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 and the parity . An intermediate coupling
representation is used in which 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 , and configurations and thirty n=3 levels of the , , , and configurations are included. The energy levels of the configurations , , and 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) symmetries with 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 which defines the so-called "QDT" (quatum defect theory) region within which we consider radiation damping of resonances, i.e. for . In order to avoid unnecessary computations where this effect is not important, we take for electron energies E in the range zero to the highest n=2 threshold (E=14.833 Ryd), and for E further up to 79 Ryd. With the quantum-defect mesh, when , that is when electron energies approach each excitation threshold but less than ( being the threshold energy), detailed resonance structures are resolved. When , 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 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 of the autoionizing states: it is negligible when the '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 , and the averaged collision strengths with radiation damping using Eq. (4) for (the Gailitis averaging or QDT region). This was done for the resonances associated with the n=2 levels of the target ion.