4 Results

The resulting effective collision strengths for the electron-temperature range $10^5\leq T/{K}\leq 10^7$ are listed in Table 6. We have used an R-matrix basis of 25 continuum orbitals, adequate for electron collision energies up to 200Ryd. In the following sections each of the effects mentioned in Sect. 2 is discussed in detail.

4.1 Energy mesh

The low-energy regime of the collision strengths for a highly ionized system such as Fe XVI is dominated by series of very narrow resonances. It would be computationally expensive to calculate such cross sections with an energy mesh fine enough to resolve all these features. However a practical choice must ensure stable integration when the rates are calculated. This is illustrated in Fig.1, where the effective collision strength for the transition 3d$\,^2$D$_{5/2}-4{\rm s}\,^2$S_1/2 has been plotted when computed with different mesh sizes. By comparing results obtained with steps of $\delta E/z^2= 5.0 \ 10^{-6}$ Ryd and $\delta E/z^2=1.0 \ 10^{-5}$ Ryd (z=15 is the residual charge of the ion), it is concluded that the latter mesh size is sufficiently fine while a mesh with a step of $\delta E/z^2=5.0 \ 10^{-5}$Ryd can lead to significant differences at the lower temperatures. Therefore cross sections are computed in the energy region below the highest threshold ($4{\rm f}\ ^2{\rm F}^{\rm o}_{7/2}$) with a step of $\delta E/z^2=1.0$ 10^-5 Ryd. In the region where all channels are open a wider step of $\delta E/z^2= 1.0 \ 10^{-3}$Ryd is more than adequate.

  

Figure 1: Effective collision strength for the 3d$\,^2$D5/2-4s$\,^2{\rm S}_{1/2}$ transition in Fe XVI computed with different energy meshes. Crosses: $\delta E/z^2=5.0 \ 10^{-5}$ Ryd. Asterisk: $\delta E/z^2=1.0 \ 10^{-5}$Ryd. Circles: $\delta E/z^2= 5.0 \ 10^{-6}$Ryd. The residual charge of the system is $z\!=\!15$. It may be seen that the latter two meshes lead to stable integration throughout the whole temperature region

4.2 Relativistic effects

As mentioned in Sect. 2, relativistic contributions are taken into account by either of two methods: by diagonalizing the Hamiltonian in intermediate coupling using a Breit-Pauli approximation, or, with less effort, by calculating reactance matrices in LS coupling before transforming to pair coupling using algebraic coefficients and TCCs. It is found that, for most transitions, effective collision strengths computed with the two methods are in good agreement. However for some transitions, particularly those arising within a term, the differences at low temperatures can be sizable as shown in Fig. 2. This is mainly caused by the neglect of the term energy splittings in the TCC method; i.e. energy-degenerate channels give rise to significantly different resonance patterns. Therefore it is advisable to adopt the full Breit-Pauli approach whenever possible.

  

Figure 2: Effective collision strength for the 3p$\,^2$P$^{\rm o}_{1/2}-3$p$\,^2$P$^{\rm o}_{3/2}$ transition in Fe XVI showing a large difference at low temperatures when the relativistic contributions are taken into account with the TCC method (crosses) and a Breit-Pauli calculation (circles)

4.3 Multipole-potential coupling

By comparing effective collision strengths computed with Options 0 and 1 in the asymptotic codes (see Sect. 2) it is possible to estimate the contributions from the long - range multipole potentials in the asymptotic region. Whe-reas for most forbidden transitions these contributions are small, it is found essential to include them in allowed transitions. For instance, it is shown in Fig. 3 that for the allowed transition 4p²P$^{\rm o}_{1/2}-4$d$\,^2$D_3/2 such differences can be fairly large throughout the temperature range of interest. A similar conclusion can be drawn when comparing the first two columns on the left in Table 5 which concerns the transitions 3s$\sp2$S$_{1/2}-\,$3p$\sp2$P$^{\rm o}_{\rm J}$.However it is found that when Option 1 is used numerical instabilities can crop up, particularly in the region just below a new threshold, causing the occasional abnormally high resonance. In the present work such features are eliminated by plotting the ratio of the two results with Options 1 and 0 in the resonance region and trimming any feature with a ratio larger than a factor of 5.

  

Figure 3: Effective collision strength for the 4p$\,^2$P$^{\rm o}_{1/2}-4$d$\,^2$D3/2 transition in Fe XVI showing the large difference throughout the temperature range resulting from different treatments of the multipole potentials in the asymptotic region, see Sect. 4.3. Crosses: Option 0, Circles: Option 1

4.4 High-l top-up

Perhaps one of the most outstanding difficulties of the present calculation is the estimate of the contribution from the high partial waves. Collision strengths for optically allowed transitions must be topped up using the Burgess sum rules in some way. These rules can only be applied when the Bethe approximation without unitarization is valid. This means that the associated radial functions must have acquired their asymptotic forms. For bound orbitals this happens at some distance after their last point of inflection. For continuum orbitals one must go beyond their first point of inflection. Last points of inflection for the present target orbitals are listed in Table 1: $r_{\rm infl}=$ 0.9a₀ for M-shell electrons, 1.7a₀ for the N-shell. One can easily obtain an estimate of the first point of inflection of the continuum orbitals by solving the asymptotic form of the ID equations for high angular momenta and a range of energies. To an accuracy well within 10% one can say that

$$r_{\rm infl}\approx l/k\mbox{\qquad if\quad}E\gg z\ \mbox{Ryd\quad and\quad}l\gg 1\,.$$ (24)

These trends are illustrated in Table 4.

 

Table 4: First point of inflection $r_{\rm infl}/a_0$of selected partial waves associated with M-shell and N-shell channels for 2 energies measured from the ground state; the respective channel energies for $t\!\gt\!1$follow from Table 2 on averaging over fine structure, as t labels a term rather than a level i. Compare with $r_{\rm infl}$ from Table 1: 0.9a₀ and 1.7a₀

 

  

Table 5: $\Omega($3s$\sp2$S$_{1/2},\,$3p$\sp2$P$^{\rm o}_{\rm J})$ in intermediate and in LS coupling: Columns denoted 0¹⁷ and 1¹⁷ give results without and with multipole coupling respectively. The following 3 columns test the validity of the top-up formula when CC calculations go out to J=17, 15 and 13 respectively. The lowest collision energy in the Table lies just above the highest target threshold

 

Table 6: Present effective collision strength $\Upsilon_{ij}(T)$ for the electron impact excitation of Fe XVI

 

 

Table 6: continued

It is borne out by Table 5 that a ratio of two for the two competing radii gives acceptable results. Convergence is excellent once the first point of inflection of the partial waves appears at three times the radius of the last point of inflection of the respective target orbital. This condition is satisfied only at the first two energies for the M-shell transitions. For transitions involving electrons with principal quantum number $n\!=\!4$ though this criterion is matched only at much higher values of angular momentum l. In the present work we had to cut short the expansion with respect to angular momenta at $J^{\rm oa}_{\max}\!=\!40$, fine at 100Ryd but somewhat tight when approaching 200Ryd (see Table 4). It suffices for most transitions in the region $E\leq 200$Ryd, although for one or two of the more difficult cases incorrect high-energy tails required truncation at the breakdown point. This situation is illustrated in Fig. 4 with the allowed transition 4p$\,^2$P$^{\rm o}_{3/2} -4$d$\,^2$D_5/2, where the collision strength is plotted using the scaling method of Burgess & Tully (1992). It may be seen that the reduced collision strength at high energies correctly approaches $\Omega_{\rm r}(1)$, but there is a point where this trend breaks down. This problem can certainly be alleviated by increasing $J^{\rm oa}_{\max}$ -- at serious computational cost. The slow partial wave convergence in some quadrupole transitions is somewhat similar but not as acute; with a value of $J^{\rm na}_{\max}\!=\!40$ and a geometric series top-up such transitions are accurately treated.

  

Figure 4: Reduced collision strength plotted as a function of the scaled energy of Burgess & Tully (1992) (see Sect. 2) for the $4{\rm p}\ ^2{\rm P}^{\rm o}_{3/2}-4{\rm d}\ ^2{\rm D}_{5/2}$ optically allowed transition in Fe XVI showing the approach towards the high-energy limit (filled circle). It may be seen the breakdown that takes place at the higher energies due to an insufficiently high $J^{\rm oa}_{\max}$ (in this calculation $J^{\rm oa}_{\max}=40$)

4.5 Comparison with previous work

The close-coupling calculation performed by Tayal (1994) for Fe XVI is very similar to the present. A striking difference lies in his treatment of the high partial waves, where for the optically allowed transitions a cut-off value of $J^{\rm oa}_{\max}=15$ was adopted and a geometric series top-up was then implemented. For the forbidden transitions, on the other hand, Tayal considered the total collision strengths converged for $J^{\rm na}_{\max}=15$. By contrast we find in the present work that reliable top-up procedures for some transitions, specially within n=4 as discussed above, could only be safely introduced at much higher values of $J^{\rm oa}_{\max}$ and $J^{\rm na}_{\max}$,namely $J^{\rm oa}_{\max}=J^{\rm na}_{\max}\sim 40$. The convergence of some quadrupole transitions was found to be unusually slow. Tayal lists collision strengths at 5 energies in the non-resonant region (22.5, 36.0, 49.5, 67.5 and 90.0 Ryd) and effective collision strengths in the electron-temperature range $8\ 10^5\leq T/{K}\leq 6\ 10^6$, thus facilitating a thorough comparison with present results. Cornille et al. (1997) have computed collision strengths for the fine-structure transitions with $n\leq 5$ at 4 energy points in the non-resonant region (26, 50, 100 and 200 Ryd). A distorted wave method with TCC recoupling is used for $L^{\rm oa}_{\max}=L^{\rm na}_{\max}=19$ at 26 and 50 Ryd, and $L^{\rm oa}_{\max}=L^{\rm na}_{\max}=24$ at 100 and 200 Ryd. For allowed transitions the Coulomb-Bethe top-up of Burgess & Shoerey (1974) is used in the range $L^{\rm oa}_{\max}<L\leq 200$.The 3s-nd quadrupole transitions are topped-up for $L\gt L^{\rm na}_{\max}$ with the program NELMA (Cornille et al. 1994) based on a distorted wave approximation without exchange.

It is found that 85% of the collision strengths listed by Tayal (1994) agree with present results to within 10%. Large differences (up to 70%) are found, however, for optically allowed transitions with large collision strengths, in particular within the n=4 terms (e.g. 4s-4p, 4p-4d). This situation is clearly illustrated in Fig. 5 with the $4{\rm s}\ ^2{\rm S}_{1/2}-4{\rm p}\ ^2{\rm P}^{\rm o}_{1/2}$ transition; it may be seen that the reduced collision strengths by Tayal show an increasing departure from the expected approach towards the high-energy limit. This finding seems to indicate that his geometric series top-up for allowed transitions can be unreliable at the high energies. On the other hand only 76% of the collision strengths by Cornille et al. (1997) are within the 10% level of agreement with the present data. Larger differences are mainly found towards the higher energies for the quadrupole transitions not arising from the ground state (i.e. 3d-4s and n=n'=4) that have not been topped by Cornille et al. beyond $L^{\rm na}_{\max}$. In Fig. 6 we show two cases where discrepancies at E=200 Ryd are greater than 30%; in the transition 3d$\,^2$D_3/2-4f$\,^2$F$^{\rm o}_{7/2}$ it is seen that the agreement is excellent except for the value at 200 Ryd, which is 50% higher; for the 4s$\,^2$S$_{1/2}-4{\rm d}\,^2$D_3/2 the situation is similar, but the high energy point is now 30% lower. The latter pattern is also found in the following transitions: 4s$\,^2$S$_{1/2}-4{\rm d}\,^2$D_5/2; 4p$\,^2$P$^{\rm o}_{1/2}-4$p$\,^2$P$^{\rm o}_{3/2}$ and 4p$\,^2$P$^{\rm o}_{1/2}-4$f$\,^2$F$^{\rm o}_{5/2}$.

  

Figure 5: Reduced collision strength for the $4{\rm s}\ ^2{\rm S}_{1/2}-4{\rm p}\ ^2{\rm P}^{\rm o}_{1/2}$ optically allowed transition in Fe XVI. Solid curve: present results. Crosses: Tayal (1994). Filled squares: Cornille et al. (1997). Filled circle: high-energy limit. The departure from the expected approach to the high-energy limit observed in the values by Tayal are believed to be due to an unreliable geometric series top-up

  

Figure 6: Comparison of present collision strength (continuous curve) with those computed by Cornille et al. (1997) (filled squares). As shown for the transitions 4-12 (3d$\,^2$D3/2-4f$\,^2$F$^{\rm o}_{7/2}$)and 6-9 ($4{\rm s}\ ^2{\rm S}_{1/2}-4{\rm d}\ ^2{\rm D}_{3/2}$), the larger discrepancies are found for the high-energy point at 200 Ryd

  

Figure 7: Comparison of present effective collision strength (circles) with those computed by Tayal (1994) (filled squares). Although good agreement is found for most transitions, there are cases showing large discrepancies: for instance transition 1-6 (3s$\,^2$S$_{1/2}-4{\rm s}\,^2{\rm S}_{1/2}$) and transition 4-5 ($3{\rm d}\,^2{\rm D}_{3/2}-3$d$\,^2$D5/2)

A comparison of the present effective collision strengths with those tabulated by Tayal (1994) in the electron-temperature range $8\ 10^5\leq T/{K}\leq 6\ 10^6$ results in only 61% of the data agreeing to within 10% (82% to within 20%). In Fig. 7 we show two transitions with significant differences: 3s$\,^2$S_1/2-4s$\,^2$S_1/2 (up to a factor of two) and 3d$\,^2$D$_{3/2}-3{\rm d}\,^2{\rm D}_{5/2}$ (37%). Regarding the former, the high values at the lower temperatures listed by Tayal are due, in our opinion, to non-physical resonances caused by the numerical instabilities discussed in connection with the use of Option 1. The differences found in the transition within the 3d$\,^2\rm D$ term are more difficult to explain. Considerable differences are also found for 3d$\,^2$D_5/2-4s$\,^2$S_1/2 and for transitions with small (< 0.01) effective collision strengths.