5 How high in energy does one need to go?

The effective collision strength ${\sl \Upsilon}$ was originally introduced by Seaton (1953) and is now almost universally used in place of the rate coefficient for presenting computational results. We call attention to the fact that the two are connected by a very simple relationship in which the factor ${\sl \Upsilon}$ should be replaced by ${\sl \Upsilon}$/$\sl \Phi$ when relativistic effects are taken into account (see Sect. 2.4). The integral defining ${\sl \Upsilon}$ extends from $E_j\, =\, 0$ to $E_j\, = \, \infty$. While we always integrate over the full range, the general practice is to replace the upper limit by the maximum value of E_j, which in the present case is 0.75 (Z₀ - 1.66)² (see ZSF). We have investigated whether this is high enough when calculating ${\sl \Upsilon}$ at temperatures where the ion in question is expected to have maximum abundance under conditions of coronal ionization equilibrium. We denote this temperature by $T_{\rm max}(Z_0)$.

   Table 7: ${\rm log}T_{\rm max}$: logarithm of the temperature at which a Li-like ion has maximum abundance under conditions of coronal ionization equilibrium as a function of atomic number Z₀. Results obtained from 5-point cubic spline fits (a) to Arnaud & Rothenflug's (1985) data points, and (b) also including the data point from Fournier et al. (1997)

Arnaud & Rothenflug's (1985) tabulation allows us to obtain $T_{\rm max}(Z_0)$ for 13 ions in the range $6 \le Z_0 \le 28$. A result for Z₀ = 42 is also available from the molybdenum ion fractions calculated and plotted by Fournier et al. (1997). Their Fig. 1a shows that $\rm Mo^{+39}$ has maximum abundance at a temperature corresponding to $kT_{\rm max} = {\rm 1.1 \ 10^4\, eV}$. Dr. Fournier (private communication) has informed us that the precise value is 10720 eV. We use the program OmeUpZ to plot and extrapolate the reduced quantity ${\rm log}\,T_{\rm max}(Z_0)\,/\,{\rm log}(Z_0)$ as a function of $Z_{\rm r} = Z/(Z+C_Z)$. We have made two fits, one including the data point for $\rm Mo^{+39}$ and the other excluding it. A graphical comparison of the spline fits and original data points is shown in Fig. 2. The fit which makes use only of the data from Arnaud & Rothenflug (1985) is a smooth monotonically decreasing curve which tends to 4.8090 at $Z_{\rm r} = 1$. From this fit we estimate that $T_{\rm max}(42) = 9.47 \ 10^7$, which is 40% lower than the temperature calculated by Fournier et al. (1997). A set of results obtained along the sequence from both fits is presented in Table 7. The most highly charged ion in the sequence considered by ZSF is $\rm U^{+89}$. According to Table 7 this ion has maximum abundance at a temperature of either $\rm 3.0 \ 10^9$ or $\rm 8.2 \ 10^9$. We obtain the higher temperature when we include the data point from Fournier et al. (1997) in the fitting process.

  

Figure 1: Reduced collision strength ${\sl \Omega}_{\rm rr} ({\rm 2s - 3s})$ for CE = 3.8 (rms error = 0.004%) and CZ = 30 (rms error = 0.15%). + points and full curves show data and spline fits for Z0 = 8, 10, 12, 14, 16, 18, 20, 22, 26. The dotted mesh shows the double fit for $3 \leq Z_0 \leq 92$

  

Figure 2: $Y_{\rm r} = {\rm log_{10}}T_{\rm max}/{\rm log_{10}}(Z+3)$ against $Z_{\rm r} = Z/(Z+9)$. + Arnaud & Rothenflug (1985) data points; * Fournier et al. (1997) data point. Full curve, fit to + data points only; dashed curve, fit to both sets of data

For the $\rm 2s \to 3s$ transition the value of ${\sl \Upsilon}(Z_0 = {\rm 92})$ at $T = {\rm 3.0 \ 10^9}$ is underestimated by more than 90% if the contribution from energies above the highest one considered by ZSF is neglected. With decreasing Z₀ the value of $T_{\rm max}(Z_0)$ falls and so does the high energy contribution. Our spline fit predicts $T_{\rm max}(36) = 6.1 \ 10^7$ and at this temperature the high energy contribution to ${\sl \Upsilon}$ is only of the order of 10%.

Our conclusion is that although $E_{j}\, = \, 0.75 (Z_0-1.66)^2$ is high enough for ions at the lower end of the sequence this is not so at the top end. Consequently one really needs to extrapolate some of ZSF's data, or interpolate them if the high energy limit points are known.

Acknowledgements

We have benefitted from many discussions over the Internet with Dr. Hong Lin Zhang (U.S.A.) and thank him for providing us with the radial orbitals of $\rm U^{+89}$. Drs. Monique Arnaud (France) and Takako Kato (Japan) were instrumental in drawing our attention to the calculations of Fournier, Pacella, May, Finkenthal and Goldstein on $\rm Mo^{+39}$. Marita C. Chidichimo is grateful for the support she has received from the Natural Sciences and Engineering Research Council of Canada (NERSC).