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2. Atomic calculations

The method of calculating electron-impact excitation collision strengths using the BP R-matrix approximation is described in detail in Paper I of this series (Hummer etal. 1993) and in Berrington etal. (1995). Basically the Hamiltonian of the (e + ion) system includes the relativistic terms, that is the mass-velocity term, the Darwin term and the spin-orbit interaction. Consequently, the energy term of the target ion is split into fine-structure levels specified by the total angular momentum tex2html_wrap_inline1060 and the parity tex2html_wrap_inline1062. An intermediate coupling representation is used in which tex2html_wrap_inline1064 is coupled to the free-electron orbital angular momentum l and spin s in the following way:
where K is an intermediate quantum number.

The target wavefunctions used in this calculation for Fe XXII are similar to those in Paper III, but with some improvements. Fifteen fine-structure energy levels dominated by tex2html_wrap_inline1072, tex2html_wrap_inline1074 and tex2html_wrap_inline1076 configurations and thirty n=3 levels of the tex2html_wrap_inline1080, tex2html_wrap_inline1082, tex2html_wrap_inline1084, tex2html_wrap_inline1086 and tex2html_wrap_inline1088 configurations are included. The energy levels of the configurations tex2html_wrap_inline1090, tex2html_wrap_inline1092, tex2html_wrap_inline1094 and tex2html_wrap_inline1096 were not included since the present 45-state calculation was already computationally very demanding. We note that this will not significantly affect the results for transitions to the n=2 levels. Table 1 (click here) lists the 45 levels and provides the key to the level indices used in tabulating the maxwellian-averaged collision strengths. The observed energies (Sugar & Corliss 1985) for all 15 n=2 levels and 6 available n=3 levels and the energies calculated here for all 45 levels are listed in Table 1 (click here) for comparison.

For collisional calculations, we include 24 total (e + ion) tex2html_wrap_inline1104 symmetries with tex2html_wrap_inline1106 for both even and odd parities. These should provide convergence of the partial wave contributions for the forbidden transitions, and the low energy part of the intercombination and the optically allowed transitions. At energies greater than the highest n=2 threshold (E=14.833 Ryd), the higher partial waves for optically allowed transitions are calculated in the Coulomb-Bethe approximation, as in most of our electron-impact excitation work (see for example, Paper III).

As is well known, coupling and resonance effects may be important in the electron-impact excitation collision strengths for some highly charged ions, as is the case for Fe XXII. In the present calculation we have included these effects, not only those due to the n=2 levels, but also those arising from the n=3 levels, with a rather large fine-structure target expansion. We have used a quantum-defect mesh of free-electron energies (see Paper III for more detail), with a maximum effective quantum number tex2html_wrap_inline1116 which defines the so-called "QDT" (quatum defect theory) region within which we consider radiation damping of resonances, i.e. for tex2html_wrap_inline1118. In order to avoid unnecessary computations where this effect is not important, we take tex2html_wrap_inline1120 for electron energies E in the range zero to the highest n=2 threshold (E=14.833 Ryd), and tex2html_wrap_inline1128 for E further up to 79 Ryd. With the quantum-defect mesh, when tex2html_wrap_inline1132, that is when electron energies approach each excitation threshold but less than tex2html_wrap_inline1134 (tex2html_wrap_inline1136 being the threshold energy), detailed resonance structures are resolved. When tex2html_wrap_inline1138, resonance averaged values are obtained (Gailitis averaging - see Paper III) for energies up to the threshold. In all, collision strengths were calculated for more than 10 000 energy points. We calculated collision strengths for tex2html_wrap_inline1140 Ryd using a coarser energy mesh since resonances are not important in this range.

The relativistic effects, which we included through the BP approximation in an ab initio manner, affect collision strengths mostly in two ways: 1) the energy splitting of a term into levels produces more complex resonance structures, and 2) the background collision strengths are affected through intermediate coupling of fine-structure levels for some transitions, mainly for the intercombination transitions.

As pointed out in Zhang & Pradhan (1995a), for highly charged ions, resonances may be subject to radiation damping. We include radiation damping by using the Bell-Seaton theory (Bell & Seaton 1985) of dielectronic recombination. As seen from Zhang & Pradhan (1995a), the radiation effect increases with the effective quantum number tex2html_wrap_inline1142 of the autoionizing states: it is negligible when the tex2html_wrap_inline1144's are small, less than 10 in the present case. We calculated detailed collision strengths with radiation damping, using Eq. (3) in Zhang & Pradhan (1995a), for tex2html_wrap_inline1146, and the averaged collision strengths with radiation damping using Eq. (4) for tex2html_wrap_inline1148 (the Gailitis averaging or QDT region). This was done for the resonances associated with the n=2 levels of the target ion.

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