden transitions within the 3s²3p³ ground configuration (Tayal et al. 1987), a large discontinuity between the $\Omega$ values at 4Ry and at 6.6Ry is evident in TH88 electric dipole collision strengths. As discussed in BMS, this is likely to be due to residual pseudo-resonance effects in the open channel energy region and, because of this, it should not be surprising that our collision strengths at 6.6Ry agree marginally better with F77, despite the more sophisticated approach adopted in the present work and by TH88.

The problems introduced by these non-physical pseudo-resonances in the open channel energy region and by the extra resonances below the highest excitation threshold, brought in by orbitals of correlation type, show up even more drastically in the effective collision strengths. A comparison between our results , tabulated in Table 6, and TH88 can be made in the $T_{\rm e}$ range 410⁵K - 310⁶K. The two sets of effective collision strengths have the same trend as a function of $T_{\rm e}$, with the noticeable exception of the 6 transitions marked by the superscript ^b in Table 6. For these transitions our $\Upsilon$'s increase with $T_{\rm e}$, whereas $\Upsilon$'s in TH88 fall off with $T_{\rm e}$, which is surprising considering that these are all optically allowed transitions and, most importantly, that for both sets of calculations the corresponding $\Omega$'s increase with E above 6.6Ry. The large drop in $\Omega$ between 4Ry and 6.6Ry, mentioned above, almost certainly accounts for this unusual behaviour resulting from the integration of the collision strengths over a Maxwellian function. Further support for this hypothesis is offered by the 3s²3p^{3 2}P$^{\rm o}_{3/2} \to$ 3s3p^{4 2}S_1/2 transition (^c in Table 6), which in TH88 shows a drop of almost a factor of 2 in the collision strength between 4Ry and 6.6Ry (their Table 2). This is clearly reflected in the drop of the effective collision strength at the lowest three temperatures tabulated in their Table 3. Starting from $T_{\rm e}\,=\,10^6$ K onwards the expected increase of $\Upsilon$ with $T_{\rm e}$ is otherwise correctly reproduced. If we make a numerical comparison of the two sets of data, we find that $69\%$ (at 410⁵K) and $40\%$ (at 310⁶K) of transitions differ by more than $50\%$, as shown by a superscript ^a in Table 6. We also note that for all such transitions $\Upsilon$(present work)<$\Upsilon$(TH88). This, along with the fact that significantly more transitions have been rejected at the lower $T_{\rm e}$, where resonance effects are more likely to contribute in the integration, forces us to believe once more that the reason for the large $\Upsilon$ values in TH88 lies in pseudo-resonance contributions due to the use of correlation orbitals in the target description.

 

Table 8: Effective collision strengths for all electric dipole fine-structure transitions between the ground 3s²3p³ and the second excited 3s²3p²3d configurations in FeXII. Temperatures in the range 410⁵K - 310⁶K

  

Table 8: continued

 

Table 9: Effective collision strengths for all electric dipole fine-structure transitions between the ground 3s²3p³ and the second excited 3s²3p²3d configurations in FeXII. Temperatures in the range 410⁶K - 10⁷K

  

Table 9: continued

3.2 3s²3p³ - 3s²3p²3d transitions

The only set of 3s²3p³$\to$3s²3p²3d electric dipole collision strengths available for comparison with the present work is by F77. Given the limitations of that computation, discussed above, we believe that our much more comprehensive target representation, proper inclusion and treatment of resonances and the provision of collisional data over a wide energy range represent the major new component of the present results. Out of 97 transitions 42 have not been included in the analysis, 16 (marked with ^b in Table 5) because of presumed identification problems to be discussed later and 26 (marked with ^a in Table 5) as not satisfying the $\pm\,50\%$ criterion introduced above. For the remaining transitions we get an average difference of $16.5\%$ between the two sets of data. We point out that of these 26 transitions marked with ^a, 11 have $\Omega\,\gt\,0.1$, 5 have $\Omega\,\gt\,0.3$ and only 2 have $\Omega\,\gt\,1$.Again, this clearly shows that the most severe errors, from a factor of 2 up to more than an order of magnitude, occur for low collision strength values, i.e. weak transitions where cancellation effects due to the particular target expansion are more likely to affect the final results.

The transitions marked by ^b in Table 5 deserve a separate discussion as we have evidence of label misassignment for the upper levels of these transitions in F77. The most striking case is for the two levels (³P)²F_5/2 and (³P)²D_5/2. By comparing the collision strength values for ²D$^{\rm o}_{3/2}$ - (³P)²F_5/2 and ²D$^{\rm o}_{3/2}$ - (³P)²D_5/2 in Table 5 it is evident that the relative strengths of these two transitions are reversed in the two calculations. By reassigning in F77 the higher collision strength to the former transition and the lower one to the latter we get a significant improvement in agreement from more than 2 orders of magnitude down to $25\%$ and $81\%$ (still a big difference, but this is a low $\Omega$value, $<\,0.1$!), respectively. Similarly for the ²P$^{\rm o}_{3/2}$ - (³P)²F_5/2, ²P$^{\rm o}_{3/2}$ - (³P)²D_5/2 transitions by interchanging the collision strengths in F77 the difference between the two sets of data drops from more than 2 orders of magnitude down to $29\%$ and $31\%$ respectively. The other suspected case of mislabelling is that of levels (¹D)²P_1/2 and (¹D)²S_1/2. The stronger transition in our computation is the ²P$^{\rm o}_{1/2}$ - (¹D)²S_1/2 rather than the ²P$^{\rm o}_{1/2}$ - (¹D)²P_1/2 as in F77. By reassigning the high $\Omega$ to the former transition in F77, the difference decreases from more than a factor of 4 down to $17\%$. Similarly, by interchanging the $\Omega$'s for the ²P$^{\rm o}_{3/2}$- (¹D)²P_1/2, ²P$^{\rm o}_{3/2}$ - (¹D)²S_1/2 transitions in F77, we get a difference of $27.5\%$ for the stronger ²P$^{\rm o}_{3/2}$ - (¹D)²P_1/2 transition, against a difference of more than a factor of 2 shown in Table 5. In F77 it is clearly stated that, due to the neglect of excited configurations of the $1^2\,2^8\,3^5$ even parity complex, the wavefunctions of some of the 3s²3p²3d excited states might be substantially wrong in that work. The strong mixing of the levels mentioned above depends critically on the target basis expansion and we propose that the mislabelling affecting the F77 data might be due to his use of an incomplete target basis set.

One last point of discussion concerns the assignment of observed energy values in F77. In that work experimental energies are derived from transition wavelengths given in Fawcett (1971), who provided data for the (¹D)²D term but not for the (¹S)²D term. In our calculation (see Table 1) the (¹S)²D$_{\rm J}$ levels are energetically lower than the (¹D)²D$_{\rm J}$ levels but F77 seems to suggest a reversed ordering in his listing, despite the lack of any clear indication of term parentages. A SUPERSTRUCTURE calculation with only 4 configurations in the basis set, performed to mimic the F77 atomic model, revealed that the (¹D)²D$_{\rm J}$ levels are indeed still energetically higher than the (¹S)²D$_{\rm J}$ levels, providing a clue to a possible misassignment in F77. By reassigning the 553855 cm^-1 and 554575 cm^-1 observed energies to the second lowest 3s²3p²3d²D doublet in his Table 1 we would get proper agreement with our results and with the updated observed energies by Corliss $\&$ Sugar (1982).