3 Results and discussion

The full set of results (Table 5) are to be found on the ftp server of the CDS (Centre de Données astronomiques de Strasbourg) in computer readable form. Here we show a small selection to illustrate some of the more important points.

One of the more interesting aspects for this ion is that an earlier calculation based on the Dirac formulation exists. We have plotted the collision strength for the 2s²2p^{2 3}P $^{\rm e}_1$ - ³P $^{\rm e}_2$ transition in Fig. 3 and for the 2s²2p^{2 1}D $^{\rm e}_2 - 2$s2p³³D$^\circ _2$ transition in Fig. 5. The scales have been chosen to allow a direct comparison with the Dirac results obtained by Aggarwal (1991) (see Figs. 1 and 2 on pa. 681 of that paper). It is obvious that the agreement is excellent. It should be noted, however, that the inclusion of the n=3 states has a pronounced effect on the overall results as the resonances at higher energies to be seen in Figs. 4 and 6 for the same cross sections are not present in the earlier calculation. In the same way, resonances converging to the 2s²2p$4\ell$ and higher thresholds are lacking in the current work. This could be remedied by extending the calculation to include these higher thresholds but the cost is prohibitive. In any case, the effect on the n=2 transitions would be small at the temperatures of interest.

There are two different sorts of top-up involved in the provision of the final data, a top-up in J, the total angular momentum and a top-up in energy. For the most part, the collision strengths for the maximum J value (59/2) calculated explicitly are sufficient to ensure convergence. In Figs. 7-9 we show the convergence in J for three different types of transition at an energy of 118 Ryd. For the 2s²2p^{2 3}P $^{\rm e}_0 - 2$s2p^{3 5}S$^\circ _2$ forbidden transition in Fig. 7, convergence is rapid even at this high energy. The 2s²2p^{2 3}P $^{\rm e}_0 - 2$s2p^{3 3}D$^\circ _1$allowed transition in Fig. 8 on the other hand varies very slowly but as can be seen in the next paragraph good results may be obtained using the Coulomb-Bethe approximation. Transitions of the type $J\rightarrow J$as illustrated in Fig. 9 for the 2s²2p^{2 3}P $^{\rm e}_0 - 2$s²2p²¹D $^{\rm e}_2$ are a real problem since they show a peak at relatively high J and a rather slow decline. This implies that the fraction of the cross section coming from the top-up is relatively large. In the present case, we have simply calculated partial wave contributions from a $J_{{\rm max}}$ high enough so that a geometric progression may be used. This phenomenon was also observed and commented on by Aggarwal (1991) and more recently by Eissner et al. ([1999]). Aggarwal set $J_{{\rm max}}$ to 29/2 which is just sufficient for an energy of 100 Ryd. At higher energies, as shown in the figure, his cross sections are not converged so that his top-up has a larger error for these few cases.

The top-up in energy has been made small by extending the calculation to large total energies. The error in this contribution is perhaps relatively large but since it only comprises a small correction at the temperatures of interest this is unimportant.

The allowed transitions have been "topped-up'' using a scheme similar to that devised by Burke and Seaton for LS-coupling, based on Coulomb-Bethe recursion laws for the collision strengths (Eissner et al. [1999]). It is accurate as long as it is carried out for values of J that are not too small or too large. For small values of J the Coulomb-Bethe approximation is not applicable while at large J values the recursion formulae become inaccurate. The collision strength displayed in Fig. 2 shows how effective this method is. Plotted is the reduced collision strength ( $\Omega(E)/\ln(E/\Delta E+e)$) versus the reduced energy ( $1-\ln(c)/\ln(E/\Delta E+c)$) as suggested by Burgess & Tully ([1992]). Here $\Delta E$ is the transition energy, E is the electron energy with respect to the reaction threshold and c is an adjustable scaling parameter. The change in scale compresses the range in energy from $0\rightarrow \infty$ to $0\rightarrow1$. The correct cross section should, on this scale, converge to the value $4gf/\Delta E$ at x=1 where gf is the weighted oscillator strength for the transition, which indeed it does.

The relativistic distorted wave calculations of Zhang & Sampson ([1996], [1997]) are also available for comparison. Overall the agreement for transitions among the n=2 states is excellent as is evidenced by Fig. 11. Here we compare all the n=2 data at an ejected electron energy of 95.3 Ryd. The same comparison for the n=2-3 common to the two calculations is made in Fig. 12. Here the agreement is poorer but there are no systematic differences. The few transitions where the discrepancies are larger are due to a few energy levels, for example the level number 36 in the present calculation, labelled j3 by Zhang & Sampson ([1997]). Here the level is strongly mixed with the 2p³3p ³F$^\circ$ state. Presumably the mixture is different in the Zhang and Sampson calculation. Such differences are bound to arise when configuration mixing is large. The question as to which value is more accurate can only be decided, if at all, by even more extensive calculations. Fortunately only a few of the more than 1300 transitions are affected so the problem should not be serious.

In summary we may say that the present results provide cross sections that are accurate to better than 20% for transitions involving only the n=2states. Collisional data for the 2p$3\ell$ are also tabulated but will be much less accurate, chiefly due to the absence of resonances converging to higher thresholds. A more complete set of data for the n=3 levels is provided by the work of Zhang & Sampson ([1996], [1997]) while data including the n=4 levels are to be found in the paper by Phillips et al. ([1996]). But note that the resonance contribution is lacking in both. Lastly, although oscillator strengths have been tabulated in Table 3 the values to be found in Froese Fischer & Saha ([1985]) are to be preferred since they have considered configuration interaction effects in much more detail.

Acknowledgements

The present calculations were carried out on the Cray T-90 and the Fujitsu VPP700 at the Leibniz-Rechenzentrum of the Bayerischen Akademie der Wissenschaften. The generous allocation of computer time and resources is gratefully acknowledged.