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.

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