A&A 407, 769-778 (2003)
DOI: 10.1051/0004-6361:20030874
K. M. Aggarwal - F. P. Keenan
Department of Pure and Applied Physics, The Queen's University of Belfast, Belfast BT7 1NN, Northern Ireland, UK
Received 17 March 2003 / Accepted 22 May 2003
Abstract
Collision strengths for transitions among the energetically lowest 46 fine-structure levels belonging to the (1s^{2}) 2s^{2}2p^{2},
2s2p^{3}, 2p^{4}, and 2s^{2}2p3
configurations of Ca XV are computed, over a wide electron energy range below 300 Ryd,
using the Dirac Atomic R-matrix Code (DARC) of Norrington & Grant (2003). Resonances in the threshold region have been resolved in a fine energy mesh, and excitation rates are
determined over a wide electron temperature range below 10^{7} K. The results are compared with those available in the literature, and the accuracy of the data is assessed.
Key words: atomic data - atomic processes
Calcium is an abundant element in the solar corona and chromosphere. Its emission lines have been observed from many ionization stages, and lines from Ca XV have particularly been useful as density diagnostics of solar flares - see, for example, Keenan et al. (1992). Additionally, its lines are also useful in laboratory and laser plasmas. Therefore, atomic data for energy levels, radiative rates, collision strengths, and rate coefficients are always in demand.
Earlier work on this ion has been performed by many workers. The prominent among these are the R-matrix calculations of Aggarwal (1992), and the Distorted-Wave (DW) results of Bhatia & Doschek (1993) and Zhang & Sampson (1996). However, the DW calculations have only been performed for values of at a few energies above thresholds, from which accurate values of rate coefficients cannot be determined, because resonances in the threshold region have not been delineated. It is well established by now that the closed-channel (Feshbach) resonances considerably enhance values of rate coefficients, even at a temperature of 10^{6.6} K, where Ca XV has its maximum ionization abundance (Mazzotta et al 1985). Additionally, the calculations of Zhang & Sampson are limited to transitions among the lowest 20 fine-structure levels of the (1s^{2}) 2s^{2}2p^{2}, 2s2p^{3} and 2p^{4} configurations, whereas Bhatia & Doschek have also included the additional 26 fine-structure levels among the (1s^{2}2s^{2}2p) 3s, 3p and 3d configurations. Similarly, our earlier calculations are also confined to the lowest 20 fine-structure levels, whereas data are also required among higher levels for diagnostic studies of solar and laboratory plasmas. Moreover, in spite of including the resonances for the computation of rates, the earlier R-matrix calculations suffer from three major deficiencies. Firstly, calculations for were restricted to partial waves with angular momentum . This affects the accuracy of the derived values of , especially at higher energies, as has already been demonstrated in our previous paper (Aggarwal & Keenan 2002). Secondly, values of were computed up to an energy of 200 Ryd only. This affects the calculations of , especially at higher temperatures. Finally, the energy mesh adopted in our earlier calculations was comparatively coarse ( Ryd). This affects the accuracy of values, particularly for the forbidden transitions for which resonances are more important. All of these restrictions were due to the computational limitations at that time. In our present work, we are extending the partial wave range up to J = 40.5, and the energy range up to 300 Ryd. Furthermore, to delineate resonances our present energy mesh is 0.002 Ryd. Therefore, apart from computing our results for a larger number of transitions among the lowest 46 fine-structure levels of Ca XV, we are attempting to make a significant overall improvement over our earlier results.
For the generation of wavefunctions, we have adopted the GRASP (General-purpose Relativistic Atomic Structure Program) code of Dyall et al. (1989), which is a fully relativistic code. Additionally, configuration interaction (CI) among the above listed six configurations was also included, and results for energy levels, radiative rates and oscillator strengths for transitions in Ca XV have already been reported and discussed (Aggarwal et al 1997). In Table 1 we list our energy levels for a ready reference and also provide an index for each level for further discussion of results. The experimentally compiled energies of Sugar & Corliss (1985) are also listed in this table. It may be noted that the experimental energies are not available for all the levels, but the agreement between the common experimental and theoretical level energies is within 2%, which is highly satisfactory. However, our 2s2p^{35}S level energy is lower than the experimental one by 4%.
Table 1: Target levels of Ca XV and their threshold energies (in Ryd).
For computations of , we have employed the fully relativistic Dirac Atomic R- matrix Code (DARC) of Norrington & Grant (2003). This program includes the relativistic effects in a systematic way, in both the target description and the scattering model. However, because of the inclusion of fine-structure in the definition of channel coupling, the matrix size of the Hamiltonian increases substantially, making the calculations computationally quite expensive. The R- matrix boundary radius has been taken to be 3.0 au, and 23 continuum orbitals have been included for each channel angular momentum, for the expansion of the wavefunction. This allows us to compute up to an energy of 300 Ryd, more than sufficient for the calculation of accurate excitation and de-excitation rate coefficients for temperatures up to 10^{7} K. The maximum number of channels for a partial wave is 178, and the corresponding size of the Hamiltonian matrix is 4106. In order to obtain converged at all energies (especially the higher ones) and for all transitions (particularly the allowed ones), we have included the contribution of all partial waves with angular momentum 40.5. Although for most of the transitions have converged due to the inclusion of such a large range of partial waves, there are some allowed transitions for which even this large range is not sufficient for convergence. Therefore, to take account of higher neglected partial waves, a top-up based on the Coulomb-Bethe approximation for allowed transitions, and geometric series for other remaining transitions has been included.
Since comparison of our collision strengths with the other available data of Aggarwal (1992), Bhatia & Doschek (1993) and Zhang & Sampson (1996) has already been discussed in our previous paper (Aggarwal & Keenan 2002), we focus our attention on the results of excitation rates in the following section.
(1) |
(2) |
(3) |
Since the threshold region is dominated by numerous resonances,
must be computed in a fine mesh of energy. Close to thresholds our mesh is 0.001 Ryd and is
0.002 Ryd in the remaining range. In total, values of
have been computed at over 21 700 energies in the threshold region below 41 Ryd.
This fine energy mesh ensures to a large
extent that neither a majority of resonances are missed, nor do the exceptionally high resonances have unreasonably large width. In Figs. 1-3 (a and b) we show resonances
for only three
transitions, namely (2s^{2}2p^{2}) ^{3}P_{0}-^{3}P_{1} (1-2), ^{3}P_{1}-^{3}P_{2} (2-3), and ^{1}D_{2}-^{1}S_{0} (4-5), respectively. These are the same transitions
for which resonances have also been shown in our earlier work (Aggarwal 1992), and hence facilitate a ready comparison, as well as provide a good idea about their density
and importance. In Figs. 1a-3a the energy and
scale are exactly the same as in Figs. 1-3 of Aggarwal (1992), but Figs. 1b-3b show values
of
in a wider energy region below 40 Ryd.
Figure 1: Collision strengths () for the (2s^{2}2p^{2}) ^{3}P_{0}-^{3}P_{1} (1-2) transition of Ca XV. a) Results below 2 Ryd and b) results in entire threshold region up to 40 Ryd. | |
Open with DEXTER |
Figure 1: Collision strengths () for the (2s^{2}2p^{2}) ^{3}P_{0}-^{3}P_{1} (1-2) transition of Ca XV. a) Results below 2 Ryd and b) results in entire threshold region up to 40 Ryd. | |
Open with DEXTER |
Table 2: Comparison between present (RM1) and earlier (RM2: Aggarwal 1992) results of effective collision strengths for transitions within the levels of the (1s^{2}2s^{2}2p^{2}) ground configuration of Ca XV. ( ).
In Table 2 we compare our present (RM1) and earlier (RM2: Aggarwal 1992) results for
for transitions among the 1s^{2}2s^{2}2p^{2} ground configuration of Ca XV, at
three temperatures, i.e. 10^{5}, 10^{6}, and 10^{7} K. It is surprising to note differences of a factor of two for many transitions (such as: 1-5 and 3-5), and over almost the
entire temperature range. Since differences between our present and the past calculations are the highest (up to a factor of four at
K) for the ^{3}P_{0}-^{3}P_{2} (1-3) transition, we have a closer look at the
values for this. In Figs. 4a,b we demonstrate its
values in the entire threshold region.
As seen in Fig. 4a, at energies below 7 Ryd (10^{6} K), this transition has many resonances whose magnitude are considerably high compared to its background value
(0.02). The higher thresholds range also demonstrates resonances for this transition as shown in Fig. 4b, but their density and magnitude are comparatively lower.
However, the magnitude of
at energies above thresholds are nearly the same (within 10%) in both of our calculations. Due to a coarser energy mesh
adopted in our earlier calculations, as explained in Sect. 1, the contribution of some of these resonances has obviously been overestimated. For example, in our earlier
calculations,
was computed at 2900 energies in the range below the 2p^{4} ^{1}S_{0} threshold (12.57 Ryd), whereas in the present calculations
we have computed
at 6700 energies in the same energy range. This and other improvements made in the present
calculations have significantly affected the earlier values of
for many transitions. Some transitions have been affected more towards the lower end of the temperature
range (such as 1-2), some at higher temperatures (such as 4-5), and some at all temperatures (such as 1-3 and 3-5). The transitions which have particularly been affected
(up to an order of magnitude) are
those whose upper levels belong to the 2p^{4} configuration (i.e. 16-20). This is because resonances arising from the n = 3 levels were not included in the earlier work.
For the same reason our present results may be underestimated for transitions with higher upper levels, because resonances arising from the n = 4 levels are
not yet accounted for.
Figure 2: Collision strengths () for the (2s^{2}2p^{2}) ^{3}P_{1}-^{3}P_{2} (2-3) transition of Ca XV. a) Results below 2 Ryd and b) results in entire threshold region up to 40 Ryd. | |
Open with DEXTER |
Figure 2: Collision strengths () for the (2s^{2}2p^{2}) ^{3}P_{1}-^{3}P_{2} (2-3) transition of Ca XV. a) Results below 2 Ryd and b) results in entire threshold region up to 40 Ryd. | |
Open with DEXTER |
Figure 3: Collision strengths () for the (2s^{2}2p^{2}) ^{1}D_{2}-^{1}S_{0} (4-5) transition of Ca XV. a) Results below 4 Ryd and b) results in entire threshold region up to 40 Ryd. | |
Open with DEXTER |
Figure 3: Collision strengths () for the (2s^{2}2p^{2}) ^{1}D_{2}-^{1}S_{0} (4-5) transition of Ca XV. a) Results below 4 Ryd and b) results in entire threshold region up to 40 Ryd. | |
Open with DEXTER |
Figure 4: Collision strengths () for the (2s^{2}2p^{2}) ^{3}P_{0}-^{3}P_{2} (1-3) transition of Ca XV. a) Results below 7 Ryd and b) results in entire threshold region up to 40 Ryd. | |
Open with DEXTER |
Figure 4: Collision strengths () for the (2s^{2}2p^{2}) ^{3}P_{0}-^{3}P_{2} (1-3) transition of Ca XV. a) Results below 7 Ryd and b) results in entire threshold region up to 40 Ryd. | |
Open with DEXTER |
In Table 3 we list our values of for transitions from the lowest 5 levels to higher excited levels over a wide electron temperature range of 10^{5} to 10^{7} K. These transitions are the most important and the temperature range included is suitable for applications in a wide variety of astrophysical, laser and fusion plasmas. The indices adopted to represent a transition have already been provided in Table 1. Our results of for the remaining 820 transitions among the 46 fine-structure levels of Ca XV are presented in Table 4. We hope the presently reported results will be helpful in understanding plasma diagnostics.
We do not see any apparent deficiency in our calculations, but past experience shows that accuracy estimates can easily be in error. Nevertheless, our calculated values of are expected to be accurate to better than 20% for a majority of transitions, especially those with upper levels below 36, i.e. 2p3d ^{3}F . This is because our calculations do not include resonances from the higher lying levels of n = 4. Similarly, results towards the lower end of the temperature range may have a comparatively lower accuracy because of the presence (or absence) of near threshold resonances. Additionally, this estimate is based on the presently adopted wavefunctions. As scope remains for improvement in our wavefunctions, the present values of may significantly vary, especially for weaker transitions, and hence may affect the accuracy of our results. However, the corresponding values of for stronger transitions are unlikely to vary by more than 20%.
Table 3: Effective collision strengths for transitions in Ca XV. ( ).
Acknowledgements
This work has been financed by the Engineering and Physical Sciences and Particle Physics and Astronomy Research Councils of the United Kingdom. A part of the computational work has been carried out on the MIRACLE Supercomputer at the HiPerSPACE Computer Centre at University College London, and we wish to thank Dr. Patrick Norrington for making his code available to us prior to publication.
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