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Subsections
The Rmatrix technique, implemented in the Queen's
University Belfast Rmatrix suite of programs (Berrington et al. 1995),
has been used for the computation of the low partial wave
contributions to the total collision strengths , being the partial collision strength for transition
, with l the orbital angular momentum of the scattered
electron. For the calculation of collision strengths for electric
dipole transitions within the Rmatrix approach we included
contributions coming from partial waves as high as ,such that
 
(1) 
This will be shown to be a proper choice in order to ensure
convergence between the low l Rmatrix results and the high l CoulombBethe
(CBe) ones (see Sect.2.2). The expansion of the total wavefunction
for the Fe^{11+} target included the lowest 19 LS coupling terms
3s^{2}3p^{3} ^{4}S, ^{2}D, ^{2}P, 3s3p^{4}
^{4}P, ^{2}D, ^{2}P, ^{2}S, 3s^{2}3p^{2}3d
(^{3}P)^{4}F, (^{3}P)^{4}D, (^{1}D)^{2}F, (^{1}D)^{2}G, (^{3}P)^{2}P,
(^{3}P)^{4}P, (^{1}S)^{2}D (^{1}D)^{2}D, (^{1}D)^{2}S, (^{1}D)^{2}P,
(^{3}P)^{2}F, (^{3}P)^{2}D. In Table 1 we list all the
finestructure levels belonging to the 3s^{2}3p^{3}, 3s3p^{4},
3s^{2}3p^{2}3d configurations along with the relative observed energy values,
taken from Corliss Sugar (1982) and Jupen et al. (1993).
Italic
type in the column listing the observed energies indicates that
experimental values are not yet available for those levels. In these
cases theoretical energies computed with the atomic structure
program SUPERSTRUCTURE (Eissner et al. 1974;
Nussbaumer Storey 1978) were empirically corrected with the procedure described in
BMS. The theoretical energies listed in Table 1 have been obtained
with a 24 configuration atomic model including those configurations
with one electron in 4s, 4p, 4d and 4f orbitals of spectroscopic type
(set3 in BMS). The energies for the 19 target LS coupling
terms listed above were obtained as weighted averages
over the finestructure observed energies given in Table 1 and were
listed in Table 5 of BMS. The 19 target terms were
represented by configuration interaction (CI) expansions constructed
with a common set of radial waves, describing the radial charge
distribution of the 1s, 2s, 2p, 3s, 3p and 3d bound electrons. These
radial functions were obtained with SUPERSTRUCTURE adopting the 12
configuration basis described in BMS. A total of 16 continuum orbitals
was included in the calculation and the Rmatrix "box'' boundary was
set at 3.09 a.u. For the expansion of the collision
complex wavefunction we combined in all possible fashions one
scattering electron having spin s=1/2 and with the LS coupling target terms listed above. In such a way we
obtained a total of 121 intermediate states specified in Table 2.
The calculation was performed with a very fine energy mesh of 1.59375
10^{3}Ry in the closed channel region (2.55 to 5.5Ry), in order to
delineate the resonance structure of the collision strength below the
highest excitation threshold. In the open channel region (5.5 to
100Ry), on the contrary, a much coarser energy mesh was chosen, namely
0.5Ry between 5.5Ry and 20Ry, 1Ry between 20Ry and 50Ry,
and 5Ry between 50Ry and 100Ry. This proved to be sufficient to describe
the variation of the collision strengths as a function of energy.
Table 1:
Energy levels (cm^{1}) for the lowest three configurations in
FeXII. For values in italic see text

Table 2:
Intermediate states () included in the expansion
of the collision complex wavefunction

A topup procedure to account for the missing partial wave
contributions to coming from was necessary in the
open channel region in order to provide final reliable results. The
application of geometric series properties to the partial collision
strengths, discussed in BMS, proved to be inappropriate to the case of
strong electric dipole transitions. For these transitions we estimated
the contributions from partial waves with with the
CoulombBethe (CBe) approximation (van Regemorter 1960;
Burgess et al. 1970). The CBe method implies the "long range'' approximation,
where only the long range potential of the target is effective in the
interaction with the scattering electron. As
stated in van Regemorter (1960) the "long range'' approximation is
only valid for sufficiently high values of l, i.e. "distant''
encounters between the target and the colliding electron.
Burgess Tully (1978) define "distant'' as those partial waves with
 
(2) 
where z is the ion net charge, r_{0} the radius of the spherical
region containing the target charge distribution and k^{2} the
scattering electron energy in Ry. This formula gives a convenient
criterion to estimate the lowest "safe'' partial wave contribution to
the total collision strength which can be computed with the CBe
technique, given a specific target model and an energy range where
collision strengths are required. Our choice of for the CBe
partial wave contributions satisfied Eq. (2) and the
convergence of Rmatrix and CoulombBethe results for the strongest electric dipole
transitions is shown in Table 3. The convergence of the two methods
at Ry, for , is in all cases better than
. After toppingup the Rmatrix
collision strengths with the CBe contributions for , great
care has been taken in checking the correct convergence to the high
energy Bethe limits of the toppedup reduced collision strengths,
, as a function of reduced energy, , as defined by
Burgess Tully (1992).
Figures 2 to 5 show a few examples of plots of
vs. , in the open channel energy region, for selected
electric dipole transitions.

Figure 2:
Reduced collision strength vs. reduced energy for the
3s^{2}3p^{3}^{2}P 3s3p^{4}^{2}S_{1/2} optically allowed transition.
The asterisk shows the high energy Bethe limit 

Figure 3:
Reduced collision strength vs. reduced energy for the
3s^{2}3p^{3}^{2}P 3s^{2}3p^{2}3d(^{3}P)^{2}D_{5/2} optically allowed transition.
The asterisk shows the high energy Bethe limit 

Figure 4:
Reduced collision strength vs. reduced energy for the
3s^{2}3p^{3}^{4}S 3s3p^{4}^{2}P_{1/2} intercombination transition.
The asterisk shows the high energy Bethe limit 

Figure 5:
Reduced collision strength vs. reduced energy for the
3s^{2}3p^{3}^{2}D 3s^{2}3p^{2}3d(^{3}P)^{4}F_{3/2} intercombination transition.
The asterisk shows the high energy Bethe limit 
Table 3:
Partial Rmatrix (Rm) and CoulombBethe (CBe) collision strengths
for at Ry

The assumption of constant at in
integrating the collision strengths over a
Maxwellian distribution of
electron energies, well justified in BMS for the forbidden
transitions, is not appropriate to the strong electric dipole
transitions we are treating here. In fact Burgess Tully (1992)
classify these transitions as type1 and give a high energy limiting
behaviour of the kind
 
(3) 
deduced from the Bethe approximation. By using the scaling rules
in Eqs.(5) and (6) of Burgess Tully (1992)
and Eq.(7) of
Burgess Tully (1978), it can be shown that, in the limit of high
E, is a linear function of . We therefore scaled our
collision strengths as suggested in Burgess Tully (1992)
for type1 transitions and performed linear interpolation of between and , given by
the Bethe limit , where is the
statistical weight of the initial level, f_{ij} the oscillator
strength and E_{ij} the excitation energy of the transition
. By applying the inverse transformation to these
interpolated values we obtained a set of values between
100Ry and 9900Ry, with an energy mesh of 100Ry, which takes
proper account of the logarithmic increase of with E. The
integration of the extended set of collision strengths is discussed
in the following section.
Toppedup final collision strengths for
all electric dipole finestructure transitions between the ground
3s^{2}3p^{3} and the first excited 3s3p^{4} configurations are compared with
previous calculations in
Table 4 at two energy values in the open channel
region. Table 5 shows a similar comparison for all electric dipole
transitions between the ground 3s^{2}3p^{3} and the second excited 3s^{2}3p^{2}3d
configurations. The complete set of collision strength values at all
electron energies considered in our scattering calculation (see
Sect.2.1), in the closed as well as open channel energy region, can
be obtained in electronic format via anonymous ftp from the IRON
Project data bank at iron.am.qub.ac.uk.
Effective, or thermally averaged collision strengths, given by
 
(4) 
where E_{j} is the colliding electron kinetic energy after excitation,
have been obtained by integrating the collision strengths using the
linear interpolation technique described in
Burgess Tully (1992). The extended set of original plus interpolated
values was used in the integration
process and it was assumed .The exponential factor in
Eq. (4) drops off rapidly with increasing energy so
that the contribution to coming from
the energy region , where is taken
constant, decreases rapidly with increasing . For example, for
K, this contribution was as high as for
, for most electric dipole transitions we
studied, dropping down to a typical (!) when the extended set
of 's is integrated (). Effective
collision strengths for all electric dipole 3s^{2}3p^{3}3s3p^{4}
finestructure transitions are given in Tables 6 and 7 for two
different temperature ranges. Similar data are presented in Tables 8
and 9 for all electric dipole 3s^{2}3p^{3}3s^{2}3p^{2}3d finestructure
transitions.
Table 4:
Collision strengths for all electric
dipole finestructure transitions between the ground 3s^{2}3p^{3} and the
first excited 3s3p^{4} configurations in FeXII. ^{a} see text

Table 5:
Collision strengths for all electric
dipole finestructure transitions between the ground 3s^{2}3p^{3} and the
second excited 3s^{2}3p^{2}3d configurations in FeXII. ^{a} and ^{b} see text

Table 6:
Effective collision strengths for all electric dipole
finestructure transitions between the ground 3s^{2}3p^{3} and the first
excited 3s3p^{4} configurations in FeXII.
Temperatures in the range 410^{5} K  310^{6} K. ^{a}, ^{b}
and ^{c} see text

Table 7:
Effective collision strengths for all electric dipole
finestructure transitions between the ground 3s^{2}3p^{3} and the first
excited 3s3p^{4} configurations in FeXII.
Temperatures in the range 410^{6}K  10^{7}K

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