4 Revised collision strength

Since there is good agreement between the 24-state results by IP-X and those from the 5-state calculation by Johnson & Kingston (1990) for electron temperatures below 24000 K, we have rerun the calculation with the original IP-X target representation. Since we are interested in the high-temperature behaviour of the effective collision strength in order to check the match with the Coulomb-Born limit quoted by BCT, we now extend the calculation of collision strengths to 100 Ryd. Convergence of the partial-wave expansion is ensured by examining the partial collision strength L profile at the highest energy point (100 Ryd) and by topping up with a simple geometric series procedure at every energy point. In Fig. 1 we plot the partial collision strength as a function of L at both 10 Ryd and 100 Ryd. This plot illustrates the computational difficulty in obtaining a converged collision strength for this transition particularly as the energy is increased. While at 10 Ryd there is a well defined peak at around L=4, at 100 Ryd the expansion becomes a flat and broad plateau for $5\leq L \leq 20$. Using a geometric series top-up in this L range would certainly lead to a significant overestimate of the total collision strength at the high energies. At 100 Ryd, say, it is only for L>30 that the top-up procedure can be safely implemented, and the latter amounts to a barely acceptable 20% of the total collision strength. In Fig. 2 the total collision strength for this transition is plotted using the reduced energy method of BT. A further difficulty with this transition becomes apparent: for the non-resonant region the collision strength, rather than flattening out as in most quadrupole transitions, displays a slow increase. As discussed before, this effect makes the management of the convergence of the partial wave expansion particularly difficult at the high energies. It also causes difficulties in determining the high-temperature trend of the effective collision strength as will be shown below. The present reduced collision strength seems to converge to the point $\Omega_{\rm r}(1)\approx 1.4$ which is in reasonable agreement ($\sim\!15$%) with the high-energy Coulomb-Born limit of 1.64 estimated by BCT.

 

Figure 1: Partial collision strength for the ${\rm 3s}\sp2{\rm 3p}\sp4\ \sp1$D $-\sp1$S transition in Ar III as a function of the total orbital angular momentum L. Squares: 10 Ryd. Circles: 100 Ryd

 

Figure 2: Total collision strength for the ${\rm 3s}\sp2{\rm 3p}\sp4\ \sp1$D $-\sp1$S transition in Ar III plotted as a function of the reduced energy with C=9.0. It is seen that the non-resonant region increases slowly and approaches the point $\Omega_{\rm r}(1)\approx 1.4$

 

Figure: Effective collision strength for the ${\rm 3s}\sp2{\rm 3p}\sp4\ \sp1$D $-\sp1$S transition in Ar III as a function of the reduced temperature with a) C=0.2 and b) C=5.0. Circles: present calculation. Squares: IP-X. Crosses: Johnson & Kingston (1990). Filled square: High-energy Coulomb-Born limit by BCT

In Fig. 3 the present reduced effective collision strength is compared with earlier work, and in Table 1 we list the present effective collision strength in the extended range of $3\leq\log(T)\leq 7$ and compare them with the data by IP-X. It may be seen that the low-temperature regime is dominated by the contribution from resonances (see Fig. 2), particularly a resonance sitting at threshold that is responsible for the high value of $\Upsilon_{\rm r}(0)=1.82$.

 

Table 1: Comparison of present effective collision strengths for the ${\rm 3s}\sp2{\rm 3p}\sp4\ \sp1$D $-\sp1$S transition in Ar III with those in IP-X. Electron temperatures are given in K. It may be seen that discrepancies are not larger than 25%

The good overall agreement (10%) at low temperatures with the work by Johnson & Kingston (1990) is reinforced. It is shown that the differences with IP-X, mainly due to the neglect of partial collision strengths for L>9 at the higher energies, are less than 10% up to $\log(T)=4.6$, they are well below 20% up to $\log(T)=4.8$ and they reach $\sim\!25$% only towards the upper limit of the temperature range considered in IP-X. In our opinion, such differences are consistent with the level of accuracy ($\sim\!20$%) claimed in IP-X for the lower members of the Si and S isoelectronic sequences. Furthermore, it is found that the fairly steep climb of the reduced effective collision strength to its high-temperature limiting value of $\Upsilon_{\rm r}(1)\approx 1.4$ only starts at relatively high temperatures, and certainly stands out from the gently oscillating patterns at intermediate temperatures. As shown in Fig. 3, the characteristic features in each temperature regime can be enhanced by a suitable choice of the scaling parameter C. For instance, in Fig. 3b we have used a C parameter similar to that used by BCT which stretches the low-temperature regime suggesting a value of $\Upsilon_{\rm r}(1)=0.8$ for IP-X; with this choice of scaling parameter the high-temperature behaviour of the present data appears almost as a vertical climb. It is worth mentioning that an estimate of the effective collision strengths for T>10^6.5 K requires an energy range greater than 100 Ryd. Therefore a top-up procedure was introduced where the collision strength for E>100 Ryd was assumed constant at $\Omega=1.40$.