We calculate the the bound-bound and bound-free radiative data for C + states from a C 2+ scattering problem. The starting point for the calculations is the C 2+ target. In our photoionization calculations, as in the OP calculations (Yu Yan & Seaton ), the C 2+ target given by Berrington et al. () was used. It includes the 2s2 1Se, 2s2p 3Po, 1Po and 2p2 3Pe, 1De, 1Se target eigenstates represented in terms of seven orthogonal basis orbitals and pseudo-orbitals: 1s, 2s, 2p, s, p, d, f. The radial parts of the basis orbitals were obtained using the CIV3 code of Hibbert ().
We have calculated ionization energies for all doublet states with total orbital angular momentum quantum number and with principal quantum number . These energies are compared with existing experimental energies in Tables 1 and 2. Note that the main 2s2nl series states are perturbed by states from the 2s2p3s and 2s2p3p electron configurations. The upper principal quantum number limit () of the present calculation was chosen to include the 2s2p3p 2S term which lies energetically between the 12s 2S and 13s 2S terms. This perturbation is not included in the OP calculation for C+, which only extends to n = 10. Such perturbations significantly modify the photoionization cross-sections and hence recombination coefficients for nearby states in the main series.
We have described most of the techniques that we use to calculate bound-bound radiative data elsewhere (Kisielius et al. ). Briefly, for , , all radiative transition probabilities are obtained from the R-matrix calculation referred to above. For most other transitions we use a combination of the Coulomb approximation and hydrogenic data, the methods and codes used having been described by Storey (). For transitions with and we use the following further technique. Using the R-matrix approach, photoionization cross-section data from a given lower term (Si Li) to a specific final symmetry (Sf Lf) can be computed for negative free electron energies, enabling us to obtain absorption oscillator strengths from the given lower term to high members of the 2s 2 nl (Sf Lf) series. This technique has the advantage over other approximate methods, such as the Coulomb approximation, that configuration interaction is fully accounted for in the wave function of the lower term. Cascading from high Rydberg states to the low-lying terms of the ion is therefore more accurately treated.
In many cases treated by the OP, the free electron energy mesh on which the photoionization cross-sections are calculated, was too coarse to accurately map narrow resonance features. This usually leads to significant contributions to the recombination coefficient being missing, since the resonance contribution is proportional to the area underneath the resonance. The photoionization cross-sections for C + generated for the OP (Yu Yan & Seaton ) have poor resolution in some of the resonance structure because they were based on a mesh in which points are equally spaced in effective quantum number relative to the next threshold (hereafter referred to as a quantum defect mesh). One hundred points were provided per integer change in effective quantum number and subsequently, this number was further reduced by removing any points that could be linearly interpolated from two adjacent points within some prescribed accuracy.
In contrast to the OP data, we use a variable step energy mesh for photoionization cross-sections that delineates all resonances to a prescribed accuracy. First, we employ quantum defect theory to determine resonance positions and widths using a coarse energy mesh. We then use this information to generate an electron energy mesh which maps the resonances to a chosen accuracy (Kisielius et al. ). In the third step we calculate photoionization cross-sections using the new variable step mesh.
|Figure 1: Comparison of the photoionization cross-section (in Mb) for 2s2p2 2D, calculated by the OP (top) and this work (bottom)|
In Fig. 1, we demonstrate the difference in resolution between an OP cross-section and one calculated by our method. The OP cross section consists of 199 points based on a quantum defect mesh whilst the latter is based on an energy mesh of 3281 points. In the OP cross-section, the resonance at 1.135 Ry consists only of 3 points, and the next one is missing completely. There is a significant difference in the peak heights of the resonances too.
|Figure 2: Comparison of the area (in Mb Ry) under the 2s2p2 2D cross-section. Top figure - OP data, bottom - present calculation|
The effects of correctly delineating the resonances are more clearly demonstrated in Fig. 2. We show the area under the cross-sections which is a more important measure of their contribution to the recombination coefficient. The total area under the 2s2p2 2D photoionization cross-section from the OP is , whereas our calculation gives for the same energy range.
This detailed treatment of the energy mesh was undertaken for the energy region from the 2s2(1S) limit up to 4/152 Ry below the 2s2p(3Po) threshold, since this region contains the main contribution to the recombination at the temperatures of interest. Above the 2s2p(3Po) threshold, a quantum defect mesh was used. A very high degree of accuracy is not required here so a step in effective quantum number was chosen that provided 1000 points per integer change in quantum number.
Finally, we note that we deal with radiative damping of resonances in the same way as described by Kisielius et al. (), in that any autoionizing states for which the radiative width is greater than the autoionization width are omitted from the list of resonances used to generate the high resolution energy mesh. As a result such resonances do not appear in the photoionization cross-sections and do not contribute to the recombination. In practice this means omitting all resonances associated with 2s2p(3Po)nh electron configurations, as well as the 2Ho terms from the 2s2p(3Po)ng configurations.
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