\documentclass{aa}
\usepackage{graphicx,txfonts,textcomp}
%

\usepackage{natbib}
\bibpunct{(}{)}{;}{a}{}{,} %for A&A style

% GDZ: this is how natbib works: 
%\cite
%\citet me (1999)
%\citep (me 1999)
%\citealt me 1999
%\citeauthor me
%\citeyear 1999

\begin{document}

\title{Atomic Data from the IRON Project\thanks{Table 2 is 
also available in electronic form
at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5)
or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.
The full datasets of wavelengths and $gf$ values (Table 4)
and of effective collision strengths (Tables 6,7) are only available at the 
same address.
The same data are also available 
at the TIPbase (http://heasarc.gsfc.nasa.gov/tipbase/home.html)
database and as an `adf04' file
at the Oak Ridge National Laboratory's database 
via http://www-cfadc.phy.ornl.gov/ }}
\subtitle{LVI. Electron excitation of Be-like Fe~{XXIII}
for the $n=2,3,4$ configurations} 

%$n=2$ ({\rm 1s^2}\,{\rm 2l\,2l'}) and  $n=3,4$ }
%(${\rm 1s^2}\,2l_2\,n_3l_3\,S'L'J'$,  for $n_3=3,4$)
  

\author{M. C. Chidichimo
        \inst{1}\fnmsep\thanks{Permanent address: Department of Applied Mathematics,
                               University of Waterloo, 
                               Waterloo, Ontario N2L 3G1, Canada}
        \and 
        G. Del Zanna 
        \inst{1}
        \and
        H. E. Mason
        \inst{1}
        \and
        N. R. Badnell
        \inst{2}
        \and
        J. A. Tully
        \inst{3}
        \and
        K. A. Berrington
        \inst{4}
        }

\institute{Department of Applied Mathematics and Theoretical Physics, 
           The Centre for Mathematical Sciences, Wilberforce Road,
           Cambridge CB3 OWA
     \and
          Department of Physics \& Applied Physics, University of Strathclyde, 
          Glasgow G4 0NG, UK
     \and
          D\'epartement Cassiop\'ee, Observatoire de la C\^ote d'Azur, 
          B.P. 4229, 06304 Nice Cedex 4, France
     \and
          School of Science and Mathematics, Sheffield Hallam University, 
          Sheffield S1 1WB, UK  
          }

\date{Received 01/06/2004; accepted 10/08/2004} 

\titlerunning{Atomic Data from the IRON Project:  Fe~{XXIII}}

\abstract{ 
Collision strengths for electron induced transitions in 
the beryllium-like ion Fe$^{+22}$ are calculated using the intermediate coupling 
frame transformation (ICFT) 
 version of the  $R$-matrix programs. Our target has 98 fine 
structure states ${\rm 1s^2}\,nl\,n'l'\,SLJ$ corresponding to 
$n = 2 $ and $n' = 2, 3, 4 $. 
 The present calculation 
is for electron impact energies in the range 3.15 to 380~Ry. 
When $T$ exceeds about ten million degrees one needs to take account of 
contributions to the thermally averaged collision strength $\sl \Upsilon$ 
coming from electrons with energies in excess of 380~Ry. We discuss a way 
of allowing for  these contributions. 
Values of $\sl \Upsilon$ for all the transitions 
between the
ground state and the excited states ${\rm 1s^2}\,2l\,n\,l\,n'l'\,S'L'J'$,
  with $n'=2,3,4$ are tabulated as a function of ${\rm log}\,T$.
 The temperature range $6.3 \le \log\, T \le 8.1$ is centred on  
${\rm log}\,T = 7.1$  which is approximately  
 where Fe$^{+22}$ has maximum  abundance in ionization
 equilibrium. To the best of our knowledge 
these are the first $R$-matrix calculations for Fe$^{+22}$ for excitations to the 
$n = 3,4$ levels.  
Good agreement with previous distored-wave  calculations is found.
However, the  resonance contributions have an important 
effect on the effective collision strengths and in turn on the level 
populations.
\keywords{Atomic data - electron - positive ion - collision strength - 
beryllium-like iron }
}

\maketitle

%
%--------------------------------------------------------------------------
\section{Introduction}

The present calculation has been carried out as part of the 
international IRON Project 
(\citealt{hummer_etal:93}), whose aim is to obtain reliable 
rate coefficients for collisional excitation of fine-structure transitions 
in positive ions induced by electron impact. 
In a previous calculation, \cite{chidichimo_etal:99} obtained level energies,  
oscillator strengths and effective collisions strengths for the $n=2$ complex 
of  Fe$^{+22}$.
They  also included a thorough discussion of
collision calculations devoted to this ion up to the time of their
own investigation. We will refer to this paper as Paper I.
Previous published calculations for transitions to the $n=3$  complex of 
Fe$^{+22}$ were  
carried out in the non-relativistic and relativistic distorted wave (DW) 
approximation. 


Unpublished non-relativistic DW results are available for 
transitions up to $n=4$ (Bhatia, priv. comm.). 

In the present paper we present  new atomic data for 
transitions up to the $n=2,3,4$ complex in Fe$^{+22}$.
The data for the $n=2$ complex represent a revision of the 
data published in Paper I.
 These are the first $R$-matrix calculations for this ion
for excitations to the 
$n = 3$ levels, and the first published calculations to $n = 4$.
A complete list of IRON Project published papers and those 
in press is available on-line\footnote{http://www.usm.uni-muenchen.de/people/ip/iron-project.html}.

This work has been carried out as part of the UK RmaX 
network, which focuses on new atomic data for X ray spectra in astrophysics. 
Earlier work by the RmaX network includes ICFT calculations up to $n=4$ for 
C-like (\cite{badnell_griffin:01}) and B-like (\cite{badnell_etal:01}) Fe ions.


Fe$^{+22}$ $n=2 \to n^{'}=3$ and $n=2 \to n^{'}=4$ spectral lines have been
observed in solar flares (see, e.g., 
\citealt{neupert_etal:67}, \citealt{neupert_etal:73}, 
\citealt{doschek_etal:73}, 
\citealt{mckenzie_etal:85}, \citealt{fawcett_etal:87}), 
and are observed with 
the current X-ray missions such as Chandra and  XMM in spectra 
from a variety of different sources.
Fe$^{+22}$ line ratios have a  temperature sensitivity  
(as shown by \citealt{bhatia_mason:81}),  and are therefore a powerful 
diagnostic tool for a variety astrophysical applications.
This paper focuses on the atomic calculations, while a follow-up 
paper (\citealt{del_zanna_etal_fexxiii}) will deal with line
identifications and astrophysical applications.


\section{Method of calculation  and target}

In what follows, we outline the steps taken in the present work which
justify our belief that the results given here are the most reliable
Fe$^{+22}$ rate coefficients currently available.
 First of all we were careful to 
obtain a good target model. 

As regards the collision calculation, we have made use of
the  $R$-matrix method 
(\citealt{hummer_etal:93}, \citealt{berrington_etal:95}) in conjunction 
with the intermediate frame coupling transformation  
(ICFT) (see  \citealt{badnell_griffin:01}, \citealt{badnell_etal:01}).

These $R$-matrix programs, which are based on approximations that take 
into account much of the collision physics responsible for resonance 
scattering and relativistic effects, are widely considered as the most 
elaborate and accurate ones in existence for this type of calculation. 


%--------------------------------------------------------------------------
\subsection{Atomic orbitals for Fe$^{+22}$}

In the present calculation, we used the same  radial orbitals 
as those given  in Paper~I (see Table~2, where the numerical parameters 
needed to generate the  orbitals are listed).
 We have used the
1s, 2s radial orbitals of \cite{clementi_roetti:74} while 2p,
3s, 3p, 3d, 4s, 4p, 4d, 4f were obtained 
by means of the CIV3 code (\citealt{hibbert:75};  \citealt{hibbert_etal:91}). 
CIV3 uses analytic radial orbitals $P_{nl}(r)$ which
are expressed as sums of Slater typer orbitals as follows:
\begin{eqnarray*}
P_{nl}= \sum_{j=1}^{k}c_{jnl}\frac{(2\zeta_{jnl})^{I_{jnl+1/2}}}
{[(2 I_{jnl})!]^{1/2}} r^{\,I_{jnl}} \exp(-\zeta_{jnl}r). 
\end{eqnarray*}

%---------------------------------------------------------------------------

\subsection{Target energy levels}

For the collisional calculation, we used  all the terms 
originating from the 17 configurations included in 
Table~\ref{level-energies} (98 levels up to $n=4$).
All 17 configurations  in our
calculation were treated as spectroscopic, i.e. there were no extra 
configurations treated as correlation. These correlation
configurations are not found to contribute significantly for the
highly charged ion Fe$^{+22}$.

In order to assess how good   our  target is, we ran 
various atomic structure calculations by adding extra 
configurations, up to $n=5$, to our basic set of configurations. 
Configurations involving double promotion from the $n=3,4,5$ complex were treated 
as correlations.
These calculations were performed with the AUTOSTRUCTURE code
(\citealt{badnell:97}) and the results are presented in Table~\ref{auto-comp}.

First, for consistency,  we checked that the  energies calculated 
with AUTOSTRUCTURE were 
basically (to within 10$^{-4}$ Ry) the same as those shown 
in Table~\ref{level-energies}, which were
calculated with the $R$-matrix codes.
Then, for each AUTOSTRUCTURE  run, we compared the level energies and the 
oscillator strengths (or the high-energy limit for the 
forbidden lines) for all the transitions with the values 
obtained from the basic set of 17 configurations.
We found variations of the order of 10\%, i.e. of the order of the 
accuracy of the calculations themselves, thus giving us 
confidence on the accuracy of the target wavefunctions.

%_____________________________________________________________
\begin{table}[!htbp]
\caption{Weighted oscillator strength $gf$(AS), 
$17$ configurations. Comparison with 
$gf$(AS4),  including  $n=4$  correlation configurations; $gf$(AS5),  
including $n=5$ correlation 
configurations. The $gf$'s were   
calculated using  AUTOSTRUCTURE.
}
\label{auto-comp}
%
%% gf, OMEGAS from AUTOSTRUCTURE File 
%F1=~mc465/FeXXIII/autostructure/OMGINFIC-KCUT=0.n=4
%% gf, OMEGAS from AUTOSTRUCTURE File 
%F2=~mc465/FeXXIII/autostructure/OMGINFIC-sto.n=5
%
%% gf, OMEGAS from AUTOSTRUCTURE File 
%F3=~mc465/FeXXIII/autostructure/OMGINFIC-sto.n=4
%
%%   i   j      gf (F1)  gf(F1)/gf(F2)  gf(F1)/gf(F3)  
%
\begin{center}
  \begin{tabular}{llll}
\hline\noalign{\smallskip}
$i-j$  & $gf$(AS) & $gf$(AS)/$gf$(AS4) & $gf$(AS)/$gf$(AS5) \\
% 1--14 in F2 and F3
1--13 & 0.25          & 1.01 & 1.02   \\
1--15 & 0.41          & 0.98  & 0.98\\
1--22 & 1.3 10$^{-2}$ & 1.07  & 1.07 \\
1--25 & 2.3 10$^{-2}$ & 1.13  & 1.14\\
1--36 & 5.0 10$^{-3}$ & 0.90  & 0.89\\
1--42 & 2.0 10$^{-4}$ & 0.86  & 0.85 \\
1--46 & 1.7 10$^{-2}$ & 0.94  & 0.93 \\
1--50 & 2.7 10$^{-2}$ & 0.98  & 0.90 \\
1--52 & 0.13          & 1.00  & 0.97 \\
1--62 & 3.1 10$^{-4}$ & 1.16  & 1.47 \\
1--70 & 2.3 10$^{-3}$ & 0.97  & 0.93 \\

%1--76 in F2 and F3 
1--75 & 3.0 10$^{-4}$ & 1.04   & 1.49 \\

%1--86 in F2 and F3 
1--87 & 6.3 10$^{-5}$ & 0.98   & 0.94 \\
1--97 & 4.6 10$^{-3}$ & 0.96   & 0.91 \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\end{tabular}
\end{center}
\end{table}

%--------------------------------------------------------------

The  theoretical target energies of Fe$^{+22}$  produced 
by the Breit-Pauli branch of the R-matrix code are 
%The AUTOSTRUCTURE code gives the 
%parity, configuration,  and values $S, L, J$ pertaining to each
%level. The energies  are 
shown in Table~\ref{level-energies} along with the configuration
identification provided by AUTOSTRUCTURE and the observed
energies for some of the levels taken from \cite{corliss_sugar:82}
 and from the National Institute of Standards and Technology (NIST) 
database\footnote{http://physics.nist.gov}. 
The good agreement  between the theoretical and observed energy levels 
gives us confidence in the target description.


There are minor differences between the level energies 
of Paper I (see Table~3) and the present calculations. This is expected since
we chose to treat all configurations as spectroscopic ones 
and avoid the inclusion of extra
configurations which were treated as correlation in Paper I. 
This approach also leads to a few
changes in the level energy order between Paper I and now Table~\ref{index}.



%-------------------------------------------------------------------------



\subsection{The quality of the target}


One way of testing the target is to calculate the corresponding 
oscillator strengths  
and see how these compare with those of other investigators and
spectroscopic observations of laser-produced plasmas 
(\citealt{fawcett_etal:79}). 
Fawcett (\citeyear{fawcett:84}, \citeyear{fawcett:85}) 
tabulates weighted oscillator strengths 
(i.e. $gf$ values) and wavelengths for transitions $n=2 \to n^{'}=3$ in many beryllium-like 
ions, including  Fe$^{+22}$. He made use of Robert~D.~Cowan's code, together
with the Slater parameter optimisation procedure, 
which is widely thought to provide reliable benchmark data. 
 Cowan's code only provides the length 
gauge oscillator strength, which in general is more reliable than the velocity 
gauge one. 


%---------------------------------------------------------------------


\begin{table*}[!htbp]
\caption {Fe$^{+22}$ level energies in rydberg units relative to the 
ground state. Theoretical results from the Breit-Pauli $R$-matrix 
program. Observed results 
from \cite{corliss_sugar:82} assuming 
1 Ry = 109737.32 $\rm cm^{-1}$. \% diff is the 
percentage difference between the theoretical and observed energies}
\label{level-energies}
\begin{center}
\begin{tabular}{lllllllllllr}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
Index & Theoretical & \% diff & Observed & Level & 
&& Index & Theoretical & \% diff & Observed & Level  \\
&&&&& & &&&&& \\
  1 & $\rm \,\,\,0.0000$&&$\rm \,\,\,0.000$ & $\rm 2s2s\,\,\,^1S_0$ & 
&& 50 & 109.6961 &      &            & $\rm 2s4p\, ^3P_1^o$   \\

 2&$\rm \,\,\,3.1503$ &($-$0.71)&$\rm \,\,\,3.173 $ & $\rm 2s2p\,^3P_0^o$&
&& 51 & 109.8039 &      &            & $\rm 2s4p\, ^3P_2^o$    \\

 3&$\rm \,\,\,3.4418$ &($-$0.38)&$\rm \,\,\,3.455 $&$\rm 2s2p\,^3P_1^o$ & 
&& 52 & 109.8492 &(+0.09)&109.753 \ \ & $\rm 2s4p\, ^1P_1^o$   \\

 4&$\rm \,\,\,4.2749$&($-$0.56)&$\rm \,\,\,4.299 $&$\rm 2s2p\, ^3P_2^o$ &  
&& 53 & 110.1453 &(+0.12)&110.017 \ \ & $\rm 2s4d\, ^3D_1$ \\

 5&$\rm \,\,\,6.8932$&(+0.48)&$\rm \,\,\,6.860$&$\rm 2s2p\, ^1P_1^o$ &  
&& 54 & 110.1605 &(+0.12)&110.035 \ \ & $\rm 2s4d\, ^3D_2$  \\

  6&$\rm \,\,\,8.6932$&($-$0.22)&$\rm \,\,\,8.713 \  $&$\rm 2p2p\, ^3P_0$ &  
&& 55 & 110.1876 &(+0.09)&110.090 \ \ & $\rm 2s4d\, ^3D_3$  \\

 7&$\rm \,\,\,9.3240$ &($-$0.39)&$\rm \,\,\,9.360 $ & $\rm 2p2p\, ^3P_1$ &  
&& 56 & 110.3691 &(+0.12)&110.245 \ \ & $\rm 2s4d\, ^1D_2$  \\

 8 & $\rm \,\,\,9.7658$ &($-$0.00)&$\rm \,\,\,9.766$&$\rm 2p2p\,^3P_2$ &  
&& 57 & 110.4055 &       &           & $\rm 2s4f\, ^3F_2^o$  \\

  9  & 10.9820 &(+0.08)& 10.973 \  & $\rm 2p2p\, ^1D_2$ &  
&& 58 & 110.4129 &       &           & $\rm 2s4f\, ^3F_3^o$  \\

 10  & 13.0046 &(+0.34)& 12.967 \ \  & $\rm 2p2p\, ^1S_0$ &  
&& 59 & 110.4266 &       &           & $\rm 2s4f\, ^3F_4^o$  \\

 11  & 81.2795 &(+0.29)& 81.048 \ \  & $\rm 2s3s\,\,\, ^3S_1$ &  
&& 60 & 110.4728 &       &           & $\rm 2s4f\, ^1F_3^o$  \\

 12  & 81.9333 &       &            & $\rm 2s3s\,\,\, ^1S_0$ &  
&& 61 & 112.8017 &       &           & $\rm 2p4s\, ^3P_0^o$  \\

 13  & 82.7951 & (+0.11) & 82.707 \ \  & $\rm 2s3p\, ^3P_1^o$  & 
&& 62 & 112.8462 &       &           & $\rm 2p4s\, ^3P_1^o$  \\

 14  & 82.7959 &    &     & $\rm 2s3p\, ^3P_0^o$ &  
&& 63 & 113.2248 &       &           & $\rm 2p4p\, ^3D_1$  \\

 15  & 83.0716 &(+0.10)& 82.989 \ \  & $\rm 2s3p\, ^1P_1^o$ &  
&& 64 & 113.4583 &       &           & $\rm 2p4p\, ^3P_1$  \\

 16  & 83.1023 &       &            & $\rm 2s3p\, ^3P_2^o$ &  
&& 65 & 113.4744 &       &           & $\rm 2p4p\, ^3D_2$  \\

 17  & 83.9218 &(+0.12)& 83.827 \ \  & $\rm 2s3d\, ^3D_1$ &  
&& 66 & 113.5026 &       &           & $\rm 2p4p\, ^3P_0$  \\

 18  & 83.9663 &(+0.07)& 83.919 \ \  & $\rm 2s3d\, ^3D_2$ &  
&& 67 & 113.6813 &       &           & $\rm 2p4d\, ^3F_2^o$  \\

 19  & 84.0368 &(+0.11)& 83.946 \ \  & $\rm 2s3d\, ^3D_3$ &  
&& 68 & 113.8231& (+0.08) & 113.735  & $\rm 2p4d\, ^3D_2^o$  \\

 20  & 84.6215 &(+0.14)& 84.502 \ \  & $\rm 2s3d\, ^1D_2$ &  
&& 69 & 113.8660 &(+0.09)&113.763 \ \ & $\rm 2p4d\, ^3F_3^o$  \\

 21  & 85.2849 &(+0.69)& 84.702 \ \  & $\rm 2p3s\, ^3P_0^o$ &  
&& 70 & 113.9168 &(+0.11)&113.799 \ \ & $\rm 2p4d\, ^3D_1^o$  \\

 22  & 85.4484 &       &            & $\rm 2p3s\, ^3P_1^o$ &  
&& 71 & 113.9473 &       &           & $\rm 2p4s\, ^3P_2^o$  \\

 23  & 86.2699 &(+0.13)& 86.160 \ \  & $\rm 2p3p\, ^3D_1 $ &  
&& 72 & 113.9672 &       &           & $\rm 2p4f\, ^3G_3$  \\

 24  & 86.3840 &       &            & $\rm 2p3s\, ^3P_2^o$ & 
&& 73 & 113.9976 &       &           & $\rm 2p4f\, ^3F_2$  \\

 25  & 86.8205 &(+0.72)& 86.297 \ \ \  & $\rm 2p3s\, ^1P_1^o$ & 
&& 74 & 114.0051 &       &           & $\rm 2p4f\, ^3F_3$  \\

 26  & 86.8622 &(+0.09)& 86.789 \ \  & $\rm 2p3p\, ^3D_2$ &  
&& 75 & 114.0074 &       &           & $\rm 2p4s\, ^1P_1^o$  \\

 27  & 86.8730 &       &            & $\rm 2p3p\, ^1P_1$ &  
&& 76 & 114.0094 & & & $\rm 2p4f\, ^3G_4$ \\

 28  & 87.0882 &       &            & $\rm 2p3p\, ^3P_0$ &  
&& 77 & 114.4139 &       &           & $\rm 2p4p\, ^1P_1$ \\

 29  & 87.3983 &       &            & $\rm 2p3d\, ^3F_2^o$ &  
&& 78 & 114.4753 & (+0.06) & 114.455 \ \ \  & $\rm 2p4p\, ^3D_3$ \\

 30  & 87.6784 &       &            & $\rm 2p3p\, ^3P_1$ &  
&& 79 & 114.4780 &      &        & $\rm 2p4p\, ^3P_2$ \\

 31  & 87.7295 &(+0.03)& 87.700 \ \  & $\rm 2p3p\, ^3D_3$ & 
&& 80 & 114.5240 &       &           & $\rm 2p4p\, ^3S_1$ \\

 32  & 87.7924 &(+0.10)& 87.709 \ \  & $\rm 2p3d\, ^3F_3^o$ &  
&& 81 & 114.7018 &       &           & $\rm 2p4p\, ^1D_2$ \\

 33  & 87.9022 &     &     & $\rm 2p3p\, ^3S_1 $ & 
&& 82 & 114.8321 &       &           & $\rm 2p4d\, ^3F_4^o$ \\

 34  & 87.9075 &       &            & $\rm 2p3d\, ^3D_2^o$ & 
&& 83 & 114.8322 &(+0.04)&114.792 \ \ & $\rm 2p4d\, ^1D_2^o$ \\

 35  & 87.9377 &(+0.06)& 87.883 \ \  & $\rm 2p3p\, ^3P_2$ &  
&& 84 & 114.9053 &(+0.06)&114.847 \ \ & $\rm 2p4d\, ^3D_3^o$ \\

 36  & 88.0896 & (+0.32) & 87.819 \ \  & $\rm 2p3d\, ^3D_1^o$ & 
&& 85 & 114.9579 &       &           & $\rm 2p4p\, ^1S_0$ \\

 37  & 88.5096 & (+0.04) &  88.475 \ \  & $\rm 2p3p\, ^1D_2$ &  
&& 86 & 114.9753 &(+0.03)&114.947 \ \ & $\rm 2p4d\, ^3P_2^o$ \\

 38  & 88.5924 &       &            & $\rm 2p3d\, ^3F_4^o$ & 
&& 87 & 114.9754 &(+0.02)&114.956 \ \ & $\rm 2p4d\, ^3P_1^o$ \\

 39  & 88.6862 & (+0.98) &  87.828 \ \  & $\rm 2p3d\, ^1D_2^o$ & 
&& 88 & 114.9851 &       &           & $\rm 2p4d\, ^3P_0^o$ \\

 40  & 88.9204 &(+0.10)& 88.839 \ \  & $\rm 2p3d\, ^3D_3^o$ & 
&& 89 & 115.0456 &       &           & $\rm 2p4f\, ^1F_3$ \\

 41  & 89.0959 & (+0.25) &  88.876 \ \  & $\rm 2p3d\, ^3P_2^o$ & 
&& 90 & 115.0704 &       &           & $\rm 2p4f\, ^3F_4$ \\

 42  & 89.1021 &       &            & $\rm 2p3d\, ^3P_1^o$ & 
&& 91 & 115.1066 &       &           & $\rm 2p4f\, ^3D_2$ \\

 43  & 89.1230 &       &            & $\rm 2p3d\, ^3P_0^o$ & 
&& 92 & 115.1119 &       &           & $\rm 2p4f\, ^3D_3$ \\

 44  & 89.2372 &    &     & $\rm 2p3p\, ^1S_0$ & 
&& 93 & 115.1254 &       &           & $\rm 2p4f\, ^3G_5$ \\

 45  & 89.6743 &(+0.19)& 89.577 \ \ \  & $\rm 2p3d\, ^1F_3^o$ & 
&& 94 & 115.1478 &       &           & $\rm 2p4f\, ^1G_4$ \\

 46  & 89.7620 &(+0.24)& 89.559 \ \  & $\rm 2p3d\, ^1P_1^o$ & 
&& 95 & 115.1587 &(+0.05)&115.102 \ \ & $\rm 2p4d\, ^1F_3^o$ \\

 47  & 109.1071  &       &            & $\rm 2s4s\, ^3S_1$ & 
&& 96 & 115.1690 &       &           & $\rm 2p4f\, ^3D_1$ \\

 48  & 109.3177  &      &            & $\rm 2s4s\, ^1S_0$ & 
&& 97 & 115.2017 &       &           & $\rm 2p4d\, ^1P_1^o$ \\

 49  & 109.6805  &      &            & $\rm 2s4p\, ^3P_0^o$ & 
&& 98 & 115.2087 &       &           & $\rm 2p4f\, ^1D_2$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\end{tabular}
\end{center}
\end{table*}

%---------------------------------------------------------------

The full set of  wavelengths $\lambda$(\AA) and transition
probabilities 
$A_{ji}$(s$^{-1}$)
for all the transitions occurring amongst the $98$ levels as
calculated with AUTOSTRUCTURE
 is provided in electronic form.
Observed energies, applying adjustments to the $LS$ energies, and 
refined $A$-values will be discussed in a forthcoming paper by 
\cite{del_zanna_etal_fexxiii}.
 In Table~\ref{oscillator} we  compare our $gf$ and wavelength values
 with those of 
\cite{bhatia_mason:81}, \cite{bhatia_mason:86}, 
\cite{guo-xin_ong:98a}, \cite{bhatia_etal:86}, \cite{murakami_kato:96},
together with the observed wavelengths from 
\cite{fawcett_etal:79}.
%
\cite{guo-xin_ong:98a} used the GRASP code
(\citealt{grasp}) with a 133-level multiconfiguration Dirack-Fock
expansion, which should provide the most accurate results. 
From Table~\ref{oscillator} we note that 
their inclusion of correlation and higher-order relativistic
effects do not seem to play a prominent role in the
1-13 and 1-15  wavelengths, our calculations and theirs agree within 
0.2\% with the observed wavelengths.  We also have good agreement, within 10\% for the
strongest transitions,  between our $gf$ results and theirs. This gives us confidence 
in our target.
  



%since the theoretical results
%agree within 0.2 \% of the experimental values. 
 
%In Table~4 we compare our $gf$ values with those of Fawcett. 
%29/11/2002 The years (1984, 1985) seem superfluous, so I've removed them



\begin{table}[!htbp] %add [h] placement to break table across pages
\caption{Past and present label of energy levels and indexing}
\label{index}
%\begin{tabular}{r@{\extracolsep{0ptplus1fil}}ccc}
\begin{center}
\begin{tabular}{lcc}
\hline\noalign{\smallskip}
 Label &  \multicolumn{2}{c}{Index} \\
       &   1999$^{\mathrm{a}}$  & 2004 \\
\hline\noalign{\smallskip}
 $\rm 2s3p\, ^3P_1^o$ & 14  &  13 \\
 $\rm 2s3p\, ^3P_0^o$ & 13  &  14 \\
 $\rm 2p3p\, ^3D_3$   & 55  &  31 \\
 $\rm 2p3p\, ^3S_1$   & 34  &  33 \\
 $\rm 2p3d\, ^3D_2^o$ & 39  &  34 \\
 $\rm 2p3d\, ^1D_2^o$ & 33  &  39 \\
 $\rm 2s4d\, ^3D_3$   & 31  &  55 \\  
 $\rm 2p4f\, ^3F_3$   & 76  &  74 \\
 $\rm 2p4f\, ^3G_4$   & 74  &  76 \\
 $\rm 2p4p\, ^3D_3$   & 79  &  78 \\
 $\rm 2p4p\, ^3P_2$   & 78  &  79 \\
 $\rm 2p4f\, ^3F_4$   & 94  &  90 \\
 $\rm 2p4f\, ^3D_3$   & 93  &  92 \\
 $\rm 2p4f\, ^3G_5$   & 92  &  93 \\
 $\rm 2p4f\, ^1G_4$   & 90  &  94 \\                     
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\end{tabular}
\begin{list}{}{}
\item[$^{\mathrm{a}}$] Paper I
\end{list}
\end{center}
\end{table}



%----------------------------------------------------------------------

\begin{table*}[!htbp]
\caption{Fe$^{+22}$: comparing the present wavelengths $\lambda$(R) and 
weighted oscillator strengths $gf$(R) from AUTOSTRUCTURE with those of 
 Bathia \& Mason (\citeyear{bhatia_mason:81},
 \citeyear{bhatia_mason:86}), 
$\lambda$(BM) and $gf$(BM);  
 \cite{bhatia_etal:86}, $\lambda$(BFS); 
\cite{fawcett:84}, $\lambda$(F) and $gf$(F); 
\cite{sampson_etal:84}; $gf$(SGC);  
\cite{murakami_kato:96}, $\lambda$(MK); 
\cite{guo-xin_ong:98a}, $\lambda$(GO) and $gf$(GO); 
Fawcett's laboratory measurements (\citealt{fawcett_etal:79}), 
$\lambda$(Exp). 
Wavelengths are in \AA.}
\label{oscillator}
%\begin{center}
\begin{tabular}{lllllllllllll}
\hline\noalign{\smallskip}
$\lambda$(R) 
& $\lambda$(BM)
&  $\lambda$(BFS)
& $\lambda$(F)
& $\lambda$(MK)
& $\lambda$(GO)
& $\lambda$(Exp)
&$gf$(R)
&$gf$(BM)
&$gf$(F)
&$gf$(SGC)
&$gf$(GO)
 & Transition \\

&&&&&&&&&&&&\\

 11.0063&
 11.00 & 11.018 
&11.017&11.006 
&11.018 & 11.022 & 2.525$^{-1}$ 
&2.4$^{-1}$ &2.7$^{-1}$&2.4$^{-1}$ & 2.679$^{-1}$ &$1-13$ \\

10.9697  & 
 10.97 &10.978
&10.979& 10.966 
&10.979 & 10.983 &4.092$^{-1}$ 
&4.06$^{-1}$ &4.5$^{-1}$&4.7$^{-1}$  & 4.110$^{-1}$ & $1-15$ \\

&&&&&&&&&&&&\\

 
10.6645 &  
& &  & & &
&1.297$^{-2}$ & && &1.290 $^{-2}$ &$1-22$ \\

 
10.4960& 
& & & & &
&2.342$^{-2}$ & && & 2.198 $^{-2}$& $1-25$ \\

&&&&&&&&&&&&\\


 
10.3448&  
&& & & &
&5.025$^{-3}$ && & & 5.810 $^{-3}$ &$1-36$ \\


 
10.2272&  
&&  & & 
&
&2.001$^{-4}$&&&& 2.336 $^{-4}$&  $1-42$ \\

 
10.1520&  
&& & &
&  
&1.745$^{-2}$&&& &1.932 $^{-2}$& $1-46$ \\

&&&&&&&&&&&&\\


 
8.3072&  & & &8.3059 & 8.316 &
&2.673$^{-2}$ && & & &$1-50$ \\

 
8.2956&  & & &8.2934 &8.303 &
&1.339$^{-1}$& && & &$1-52$ \\

&&&&&&&&&&&&\\

 
8.0753&  
& & & &
&
&3.112$^{-4}$ & && & & $1-62$ \\



 
7.9931&  
&& & &
& 
&2.987$^{-4}$&&&& &$1-75$ \\                         


&&&&&&&&&&&&\\

 
7.9994&  & & & & & & 2.347$^{-3}$ & & && & $1-70$ \\

&&&&&&&&&&&\\

 
7.9258&
&& & &
&  
&6.297$^{-5}$&& && &$1-87$ \\

 
7.9102&  
&& & &
&  
&4.574$^{-3}$&& && &$1-97$ \\

\noalign{\smallskip}
\hline
\noalign{\smallskip}
\end{tabular}
%\end{center}
\end{table*}

%----------------------------------------------------------------------


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{The collision and effective collision strengths}

%The atomic data were calculated using the $R$-matrix method 
%(Hummer et al. \cite{hbe}, Berrington  et al. \cite{ben}) in conjuction 
%with the intermediate frame coupling transformation 
%(Badnell and Griffin \cite{bad1}, Badnell et al. \cite{bad2})

Our ultimate aim is to tabulate the temperature-dependent effective 
collision strength $\Upsilon(i-j)$, defined by~: 

$$ \Upsilon(i-j) = \int_0^\infty \Omega(i-j)\, {\rm exp}(-E_j/kT)\,
{\rm d}(E_j/kT) $$
where $E_j$ is the energy of the colliding electron after excitation has 
occurred and $k$ is the Boltzmann 
constant. 
We performed the numerical integration by linearly interpolating 
the $\Omega(i,j)\,{\rm exp}(-E_j/kT)$ data points as suggested 
by \cite{burgess_tully:92}. In order to try and delineate the resonance 
structure in the collision strengths we calculated each $\Omega(i-j)$ at 
$116847$ energy points,  i.e. with an extremely fine mesh.

For electrons incident with kinetic 
energies relative to the ground state of the target less than or equal to 
380~Ry we  used  the $R$-matrix method based on the close coupling 
approximation. This allows us to take account of channel 
coupling up to the $n=4$ levels.  
%
In order to delineate the multitude of resonance peaks we ran the  
code at $116746$ values of the collision energy starting at 3.1521~Ry 
(relative to the ground state), and up to 116 ~Ry.
In this resonance region  we used an energy step-length of 
$2 \times 10^{-6} z^2 = 9.68 \times 10^{-4}$ Ry. 
%
In  the interval 
$116 < E_i < 382 $ Ry, in which there are no resonances, we 
calculated collision strengths at $101$ energy points,
with  a step-length of $5.5 \times 10^{-3} z^2 = 2.662$ Ry. 


In Paper I,  the Breit-Pauli
code was run at 7704 values of the collision energy starting at 3.1521~Ry,
relative to the ground state and going up to 103.05816~Ry. 
The interval
between 103.05816 and 116 Ry was covered by making a linear extrapolation
backwards using the values of the collision strength at 116 and 127.5 Ry.
Also by an unfortunate and regretful oversight, 
at low energies the higher partial wave contribution to the
collision strength for dipole transitions was not 
included\footnote{This fact was brought to our 
attention by Dr. C. P. Ballance}.


The  R-matrix code also
 attends  to the crucial issue of a top-up procedure in order 
to account for the higher partial waves in the collision strengths.
With increasing energy, more and more partial waves need to be
 calculated. Details of how this {\sl top-up} was done for 
C-like (\citealt{badnell_griffin:01}) and B-like (\citealt{badnell_etal:01})
 Fe ions are given in those publications.

 
In the present calculations, in order to ensure convergence of the
expansion,  we let the partial wave 
quantum number $J$ extend up to $J = 41.5$ and then carried out  a 
top-up procedure by estimating the contributions from higher partial waves.


In order to calculate reliable effective 
collision strengths at temperatures which are more than about 
$2 \times 10^7\,{\rm K}$, we need  to calculate, or estimate, 
the value of the collision strengths   
$\Omega(i,j)$ for energies even higher than 380~Ry,
 up to several thousand Rydbergs.
%
We extended our $R$-matrix collision strengths beyond 382~Ry
by using the method of scaling and extrapolating the 
collision strengths to  the 
appropriate high-energy limits as described in \cite{burgess_tully:92}. 
 For optically allowed transitions,
the  high-energy limits are directly obtained from the 
 oscillator strengths.
We used the gf values calculated with 
AUTOSTRUCTURE (see  Table~\ref{oscillator}).
%
%the results are given along with line strengths $S$ and $gf$ 
%values in Table~\ref{oscillator}. 
For optically forbidden transitions between levels with 
the same parity and spin we calculated the high-energy Born limits in the 
manner described by \cite{burgess_etal:97}. 
For optically forbidden intersystem 
transitions we  used  a comparable method developed by one of us (MCC)  
(see Appendix, \citealt{chidichimo_etal:03}). 
NRB has included this method in his  AUTOSTRUCTURE 
code. The high-energy limits for the 
optically forbidden lines used for the extrapolation 
procedure  are given in Table~\ref{born}.


We developed an IDL graphical interactive program (a modified form (GDZ) of 
the codes used by the CHIANTI team) to  
inspect visually  all the data. It is based on the
program OmeUps of \cite{burgess_tully:92}, whereby
 spline fits are performed on the  
$101$ $\Omega(i-j)$ data points  and the high-energy limit (see Fig.
 \ref{fig-2}). 
We then perform an interpolation to 
obtain $\Omega(i,j)$  values  at  energies ranging from 382~Ry 
to $10^6$~Ry, which we deem sufficient for the purpose of thermal
averaging. 
The effective 
collision strength $\Upsilon(i-j)$ are then calculated.
This procedure gives added reliability at
 high temperatures.


\begin{table}[!htbp]
\caption{High energy Born limits for forbidden transitions. 
$\,(1.640^{-2} \equiv \, 1.640 \times 10^{-2})$}
\label{born}
\begin{center}
\begin{tabular}{clcl}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
$i-j$&${\sl \Omega}(i-j)$ & $i-j$ & ${\sl \Omega}(i-j)$ \\
  & \\
$1-12$& 1.640$^{-2}$  & $1-60$& 2.573$^{-3}$ \\
$1-18$& 2.115$^{-4}$  & $1-65$& 4.455$^{-6}$ \\
$1-20$& 4.784$^{-2}$  & $1-66$& 1.050$^{-5}$ \\
$1-26$& 9.233$^{-5}$  & $1-69$& 8.712$^{-6}$ \\
$1-28$& 5.990$^{-6}$  & $1-73$& 3.922$^{-5}$ \\
$1-32$& 2.413$^{-5}$  & $1-76$& 5.013$^{-6}$ \\
$1-35$& 1.099$^{-4}$  & $1-79$& 2.314$^{-6}$ \\
$1-37$& 2.369$^{-4}$  & $1-81$& 6.350$^{-6}$ \\
$1-40$& 8.954$^{-6}$  & $1-84$& 1.230$^{-6}$ \\
$1-44$& 3.779$^{-5}$  & $1-85$& 2.907$^{-5}$ \\
$1-45$& 1.548$^{-4}$  & $1-90$& 1.864$^{-7}$ \\
$1-48$& 2.995$^{-3}$  & $1-91$& 1.338$^{-5}$ \\
$1-54$& 6.059$^{-5}$  & $1-94$& 7.422$^{-6}$ \\
$1-56$& 7.435$^{-3}$  & $1-95$& 1.964$^{-5}$ \\
$1-58$& 7.548$^{-5}$  & $1-98$& 4.049$^{-5}$ \\
%
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\end{tabular}
\end{center}
\end{table}



%--------------------------------------------------------------------------------

\begin{figure*}[!htbp]
\centering
\includegraphics[height=6.1cm,width=6cm]{H1358f1a.ps}
\includegraphics[height=6.1cm,width=6cm]{H1358f1b.ps}
\includegraphics[height=6.1cm,width=6cm]{H1358f1c.ps}
\includegraphics[height=6.1cm,width=6cm]{H1358f1d.ps}
\includegraphics[height=6.1cm,width=6cm]{H1358f1e.ps}
\includegraphics[height=6.1cm,width=6cm]{H1358f1f.ps}
\includegraphics[height=6.1cm,width=6cm]{H1358f1g.ps}

\caption{$\log \Omega(i-j)$ as function of the final energy $E_j$ in
  rydbergs. Circles indicate the DW values of \cite{bhatia_mason:86}}.
\label{fig-1}
\end{figure*}


\begin{figure}[!htbp]
\centering
\includegraphics[width=7cm]{H1358f2.ps}
\caption{Top figure shows $\Omega(1-15)$  as function of 
the incident energy in rydbergs. Bottom figure shows the
collision strength high energy points in the scaled form and the fit
to the infinity energy point}
\label{fig-2}
\end{figure}

%--------------------------------------------------------------------------------

\section{New collisional data}

Collision strengths $\Omega(i-j)$ between all 
4753 transitions among the 98 levels were calculated.


%\subsection{Effective collision strengths}

Thermal averaging of the collision strengths was done using the 
linear interpolation method described by \cite{burgess_tully:92}. 
%
The  effective collision strengths $\sl \Upsilon$ 
were calculated for the temperature range
$6.3 \leq {\rm log}\,T \leq 8.1$, 
 centred on the temperature where Fe$^{+22}$ is abundant 
under conditions of coronal ionization equilibrium 
(see, e.g., \citealt{arnaud_raymond:92}).
%
For temperatures below five million degrees the abundance 
of Fe$^{+22}$ will be negligible in equilibrium conditions. 
Astrophysical situations may exist where 
Fe$^{+22}$ is abundant at temperatures lower than this; in these cases one 
would need to extend the temperature range below $10^{6.3}\,$ $^{\rm o}$K. 

 Tables~\ref{upsilon1},\ref{upsilon2} 
present the effective collision strengths $\sl \Upsilon$ for excitations
 from the ground state only, which are the dominant contributors for 
astrophysical plasmas (see below).
A more comprehensive dataset, that includes an adf04 file (see ref. 
for definition) and 
collision strengths for excitations from all the $n=2$ states
is provided on-line for completeness.
Also the data with the target energies and the $A$ values
are available on-line in ascii form. 
% now from the A\&A databank at CDS. 
Other data mentioned in this paper 
(e.g. collision strengths) are available upon request from one
of the  authors (GDZ).


\begin{table*}[!htbp]
\caption{$\rm Fe^{+22}$ effective collision strengths ${\sl \Upsilon}(i-j)$ 
to $n=2,3$ levels for $6.3 \le \log T \le 8.1$. $\,(2.421^{-3} = \, 2.421 \times 10^{-3})$}
\label{upsilon1}
\begin{flushleft}
\begin{tabular}{llllllllllll}
\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
$i-j$   & 6.3 & 6.5 & 6.7 & 6.9 & 7.1 & 7.3 & 7.5 & 7.7 & 7.9 & 8.1 \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\\
1-2  & 2.619$^{-3}$ & 2.668$^{-3}$ & 2.590$^{-3}$ & 2.339$^{-3}$ & 1.963$^{-3}$ & 1.551$^{-3}$ & 1.169$^{-3}$ & 8.510$^{-4}$ & 6.032$^{-4}$ & 4.192$^{-4}$ \\
1-3  & 1.549$^{-2}$ & 1.590$^{-2}$ & 1.614$^{-2}$ & 1.604$^{-2}$ & 1.569$^{-2}$ & 1.532$^{-2}$ & 1.506$^{-2}$ & 1.498$^{-2}$ & 1.505$^{-2}$ & 1.526$^{-2}$ \\
1-4  & 1.269$^{-2}$ & 1.303$^{-2}$ & 1.269$^{-2}$ & 1.145$^{-2}$ & 9.595$^{-3}$ & 7.564$^{-3}$ & 5.689$^{-3}$ & 4.132$^{-3}$ & 2.924$^{-3}$ & 2.029$^{-3}$ \\
1-5  & 3.280$^{-1}$ & 3.417$^{-1}$ & 3.596$^{-1}$ & 3.820$^{-1}$ & 4.094$^{-1}$ & 4.420$^{-1}$ & 4.795$^{-1}$ & 5.213$^{-1}$ & 5.664$^{-1}$ & 6.136$^{-1}$ \\
1-6  & 1.873$^{-4}$ & 2.191$^{-4}$ & 2.460$^{-4}$ & 2.508$^{-4}$ & 2.313$^{-4}$ & 1.986$^{-4}$ & 1.638$^{-4}$ & 1.333$^{-4}$ & 1.094$^{-4}$ & 9.159$^{-5}$ \\
1-7  & 2.564$^{-4}$ & 3.038$^{-4}$ & 3.370$^{-4}$ & 3.329$^{-4}$ & 2.920$^{-4}$ & 2.328$^{-4}$ & 1.732$^{-4}$ & 1.228$^{-4}$ & 8.424$^{-5}$ & 5.654$^{-5}$ \\
1-8  & 7.381$^{-4}$ & 8.287$^{-4}$ & 8.889$^{-4}$ & 8.869$^{-4}$ & 8.292$^{-4}$ & 7.477$^{-4}$ & 6.700$^{-4}$ & 6.092$^{-4}$ & 5.670$^{-4}$ & 5.400$^{-4}$ \\
1-9  & 1.262$^{-3}$ & 1.365$^{-3}$ & 1.426$^{-3}$ & 1.422$^{-3}$ & 1.366$^{-3}$ & 1.294$^{-3}$ & 1.232$^{-3}$ & 1.192$^{-3}$ & 1.171$^{-3}$ & 1.164$^{-3}$ \\
1-10  & 8.662$^{-4}$ & 9.585$^{-4}$ & 1.012$^{-3}$ & 9.971$^{-4}$ & 9.236$^{-4}$ & 8.223$^{-4}$ & 7.190$^{-4}$ & 6.272$^{-4}$ & 5.515$^{-4}$ & 4.923$^{-4}$ \\
\\
1-11  & 2.916$^{-3}$ & 2.275$^{-3}$ & 1.759$^{-3}$ & 1.346$^{-3}$ & 1.018$^{-3}$ & 7.587$^{-4}$ & 5.567$^{-4}$ & 4.015$^{-4}$ & 2.849$^{-4}$ & 1.993$^{-4}$ \\
1-12  & 1.344$^{-2}$ & 1.338$^{-2}$ & 1.340$^{-2}$ & 1.352$^{-2}$ & 1.376$^{-2}$ & 1.409$^{-2}$ & 1.448$^{-2}$ & 1.487$^{-2}$ & 1.523$^{-2}$ & 1.554$^{-2}$ \\
1-13  & 5.015$^{-3}$ & 5.060$^{-3}$ & 5.278$^{-3}$ & 5.760$^{-3}$ & 6.598$^{-3}$ & 7.873$^{-3}$ & 9.648$^{-3}$ & 1.195$^{-2}$ & 1.478$^{-2}$ & 1.810$^{-2}$ \\
1-14  & 4.793$^{-4}$ & 4.202$^{-4}$ & 3.547$^{-4}$ & 2.897$^{-4}$ & 2.298$^{-4}$ & 1.771$^{-4}$ & 1.327$^{-4}$ & 9.685$^{-5}$ & 6.902$^{-5}$ & 4.825$^{-5}$ \\
1-15  & 7.037$^{-3}$ & 7.256$^{-3}$ & 7.735$^{-3}$ & 8.622$^{-3}$ & 1.007$^{-2}$ & 1.221$^{-2}$ & 1.514$^{-2}$ & 1.892$^{-2}$ & 2.354$^{-2}$ & 2.894$^{-2}$ \\
1-16  & 2.419$^{-3}$ & 2.133$^{-3}$ & 1.801$^{-3}$ & 1.469$^{-3}$ & 1.162$^{-3}$ & 8.940$^{-4}$ & 6.688$^{-4}$ & 4.872$^{-4}$ & 3.467$^{-4}$ & 2.419$^{-4}$ \\
1-17  & 1.941$^{-3}$ & 1.774$^{-3}$ & 1.555$^{-3}$ & 1.313$^{-3}$ & 1.071$^{-3}$ & 8.451$^{-4}$ & 6.455$^{-4}$ & 4.786$^{-4}$ & 3.459$^{-4}$ & 2.449$^{-4}$ \\
1-18  & 3.354$^{-3}$ & 3.079$^{-3}$ & 2.713$^{-3}$ & 2.310$^{-3}$ & 1.909$^{-3}$ & 1.539$^{-3}$ & 1.215$^{-3}$ & 9.473$^{-4}$ & 7.370$^{-4}$ & 5.788$^{-4}$ \\
1-19  & 4.613$^{-3}$ & 4.220$^{-3}$ & 3.697$^{-3}$ & 3.118$^{-3}$ & 2.540$^{-3}$ & 2.001$^{-3}$ & 1.527$^{-3}$ & 1.131$^{-3}$ & 8.172$^{-4}$ & 5.782$^{-4}$ \\
1-20  & 1.859$^{-2}$ & 1.940$^{-2}$ & 2.056$^{-2}$ & 2.218$^{-2}$ & 2.434$^{-2}$ & 2.698$^{-2}$ & 2.994$^{-2}$ & 3.299$^{-2}$ & 3.589$^{-2}$ & 3.849$^{-2}$ \\
1-21  & 1.862$^{-5}$ & 1.604$^{-5}$ & 1.286$^{-5}$ & 9.767$^{-6}$ & 7.157$^{-6}$ & 5.119$^{-6}$ & 3.597$^{-6}$ & 2.492$^{-6}$ & 1.706$^{-6}$ & 1.157$^{-6}$ \\
1-22  & 2.800$^{-4}$ & 2.798$^{-4}$ & 2.853$^{-4}$ & 3.036$^{-4}$ & 3.404$^{-4}$ & 4.002$^{-4}$ & 4.859$^{-4}$ & 5.988$^{-4}$ & 7.383$^{-4}$ & 9.026$^{-4}$ \\
1-23  & 1.060$^{-4}$ & 9.512$^{-5}$ & 8.010$^{-5}$ & 6.449$^{-5}$ & 5.033$^{-5}$ & 3.836$^{-5}$ & 2.862$^{-5}$ & 2.095$^{-5}$ & 1.507$^{-5}$ & 1.068$^{-5}$ \\
1-24  & 7.468$^{-5}$ & 6.212$^{-5}$ & 4.879$^{-5}$ & 3.669$^{-5}$ & 2.678$^{-5}$ & 1.915$^{-5}$ & 1.347$^{-5}$ & 9.345$^{-6}$ & 6.402$^{-6}$ & 4.339$^{-6}$ \\
1-25  & 4.670$^{-4}$ & 4.685$^{-4}$ & 4.844$^{-4}$ & 5.238$^{-4}$ & 5.950$^{-4}$ & 7.050$^{-4}$ & 8.589$^{-4}$ & 1.059$^{-3}$ & 1.306$^{-3}$ & 1.596$^{-3}$ \\
1-26  & 1.824$^{-4}$ & 1.687$^{-4}$ & 1.496$^{-4}$ & 1.307$^{-4}$ & 1.149$^{-4}$ & 1.034$^{-4}$ & 9.573$^{-5}$ & 9.122$^{-5}$ & 8.897$^{-5}$ & 8.820$^{-5}$ \\
1-27  & 8.631$^{-5}$ & 7.675$^{-5}$ & 6.327$^{-5}$ & 4.953$^{-5}$ & 3.751$^{-5}$ & 2.778$^{-5}$ & 2.023$^{-5}$ & 1.452$^{-5}$ & 1.030$^{-5}$ & 7.229$^{-6}$ \\
1-28  & 4.208$^{-5}$ & 3.715$^{-5}$ & 3.075$^{-5}$ & 2.449$^{-5}$ & 1.925$^{-5}$ & 1.523$^{-5}$ & 1.229$^{-5}$ & 1.022$^{-5}$ & 8.798$^{-6}$ & 7.837$^{-6}$ \\
1-29  & 1.312$^{-4}$ & 1.182$^{-4}$ & 9.872$^{-5}$ & 7.813$^{-5}$ & 5.955$^{-5}$ & 4.414$^{-5}$ & 3.199$^{-5}$ & 2.275$^{-5}$ & 1.593$^{-5}$ & 1.102$^{-5}$ \\
1-30  & 4.841$^{-5}$ & 4.076$^{-5}$ & 3.282$^{-5}$ & 2.558$^{-5}$ & 1.952$^{-5}$ & 1.467$^{-5}$ & 1.089$^{-5}$ & 7.993$^{-6}$ & 5.801$^{-6}$ & 4.167$^{-6}$ \\
1-31  & 1.185$^{-4}$ & 1.048$^{-4}$ & 8.873$^{-5}$ & 7.260$^{-5}$ & 5.782$^{-5}$ & 4.491$^{-5}$ & 3.403$^{-5}$ & 2.517$^{-5}$ & 1.821$^{-5}$ & 1.294$^{-5}$ \\
1-32  & 1.773$^{-4}$ & 1.601$^{-4}$ & 1.348$^{-4}$ & 1.086$^{-4}$ & 8.565$^{-5}$ & 6.718$^{-5}$ & 5.319$^{-5}$ & 4.306$^{-5}$ & 3.604$^{-5}$ & 3.136$^{-5}$ \\
1-33  & 6.090$^{-5}$ & 5.078$^{-5}$ & 4.015$^{-5}$ & 3.056$^{-5}$ & 2.267$^{-5}$ & 1.652$^{-5}$ & 1.186$^{-5}$ & 8.403$^{-6}$ & 5.885$^{-6}$ & 4.081$^{-6}$ \\
1-34  & 1.649$^{-4}$ & 1.467$^{-4}$ & 1.213$^{-4}$ & 9.529$^{-5}$ & 7.235$^{-5}$ & 5.358$^{-5}$ & 3.891$^{-5}$ & 2.780$^{-5}$ & 1.960$^{-5}$ & 1.367$^{-5}$ \\
1-35  & 1.111$^{-4}$ & 1.018$^{-4}$ & 9.238$^{-5}$ & 8.521$^{-5}$ & 8.133$^{-5}$ & 8.070$^{-5}$ & 8.261$^{-5}$ & 8.607$^{-5}$ & 9.015$^{-5}$ & 9.419$^{-5}$ \\
1-36  & 3.165$^{-4}$ & 3.139$^{-4}$ & 3.105$^{-4}$ & 3.124$^{-4}$ & 3.234$^{-4}$ & 3.454$^{-4}$ & 3.792$^{-4}$ & 4.246$^{-4}$ & 4.808$^{-4}$ & 5.466$^{-4}$ \\
1-37  & 1.597$^{-4}$ & 1.502$^{-4}$ & 1.428$^{-4}$ & 1.399$^{-4}$ & 1.426$^{-4}$ & 1.503$^{-4}$ & 1.615$^{-4}$ & 1.743$^{-4}$ & 1.870$^{-4}$ & 1.985$^{-4}$ \\
1-38  & 1.338$^{-4}$ & 1.155$^{-4}$ & 9.507$^{-5}$ & 7.539$^{-5}$ & 5.799$^{-5}$ & 4.342$^{-5}$ & 3.170$^{-5}$ & 2.260$^{-5}$ & 1.579$^{-5}$ & 1.085$^{-5}$ \\
1-39  & 9.381$^{-5}$ & 7.808$^{-5}$ & 6.213$^{-5}$ & 4.784$^{-5}$ & 3.601$^{-5}$ & 2.667$^{-5}$ & 1.952$^{-5}$ & 1.415$^{-5}$ & 1.018$^{-5}$ & 7.265$^{-6}$ \\
1-40  & 9.783$^{-5}$ & 8.259$^{-5}$ & 6.611$^{-5}$ & 5.108$^{-5}$ & 3.875$^{-5}$ & 2.928$^{-5}$ & 2.233$^{-5}$ & 1.743$^{-5}$ & 1.411$^{-5}$ & 1.195$^{-5}$ \\
1-41  & 1.392$^{-4}$ & 1.159$^{-4}$ & 9.270$^{-5}$ & 7.189$^{-5}$ & 5.438$^{-5}$ & 4.025$^{-5}$ & 2.920$^{-5}$ & 2.078$^{-5}$ & 1.454$^{-5}$ & 1.003$^{-5}$ \\
1-42  & 1.063$^{-4}$ & 9.017$^{-5}$ & 7.437$^{-5}$ & 6.051$^{-5}$ & 4.923$^{-5}$ & 4.061$^{-5}$ & 3.442$^{-5}$ & 3.038$^{-5}$ & 2.815$^{-5}$ & 2.740$^{-5}$ \\
1-43  & 3.411$^{-5}$ & 2.907$^{-5}$ & 2.371$^{-5}$ & 1.870$^{-5}$ & 1.435$^{-5}$ & 1.076$^{-5}$ & 7.888$^{-6}$ & 5.659$^{-6}$ & 3.983$^{-6}$ & 2.760$^{-6}$ \\
1-44  & 9.385$^{-5}$ & 8.394$^{-5}$ & 7.335$^{-5}$ & 6.395$^{-5}$ & 5.650$^{-5}$ & 5.096$^{-5}$ & 4.698$^{-5}$ & 4.418$^{-5}$ & 4.221$^{-5}$ & 4.085$^{-5}$ \\
1-45  & 1.679$^{-4}$ & 1.529$^{-4}$ & 1.377$^{-4}$ & 1.257$^{-4}$ & 1.181$^{-4}$ & 1.150$^{-4}$ & 1.153$^{-4}$ & 1.182$^{-4}$ & 1.227$^{-4}$ & 1.278$^{-4}$ \\
1-46  & 7.938$^{-4}$ & 8.077$^{-4}$ & 8.341$^{-4}$ & 8.800$^{-4}$ & 9.512$^{-4}$ & 1.052$^{-3}$ & 1.185$^{-3}$ & 1.350$^{-3}$ & 1.548$^{-3}$ & 1.775$^{-3}$ \\
\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\end{tabular}
\end{flushleft}
\end{table*}


%----------------------------------------------------------
\begin{table*}[!htbp]
\caption{$\rm Fe^{+22}$ effective collision strengths ${\sl \Upsilon}(i-j)$ 
to $n=4$ levels for $6.3 \le \log T \le 8.1$. $\,(2.421^{-3} = \, 2.421 \times 10^{-3})$}
\label{upsilon2}
\begin{flushleft}
\begin{tabular}{llllllllllll}
\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
$i-j$ &  6.3 & 6.5 &  6.7 & 6.9 & 7.1 & 7.3 & 7.5 & 7.7 & 7.9 & 
8.1  \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\\
1-47  & 4.092$^{-4}$ & 3.497$^{-4}$ & 2.986$^{-4}$ & 2.518$^{-4}$ & 2.077$^{-4}$ & 1.666$^{-4}$ & 1.296$^{-4}$ & 9.797$^{-5}$ & 7.218$^{-5}$ & 5.208$^{-5}$ \\
1-48  & 2.320$^{-3}$ & 2.327$^{-3}$ & 2.357$^{-3}$ & 2.408$^{-3}$ & 2.473$^{-3}$ & 2.548$^{-3}$ & 2.626$^{-3}$ & 2.699$^{-3}$ & 2.765$^{-3}$ & 2.820$^{-3}$ \\
1-49  & 1.100$^{-4}$ & 9.970$^{-5}$ & 8.869$^{-5}$ & 7.680$^{-5}$ & 6.430$^{-5}$ & 5.187$^{-5}$ & 4.033$^{-5}$ & 3.031$^{-5}$ & 2.214$^{-5}$ & 1.581$^{-5}$ \\
1-50  & 6.792$^{-4}$ & 6.721$^{-4}$ & 6.819$^{-4}$ & 7.128$^{-4}$ & 7.714$^{-4}$ & 8.645$^{-4}$ & 9.973$^{-4}$ & 1.173$^{-3}$ & 1.390$^{-3}$ & 1.646$^{-3}$ \\
1-51  & 5.506$^{-4}$ & 5.023$^{-4}$ & 4.488$^{-4}$ & 3.896$^{-4}$ & 3.268$^{-4}$ & 2.638$^{-4}$ & 2.050$^{-4}$ & 1.539$^{-4}$ & 1.121$^{-4}$ & 7.978$^{-5}$ \\
1-52  & 1.957$^{-3}$ & 2.073$^{-3}$ & 2.269$^{-3}$ & 2.575$^{-3}$ & 3.022$^{-3}$ & 3.638$^{-3}$ & 4.443$^{-3}$ & 5.445$^{-3}$ & 6.637$^{-3}$ & 8.005$^{-3}$ \\
1-53  & 4.497$^{-4}$ & 4.179$^{-4}$ & 3.786$^{-4}$ & 3.324$^{-4}$ & 2.815$^{-4}$ & 2.294$^{-4}$ & 1.800$^{-4}$ & 1.366$^{-4}$ & 1.007$^{-4}$ & 7.253$^{-5}$ \\
1-54  & 7.678$^{-4}$ & 7.165$^{-4}$ & 6.533$^{-4}$ & 5.791$^{-4}$ & 4.976$^{-4}$ & 4.148$^{-4}$ & 3.368$^{-4}$ & 2.686$^{-4}$ & 2.127$^{-4}$ & 1.692$^{-4}$ \\
1-55  & 1.046$^{-3}$ & 9.739$^{-4}$ & 8.839$^{-4}$ & 7.768$^{-4}$ & 6.582$^{-4}$ & 5.367$^{-4}$ & 4.214$^{-4}$ & 3.198$^{-4}$ & 2.359$^{-4}$ & 1.700$^{-4}$ \\
1-56  & 2.937$^{-3}$ & 3.078$^{-3}$ & 3.276$^{-3}$ & 3.543$^{-3}$ & 3.884$^{-3}$ & 4.289$^{-3}$ & 4.736$^{-3}$ & 5.191$^{-3}$ & 5.625$^{-3}$ & 6.013$^{-3}$ \\
1-57  & 3.334$^{-4}$ & 3.055$^{-4}$ & 2.713$^{-4}$ & 2.322$^{-4}$ & 1.909$^{-4}$ & 1.505$^{-4}$ & 1.140$^{-4}$ & 8.333$^{-5}$ & 5.907$^{-5}$ & 4.087$^{-5}$ \\
1-58  & 4.874$^{-4}$ & 4.506$^{-4}$ & 4.061$^{-4}$ & 3.558$^{-4}$ & 3.034$^{-4}$ & 2.528$^{-4}$ & 2.078$^{-4}$ & 1.706$^{-4}$ & 1.417$^{-4}$ & 1.204$^{-4}$ \\
1-59  & 6.012$^{-4}$ & 5.511$^{-4}$ & 4.895$^{-4}$ & 4.190$^{-4}$ & 3.444$^{-4}$ & 2.715$^{-4}$ & 2.057$^{-4}$ & 1.504$^{-4}$ & 1.068$^{-4}$ & 7.407$^{-5}$ \\
1-60  & 1.094$^{-3}$ & 1.153$^{-3}$ & 1.238$^{-3}$ & 1.351$^{-3}$ & 1.492$^{-3}$ & 1.653$^{-3}$ & 1.820$^{-3}$ & 1.978$^{-3}$ & 2.118$^{-3}$ & 2.234$^{-3}$ \\
1-61  & 2.046$^{-6}$ & 1.697$^{-6}$ & 1.415$^{-6}$ & 1.173$^{-6}$ & 9.574$^{-7}$ & 7.640$^{-7}$ & 5.945$^{-7}$ & 4.514$^{-7}$ & 3.354$^{-7}$ & 2.449$^{-7}$ \\
1-62  & 1.467$^{-5}$ & 1.356$^{-5}$ & 1.298$^{-5}$ & 1.286$^{-5}$ & 1.320$^{-5}$ & 1.404$^{-5}$ & 1.541$^{-5}$ & 1.734$^{-5}$ & 1.980$^{-5}$ & 2.274$^{-5}$ \\
1-63  & 1.135$^{-5}$ & 1.004$^{-5}$ & 8.813$^{-6}$ & 7.589$^{-6}$ & 6.365$^{-6}$ & 5.178$^{-6}$ & 4.085$^{-6}$ & 3.132$^{-6}$ & 2.342$^{-6}$ & 1.718$^{-6}$ \\
1-64  & 9.828$^{-6}$ & 8.833$^{-6}$ & 7.827$^{-6}$ & 6.775$^{-6}$ & 5.693$^{-6}$ & 4.630$^{-6}$ & 3.645$^{-6}$ & 2.786$^{-6}$ & 2.077$^{-6}$ & 1.518$^{-6}$ \\
1-65  & 1.498$^{-5}$ & 1.374$^{-5}$ & 1.249$^{-5}$ & 1.120$^{-5}$ & 9.908$^{-6}$ & 8.702$^{-6}$ & 7.647$^{-6}$ & 6.783$^{-6}$ & 6.113$^{-6}$ & 5.615$^{-6}$ \\
1-66  & 1.098$^{-5}$ & 1.036$^{-5}$ & 9.954$^{-6}$ & 9.694$^{-6}$ & 9.538$^{-6}$ & 9.465$^{-6}$ & 9.463$^{-6}$ & 9.521$^{-6}$ & 9.624$^{-6}$ & 9.755$^{-6}$ \\
1-67  & 1.962$^{-5}$ & 1.805$^{-5}$ & 1.617$^{-5}$ & 1.402$^{-5}$ & 1.171$^{-5}$ & 9.396$^{-6}$ & 7.257$^{-6}$ & 5.414$^{-6}$ & 3.923$^{-6}$ & 2.778$^{-6}$ \\
1-68  & 2.253$^{-5}$ & 2.071$^{-5}$ & 1.862$^{-5}$ & 1.625$^{-5}$ & 1.370$^{-5}$ & 1.112$^{-5}$ & 8.714$^{-6}$ & 6.605$^{-6}$ & 4.871$^{-6}$ & 3.514$^{-6}$ \\
1-69  & 2.281$^{-5}$ & 2.124$^{-5}$ & 1.943$^{-5}$ & 1.743$^{-5}$ & 1.535$^{-5}$ & 1.338$^{-5}$ & 1.167$^{-5}$ & 1.034$^{-5}$ & 9.399$^{-6}$ & 8.798$^{-6}$ \\
1-70  & 9.522$^{-5}$ & 9.781$^{-5}$ & 1.017$^{-4}$ & 1.075$^{-4}$ & 1.156$^{-4}$ & 1.266$^{-4}$ & 1.409$^{-4}$ & 1.586$^{-4}$ & 1.796$^{-4}$ & 2.037$^{-4}$ \\
1-71  & 9.074$^{-6}$ & 7.861$^{-6}$ & 6.738$^{-6}$ & 5.651$^{-6}$ & 4.602$^{-6}$ & 3.624$^{-6}$ & 2.759$^{-6}$ & 2.037$^{-6}$ & 1.465$^{-6}$ & 1.033$^{-6}$ \\
1-72  & 5.620$^{-6}$ & 5.071$^{-6}$ & 4.438$^{-6}$ & 3.751$^{-6}$ & 3.056$^{-6}$ & 2.401$^{-6}$ & 1.824$^{-6}$ & 1.346$^{-6}$ & 9.706$^{-7}$ & 6.869$^{-7}$ \\
1-73  & 1.297$^{-5}$ & 1.314$^{-5}$ & 1.358$^{-5}$ & 1.441$^{-5}$ & 1.573$^{-5}$ & 1.758$^{-5}$ & 1.990$^{-5}$ & 2.254$^{-5}$ & 2.529$^{-5}$ & 2.794$^{-5}$ \\
1-74  & 9.262$^{-6}$ & 8.418$^{-6}$ & 7.416$^{-6}$ & 6.299$^{-6}$ & 5.145$^{-6}$ & 4.040$^{-6}$ & 3.057$^{-6}$ & 2.239$^{-6}$ & 1.596$^{-6}$ & 1.115$^{-6}$ \\
1-75  & 1.608$^{-5}$ & 1.503$^{-5}$ & 1.454$^{-5}$ & 1.454$^{-5}$ & 1.505$^{-5}$ & 1.608$^{-5}$ & 1.765$^{-5}$ & 1.975$^{-5}$ & 2.235$^{-5}$ & 2.537$^{-5}$ \\
1-76  & 6.747$^{-6}$ & 6.304$^{-6}$ & 5.831$^{-6}$ & 5.364$^{-6}$ & 4.952$^{-6}$ & 4.636$^{-6}$ & 4.436$^{-6}$ & 4.348$^{-6}$ & 4.349$^{-6}$ & 4.407$^{-6}$ \\
1-77  & 6.073$^{-6}$ & 5.197$^{-6}$ & 4.461$^{-6}$ & 3.793$^{-6}$ & 3.166$^{-6}$ & 2.578$^{-6}$ & 2.043$^{-6}$ & 1.577$^{-6}$ & 1.190$^{-6}$ & 8.804$^{-7}$ \\
1-78  & 1.461$^{-5}$ & 1.345$^{-5}$ & 1.214$^{-5}$ & 1.067$^{-5}$ & 9.075$^{-6}$ & 7.443$^{-6}$ & 5.891$^{-6}$ & 4.514$^{-6}$ & 3.365$^{-6}$ & 2.454$^{-6}$ \\
1-79  & 7.663$^{-6}$ & 7.000$^{-6}$ & 6.358$^{-6}$ & 5.712$^{-6}$ & 5.075$^{-6}$ & 4.483$^{-6}$ & 3.964$^{-6}$ & 3.538$^{-6}$ & 3.203$^{-6}$ & 2.950$^{-6}$ \\
1-80  & 7.865$^{-6}$ & 7.027$^{-6}$ & 6.220$^{-6}$ & 5.401$^{-6}$ & 4.566$^{-6}$ & 3.742$^{-6}$ & 2.970$^{-6}$ & 2.287$^{-6}$ & 1.717$^{-6}$ & 1.261$^{-6}$ \\
1-81  & 6.570$^{-6}$ & 6.174$^{-6}$ & 5.869$^{-6}$ & 5.650$^{-6}$ & 5.529$^{-6}$ & 5.516$^{-6}$ & 5.594$^{-6}$ & 5.726$^{-6}$ & 5.872$^{-6}$ & 6.005$^{-6}$ \\
1-82  & 2.143$^{-5}$ & 1.986$^{-5}$ & 1.788$^{-5}$ & 1.555$^{-5}$ & 1.300$^{-5}$ & 1.043$^{-5}$ & 8.027$^{-6}$ & 5.957$^{-6}$ & 4.283$^{-6}$ & 3.004$^{-6}$ \\
1-83  & 9.578$^{-6}$ & 8.801$^{-6}$ & 7.900$^{-6}$ & 6.883$^{-6}$ & 5.797$^{-6}$ & 4.715$^{-6}$ & 3.708$^{-6}$ & 2.831$^{-6}$ & 2.107$^{-6}$ & 1.537$^{-6}$ \\
1-84  & 1.046$^{-5}$ & 9.696$^{-6}$ & 8.751$^{-6}$ & 7.655$^{-6}$ & 6.473$^{-6}$ & 5.301$^{-6}$ & 4.234$^{-6}$ & 3.341$^{-6}$ & 2.647$^{-6}$ & 2.143$^{-6}$ \\
1-85  & 2.547$^{-5}$ & 2.502$^{-5}$ & 2.482$^{-5}$ & 2.479$^{-5}$ & 2.490$^{-5}$ & 2.513$^{-5}$ & 2.546$^{-5}$ & 2.588$^{-5}$ & 2.634$^{-5}$ & 2.683$^{-5}$ \\
1-86  & 1.576$^{-5}$ & 1.463$^{-5}$ & 1.322$^{-5}$ & 1.155$^{-5}$ & 9.714$^{-6}$ & 7.855$^{-6}$ & 6.114$^{-6}$ & 4.600$^{-6}$ & 3.362$^{-6}$ & 2.403$^{-6}$ \\
1-87  & 1.361$^{-5}$ & 1.289$^{-5}$ & 1.199$^{-5}$ & 1.094$^{-5}$ & 9.831$^{-6}$ & 8.748$^{-6}$ & 7.796$^{-6}$ & 7.051$^{-6}$ & 6.553$^{-6}$ & 6.311$^{-6}$ \\
1-88  & 4.959$^{-6}$ & 4.617$^{-6}$ & 4.183$^{-6}$ & 3.666$^{-6}$ & 3.091$^{-6}$ & 2.499$^{-6}$ & 1.938$^{-6}$ & 1.445$^{-6}$ & 1.042$^{-6}$ & 7.316$^{-7}$ \\
1-89  & 2.584$^{-6}$ & 2.358$^{-6}$ & 2.088$^{-6}$ & 1.791$^{-6}$ & 1.487$^{-6}$ & 1.197$^{-6}$ & 9.373$^{-7}$ & 7.169$^{-7}$ & 5.375$^{-7}$ & 3.964$^{-7}$ \\
1-90  & 2.632$^{-6}$ & 2.390$^{-6}$ & 2.102$^{-6}$ & 1.785$^{-6}$ & 1.464$^{-6}$ & 1.163$^{-6}$ & 9.030$^{-7}$ & 6.933$^{-7}$ & 5.341$^{-7}$ & 4.191$^{-7}$ \\
1-91  & 6.271$^{-6}$ & 6.163$^{-6}$ & 6.119$^{-6}$ & 6.182$^{-6}$ & 6.402$^{-6}$ & 6.809$^{-6}$ & 7.401$^{-6}$ & 8.135$^{-6}$ & 8.943$^{-6}$ & 9.750$^{-6}$ \\
1-92  & 5.423$^{-6}$ & 4.953$^{-6}$ & 4.382$^{-6}$ & 3.738$^{-6}$ & 3.069$^{-6}$ & 2.424$^{-6}$ & 1.846$^{-6}$ & 1.363$^{-6}$ & 9.805$^{-7}$ & 6.912$^{-7}$ \\
1-93  & 5.478$^{-6}$ & 4.970$^{-6}$ & 4.358$^{-6}$ & 3.681$^{-6}$ & 2.988$^{-6}$ & 2.332$^{-6}$ & 1.754$^{-6}$ & 1.277$^{-6}$ & 9.043$^{-7}$ & 6.271$^{-7}$ \\
1-94  & 5.202$^{-6}$ & 5.083$^{-6}$ & 4.969$^{-6}$ & 4.893$^{-6}$ & 4.892$^{-6}$ & 4.987$^{-6}$ & 5.178$^{-6}$ & 5.442$^{-6}$ & 5.745$^{-6}$ & 6.053$^{-6}$ \\
1-95  & 1.203$^{-5}$ & 1.190$^{-5}$ & 1.179$^{-5}$ & 1.173$^{-5}$ & 1.179$^{-5}$ & 1.202$^{-5}$ & 1.246$^{-5}$ & 1.310$^{-5}$ & 1.391$^{-5}$ & 1.479$^{-5}$ \\
1-96  & 3.593$^{-6}$ & 3.280$^{-6}$ & 2.898$^{-6}$ & 2.466$^{-6}$ & 2.016$^{-6}$ & 1.581$^{-6}$ & 1.193$^{-6}$ & 8.686$^{-7}$ & 6.144$^{-7}$ & 4.247$^{-7}$ \\
1-97  & 1.545$^{-4}$ & 1.603$^{-4}$ & 1.687$^{-4}$ & 1.806$^{-4}$ & 1.968$^{-4}$ & 2.183$^{-4}$ & 2.457$^{-4}$ & 2.796$^{-4}$ & 3.197$^{-4}$ & 3.657$^{-4}$ \\
1-98  & 1.326$^{-5}$ & 1.356$^{-5}$ & 1.410$^{-5}$ & 1.502$^{-5}$ & 1.640$^{-5}$ & 1.829$^{-5}$ & 2.064$^{-5}$ & 2.330$^{-5}$ & 2.609$^{-5}$ & 2.878$^{-5}$ \\
\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\end{tabular}
\end{flushleft}
\end{table*}


%----------------------------------------------------------




%------------------------------------------------------------------------

\begin{table*}[!htbp]
%\caption[]{$\Upsilon(i,j)$ for $T = 10^{7.1}$ $^{\mathrm{o}}$K.
\caption[]{$\Upsilon(i,j)$ for $T = 10^{7.1}$ \textdegree K.
CDMB, present results; 
BM, Bhatia \& Mason (\citeyear{bhatia_mason:81}, \citeyear{bhatia_mason:86}); 
GO, \cite{guo-xin_ong:98b}; 
B$^{\mathrm{a}}$, Bhatia; 
SGC, \cite{sampson_etal:84}}
\label{others}
\begin{center}
\begin{tabular}{ccccc@{\quad\qquad}ccc@{\quad\qquad}cccc}
\hline\noalign{\smallskip}
$i,j$ & CDMB & BM & GO &SGC & $i,j$ & CDMB & SGC & 
$i,j$ & CDMB & B$^{\mathrm{a}}$ & SGC  \\
\hline\noalign{\smallskip}
\\
$1,11$ & 1.018$^{-3}$ & 6.103$^{-4}$ & 6.487$^{-4}$ & 6.196$^{-4}$ &
$1,27$ & 3.751$^{-5}$ & 1.326$^{-5}$ & 
$1,43$ & 1.435$^{-5}$ &              & 8.123$^{-6}$   \\

$1,12$ & 1.376$^{-2}$ & 1.297$^{-2}$ & 1.507$^{-2}$ & 1.560$^{-2}$ &
$1,28$ & 1.925$^{-5}$ & 6.106$^{-6}$ & 
$1,44$ & 5.650$^{-5}$ &              & 1.642$^{-5}$ \\

$1,13$ & 6.598$^{-3}$ & 5.485$^{-3}$ & 6.615$^{-3}$ & 5.920$^{-3}$ & 
$1,29$ & 5.955$^{-5}$ & 2.855$^{-5}$ & 
$1,45$ & 1.181$^{-4}$ &              & 5.230$^{-5}$  \\

$1,14$ & 2.298$^{-4}$ & 1.590$^{-4}$ & 1.629$^{-4}$ & 1.611$^{-4}$ & 
$1,30$ & 1.952$^{-5}$ & 7.159$^{-6}$ & 
$1,46$ & 9.512$^{-4}$ &              & 1.181$^{-3}$  \\

$1,15$ & 1.007$^{-2}$ & 9.167$^{-3}$ & 9.542$^{-3}$ & 1.075$^{-2}$ & 
$1,31$ & 5.782$^{-5}$ & 4.069$^{-5}$ & 
$1,47$ & 2.077$^{-4}$ & 2.259$^{-4}$ &               \\

$1,16$ & 1.162$^{-3}$ & 8.050$^{-4}$ & 7.969$^{-4}$ & 8.112$^{-4}$ & 
$1,32$ & 8.565$^{-6}$ & 4.358$^{-5}$ & 
$1,48$ & 2.473$^{-3}$ & 2.287$^{-3}$ &                \\

$1,17$ & 1.071$^{-3}$ & 8.970$^{-4}$ & 8.878$^{-4}$ & 8.991$^{-4}$ & 
$1,33$ & 2.267$^{-5}$ & 6.116$^{-5}$ & 
$1,49$ & 6.430$^{-5}$ & 6.004$^{-5}$ &                \\

$1,18$ & 1.909$^{-3}$ & 1.551$^{-3}$ & 1.585$^{-3}$ & 1.644$^{-3}$ &
$1,34$ & 7.235$^{-5}$ & 3.424$^{-5}$ & 
$1,50$ & 7.714$^{-4}$ & 6.249$^{-4}$ &                \\

$1,19$ & 2.540$^{-3}$ & 2.072$^{-3}$ & 2.079$^{-3}$ & 2.121$^{-3}$ & 
$1,35$ & 8.133$^{-5}$ & 8.430$^{-5}$ & 
$1,51$ & 3.268$^{-4}$ & 3.310$^{-4}$ &                \\

$1,20$ & 2.434$^{-2}$ & 1.801$^{-2}$ & 2.506$^{-2}$ & 2.564$^{-2}$ & 
$1,36$ & 3.234$^{-4}$ & 4.093$^{-4}$ & 
$1,52$ & 3.022$^{-3}$ & 2.984$^{-3}$ &                \\

$1,21$ & 7.157$^{-6}$ &              &              & 2.355$^{-6}$ & 
$1,37$ & 1.404$^{-4}$ & 1.473$^{-4}$ & 
$1,53$ & 2.815$^{-4}$ & 2.830$^{-4}$ &                 \\

$1,22$ & 3.404$^{-4}$ &              &              & 5.228$^{-4}$ & 
$1,38$ & 5.799$^{-5}$ & 3.381$^{-5}$ & 
$1,54$ & 4.976$^{-4}$ & 5.240$^{-4}$ &                  \\

$1,23$ & 5.033$^{-5}$ &              &              & 3.014$^{-5}$ & 
$1,39$ & 3.601$^{-5}$ & 1.465$^{-5}$ & 
$1,55$ & 6.582$^{-4}$ & 6.949$^{-4}$ &                  \\

$1,24$ & 2.678$^{-5}$ &              &              & 8.776$^{-6}$ & 
$1,40$ & 3.875$^{-5}$ & 1.771$^{-5}$ & 
$1,56$ & 3.884$^{-3}$ & 3.887$^{-3}$ &                  \\

$1,25$ & 5.950$^{-4}$ &              &              & 8.420$^{-4}$ &
$1,41$ & 5.438$^{-5}$ & 2.772$^{-5}$ \\ 

$1,26$ & 1.149$^{-4}$ &              &              & 1.256$^{-4}$ &       
$1,42$ & 4.923$^{-5}$ & 3.391$^{-5}$ \\

\\  
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\end{tabular}
\begin{list}{}{}
\item[$^{\mathrm{a}}$] Private communication
\end{list}
\end{center}
\end{table*}

%-------------------------------------------------------------------------

\subsection{Comparisons with previous calculations} 


For the $n=2 \to n^{'}=2$ transitions, the present
 effective collision strengths only differ  
 from the values published in Paper I by about 10\%,
with the exception of the 1-6 (2s$^2$ $^1$S$_{0}^{\rm e}$--2p$^2$ $^3$P$_{0}^{\rm e}$)  and
 1-7 (2s$^2$ $^1$S$_{0}^{\rm e}$--2p$^2$ $^3$P$_{1}^{\rm e}$ ) 
where differences are larger (up to 60\%).


Since these are the first calculations for the 
$n=2 \to n^{'}=3$ and $n=2 \to n^{'}=4$ transitions that take into 
account the resonance effects, it is interesting to compare 
both the $\Omega(i,j)$ and $\Upsilon(i,j)$ values with the 
previous relativistic and non-relativistic DW results, to assess the importance of 
the resonances.



\subsubsection{Allowed and forbidden $n=2 \to n^{'}=3,4$ transitions in
  comparison with DW }

We give illustrations of the different types of collision strength 
encountered in the present investigation by plotting ${\sl \Omega}(i - j)$ 
versus $E_{\sl j}$, the final electron energy in~rydbergs. 
%In the plots we also include for comparison the distorted-wave 
%results of previous studies.
%
We included comparisons with  the published collision strengths for
the $n=2 \to n^{'}=3$ transitions, from the 
 papers by Bhatia \& Mason (\citeyear{bhatia_mason:81},
 \citeyear{bhatia_mason:86};  hereafter BM).
%, Sampson et al.
%(\citeyear{sampson_etal:84}; hereafter SGC), and 
%\cite{guo-xin_ong:98b}.
This study  provided 
 collision strengths for the $n=2 \to n^{'}=3$ transitions,
and  used distorted wave
approximations which do not take account of  resonance effects. 
%  deal with the transitions in Table~\ref{others}. 
%\cite{guo-xin_ong:98b} and SGC used a relativistic DW method to calculate 
%the collision strengths.

Fig~\ref{fig-1}  shows  $ \Omega(1 - 12)$, $\Omega(1 - 13)$, 
$ \Omega(1 - 15)$, $ \Omega(1 - 19)$,  $ \Omega(1 - 20)$, 
$\Omega(1 - 50)$ and $ \Omega(1 - 52)$. The collision strengths 
$ \Omega(1 - 15)$ and $ \Omega(1 - 52)$ 
 correspond to   optically allowed 
transitions which increase logarithmically 
with energy as $E_j \to \infty$. 
%In order to delineate the low energy 
%resonances in Fig.~1 we use a magnified energy scale there compared to 
%the one in Fig.~2. 
The collision strengths $\Omega(1 - 13)$ and $\Omega(1 - 50)$ correspond to   
intersystem transitions 
that behave as though they were optically allowed owing to the breakdown of 
$LS$ coupling. For this to happen the initial and final levels must have 
different parities and $\Delta J = 0, \pm 1$, subject to the condition that 
$J = 0 \not\to J' = 0$. 
 $ \Omega(1 - 12)$ and $ \Omega(1 - 20)$ are forbidden transitions 
in which neither the parity nor the spin change. The collision strength 
for this type of transition tends to a finite limiting value as 
$E_j \to \infty$. $ \Omega(1 - 19)$ is a forbidden intersystem 
transition for which the collision strength normally falls off like  
$E_j^{-2}$ in the high energy limit. 

The graphs  show that there is  excellent 
agreement between our background collision strengths and the distorted wave 
results.  
%
Only for the forbidden transitions $1-20$ (electric quadrupole) is the 
difference more noticeable. Fig.~\ref{fig-1} shows that the BM 
 distorted wave collision strength $\Omega(1-20)$ is 
lower than ours. Our calculations show that the contribution, from partial
waves $13 \le L \le 40$ at $E_i = 282 Ry$,  to  the total collision
strength $\Omega(1-20)$ is 17 \%. For $L>40$ the contribution is
negligible.
  Therefore, it is likely that  insufficient high
  partial wave contribution was taken into account, for this type of
  transition, in  BM's calculations. Note the prominent resonance structure 
at low energies which has been neglected in all previous calculations.
% \cite{sampson_etal:84} and \cite{guo-xin_ong:98b} results are 
%in better agreement with our data 
%for this transition (see Fig.~6). 




%-------------------------------------------





\subsubsection{Comparison of  $\Upsilon(i,j)$ for the $n=2 \to n^{'}=3,4$ }


Sampson et al.
(\citeyear{sampson_etal:84}; hereafter SGC), and 
(\citealt{guo-xin_ong:98b}; hereafter GO) used a relativistic DW method to calculate 
the collision strengths.
BM and GO give collision strengths for 
transitions between the lowest 20 levels, at the energies $E_i$ =
85.0, 127.5, 170.0, 250.0, 350~Ry. 
 We used the \cite{burgess_tully:92} method 
(with a linear extrapolation to estimate the value of $\Omega(i,j)$ at
 $E_j = 0$)
to calculate the thermally-averaged $\Upsilon(i,j)$ 
 for these two datasets.

%
The $\Upsilon(i,j)$ of  SGC  
were obtained from the 
CHIANTI\footnote{http://www.chianti.rl.ac.uk/}
 atomic database version 4 (\citealt{young_etal:03}).
%
% contains upsilons for transitions between 
%the ground state and $n=3$ and $n=4$, respectively. 
%
For the $n=2 \to n^{'}=4$ transitions, we have added to the
comparison the distorted-wave unpublished results kindly provided by A. Bhatia,
 included in the CHIANTI database (hereafter B).


Table~\ref{others} presents a  comparison of the 
$\Upsilon(i,j)$ calculated at $T = {\rm 1.2 \times 10^7}$ \textdegree K
with the present study (CDMB) with those of BM, B and SGC. 
%
From Table~\ref{others} we see that for optically allowed transitions between states 
with the same spin multiplicity the agreement is almost perfect in the case of (1,52)
where CDMB/B = 1.01  but 
otherwise varies from between CDMB/SGC = 0.93 for (1,15) up to CDMB/SGC = 0.70 
for (1,25). For optically allowed transitions between singlet and triplet states the 
agreement varies from between CDMB/SGC = 1.11 for (1,13) up to CDMB/SGC = 1.37 
for (1,42) and  CDMB/B = 1.23 for (1,50).

%---------------------------------------
\begin{table*}[!htbp]
\caption{Fractional level population $N_{j}$ 
for the $n=2,3$ levels, calculated at 10$^{9}$, 10$^{14}$ (cm$^{-3}$)
electron  densities and the temperature $T$= 1.3 MK.
(R): computed with all the resonances and all the levels up to $n=4$;
(NR): computed with all the levels up to $n=4$, but neglecting the
contribution from the resonances;
(BM): computed with the DW collision strengths of \cite{bhatia_mason:86}, which
included only the $n=2,3$ levels.
}
\label{fractional-level}
\begin{center}
\begin{tabular}[c]{@{}rccccccc@{}}

\noalign{\smallskip}
\hline\noalign{\smallskip}
 i    & Level &  10$^{9}$ (R) &  10$^{9}$ (NR) &   10$^{9}$ (BM)  & 
10$^{14}$ (R) & 10$^{14}$ (NR) & 10$^{14}$ (BM)  \\ 

\hline\noalign{\smallskip} 
1 & 2s$^2$ $^1$S$_{0}^{\rm e}$ & 0.98554 &  0.99104 & 0.99019 &
 0.90014 & 0.93809 & 0.93757  \\ 
2 & 2s 2p $^3$P$_{0}^{\rm o}$ &  0.01446 & 9.0 10$^{-3}$ & 9.8 10$^{-3}$ &
  0.02465 & 0.01600 &  0.01754 \\ 
3 & 2s 2p $^3$P$_{1}^{\rm o}$ & 1.6 10$^{-9}$ & 1.1 10$^{-9}$ & 1.0 10$^{-9}$ & 
1.3 10$^{-4}$ &  9.5 10$^{-5}$ & 9.0 10$^{-5}$ \\ 
4 & 2s 2p $^3$P$_{2}^{\rm o}$ & 3.2 10$^{-6}$ & 1.6 10$^{-6}$ &  1.5 10$^{-6}$ &
 0.07508 & 0.04581 & 0.04479  \\ 
5 & 2s 2p $^1$P$_{1}^{\rm o}$ & 4.7 10$^{-11}$ & 4.5 10$^{-11}$ & 4.3 10$^{-11}$ &
4.3 10$^{-6}$ &  4.3 10$^{-6}$ & 4.1 10$^{-6}$   \\ 

6 & 2p$^2$ $^3$P$_{0}^{\rm e}$ &  7.8 10$^{-14}$ & 5.3 10$^{-14}$ & 1.8 10$^{-14}$ & 
 8.1 10$^{-9}$ &  5.3 10$^{-9}$ & 2.0 10$^{-9}$ \\ 
7 & 2p$^2$ $^3$P$_{1}^{\rm e}$ & 5.4 10$^{-13}$ & 3.1 10$^{-13}$ & 3.0 10$^{-13}$ &
 1.5 10$^{-7}$ & 8.6 10$^{-8}$ & 8.4 10$^{-8}$  \\ 
8 & 2p$^2$ $^3$P$_{2}^{\rm e}$ & 2.2 10$^{-13}$ & 1.3 10$^{-13}$ &  1.0 10$^{-13}$ &
 1.7 10$^{-7}$ & 9.9 10$^{-8}$ & 9.3 10$^{-8}$ \\ 
9 & 2p$^2$ $^1$D$_{2}^{\rm e}$ &  4.0 10$^{-13}$ & 2.9 10$^{-13}$ & 2.2 10$^{-13}$ &
  1.1 10$^{-7}$ &  6.8 10$^{-8}$ &  5.7 10$^{-8}$ \\ 
10 & 2p$^2$ $^1$S$_{0}^{\rm e}$ & 9.0 10$^{-14}$ &  6.8 10$^{-14}$ & 2.8 10$^{-14}$ &
8.6 10$^{-9}$ & 6.5 10$^{-9}$ & 2.8 10$^{-9}$ \\ 


11 & 2s 3s $^3$S$_{1}^{\rm e}$ &  3.7 10$^{-16}$ & 2.6 10$^{-16}$ &  1.5 10$^{-16}$ &
4.0 10$^{-11}$ & 2.8 10$^{-11}$ & 1.4 10$^{-11}$  \\ 
12 & 2s 3s $^1$S$_{0}^{\rm e}$ & 9.3 10$^{-15}$ & 9.1 10$^{-15}$ & 8.6 10$^{-15}$ &
8.5 10$^{-10}$ & 8.6 10$^{-10}$ &  8.2 10$^{-10}$ \\ 
13 & 2s 3p $^3$P$_{1}^{\rm o}$ &  1.3 10$^{-15}$ & 1.3 10$^{-15}$ & 1.0 10$^{-15}$ & 
1.2 10$^{-10}$ & 1.2 10$^{-10}$ &  9.7 10$^{-11}$ \\ 
14 & 2s 3p $^3$P$_{0}^{\rm o}$ & 6.6 10$^{-15}$ & 4.6 10$^{-15}$ & 2.7 10$^{-15}$ &
 8.3 10$^{-10}$ &  5.8 10$^{-10}$ & 2.6 10$^{-10}$ \\ 
15 & 2s 3p $^1$P$_{1}^{\rm o}$ & 1.2 10$^{-15}$ & 1.2 10$^{-15}$ & 1.0 10$^{-15}$ &
1.1 10$^{-10}$ & 1.1 10$^{-10}$ & 9.7 10$^{-11}$ \\ 
16 & 2s 3p $^3$P$_{2}^{\rm o}$ &  1.3 10$^{-14}$ &  9.6 10$^{-15}$ & 7.8 10$^{-15}$ &
2.0 10$^{-9}$ & 1.4 10$^{-9}$ & 7.4 10$^{-10}$ \\ 
17 & 2s 3d $^3$D$_{1}^{\rm e}$ & 6.4 10$^{-17}$ & 4.9 10$^{-17}$ & 3.3 10$^{-17}$ &
7.8 10$^{-12}$ &  5.9 10$^{-12}$ & 3.1 10$^{-12}$ \\ 
18 & 2s 3d $^3$D$_{2}^{\rm e}$ & 8.1 10$^{-17}$ & 6.9 10$^{-17}$ & 5.8 10$^{-17}$ &
9.6 10$^{-12}$ & 7.8 10$^{-12}$ & 5.5 10$^{-12}$ \\ 
19 & 2s 3d $^3$D$_{3}^{\rm e}$ & 1.1 10$^{-16}$ & 9.2 10$^{-17}$ & 7.9 10$^{-17}$ &
2.0 10$^{-11}$ & 1.5 10$^{-11}$ & 7.5 10$^{-12}$ \\ 
20 & 2s 3d $^1$D$_{2}^{\rm e}$ & 1.4 10$^{-15}$ & 1.3 10$^{-15}$ & 9.5 10$^{-16}$ &
 1.2 10$^{-10}$ & 1.3 10$^{-10}$ & 9.0 10$^{-11}$  \\ 

\noalign{\smallskip}
\hline
\noalign{\smallskip}
\end{tabular}
\end{center}
\end{table*}



For the transitions $n=2 \to n^{'}=3$ (1,20), (1,32), (1,35), (1,36), (1,37) 
the differences between the present and other calculations are not larger than
$35 \% $;  for the transition 
(1,12) we notice that CDMB/SGC =0.19. For most of the remaining 
$n=2 \to n^{'}=3$ transitions $\Upsilon$(CDMB) 
 exceeds $\Upsilon$(SGC) 
and $\Upsilon$(BM) by up to a factor 3.
 The reason for this is 
undoubtedly the neglect by 
SGC and BM of resonances, which in our calculation occur in great profusion 
and have a profound effect on the collision strengths of optically forbidden 
transitions. 


For the optically forbidden transitions up to $n=4$ the agreement between CDMB and B 
is strikingly good, which seems to indicate that the effect of 
resonances is negligible for these transitions. 


%The excellent agreement between CDMB and SGC for (?,?) is not 
%duplicated by BM since their result, curiously enough, is almost a factor 
%of 4 smaller. 

%--------------------------------------------------------------

\subsection{Level populations}


Transition probabilities were calculated with 
AUTOSTRUCTURE and the theoretical energies of Table
\ref{level-energies}, 
and used together with the 
$\Upsilon$ values to calculate the fractional level population $N_j$. 
Magnetic quadrupole transition probabilities were calculated 
with SUPERSTRUCTURE (see
 \citealt{eissner_etal:74}; \citealt{nussbaumer_storey:78}).
The level population equations were solved including the excitation and radiative 
decay  between all levels.  We obtain 
$N_j (N_{\rm e}, T_{\rm e})$, the 
population of level $j$ relative to the total 
number density of the ion, as  a function of the electron
temperature and density.

The values corresponding to the $n=2,3$ levels 
calculated at the temperature of maximum ion fraction ($T$= 1.3 MK)
and at two densities  are shown in Table~\ref{fractional-level}.
%--------------------------------------
At electron densities of $10^{14}$ cm$^{-3}$ most of the 
ion population is still in the $^1S_0$ ground state,
with only a small fraction in the first  $^3P$ excited states. 
This is a very high electron density for astrophysical plasmas.
Therefore, for astrophysical plasmas, almost all of the excitations 
to the higher states 
will come from the ground state, even if collision 
strengths from excited states are much larger than the 
values from the ground.


%-------------------------------------------------------------------
In order to estimate the effects that resonances have on the 
level populations (hence on line intensities), we have 
also calculated the  effective collision strengths using 
the spline fits to the $101$ $\Omega(i-j)$ data points, i.e. by
neglecting the contribution from the resonances. 
These values (NR) are also shown in Table~\ref{fractional-level}.
It is quite clear that the resonances have a dramatic 
effect on the populations of the $n=2$ levels, while 
the effect is much reduced for the $n=3$ levels.

For comparison, we have also calculated the level populations
using the same set of transition probabilities and the 
 effective collision strengths which we obtained 
from the DW data published in \cite{bhatia_mason:86}.
Good agreement  is found between these DW results 
(BM, Table~\ref{fractional-level}) and the 
present ones that neglect the resonances (NR). 
Note that the original level populations published by
\cite{bhatia_mason:81} differ from the present result.
This is due to two causes. First, the collision strengths
were revised in \cite{bhatia_mason:86}. Second, 
\cite{bhatia_mason:81} used a different set of transition
probabilities.


\section{Summary and conclusions}


The background values of the collision strengths shown in
Fig.~\ref{fig-1} confirm the  
reliability of previous DW calculations (\citealt{bhatia_mason:81};
 \citealt{bhatia_mason:86}). 
 However, the  resonance contributions have an important 
effect on the effective collision strengths and in turn on the level 
populations.
For the $n=2 \to 2$ transitions, the contribution of the
resonances is large.
The resonance contributions of optically forbidden transitions 
are significant for the $n=2 \to 3$ complex, particularly at low temperatures, but not
so important for the $n=2 \to 4$ transitions.

It is very important in these calculations to take account 
of the infinity energy limit to
obtain  effective collision strengths. This is now facilitated within the 
AUTOSTRUCTURE code for use with the ICFT version of the  R-matrix programs.

The accurate calculations for Fe$^{+22}$ presented in this paper
provide 
an opportunity for reliably analysing the X-ray
spectra from astrophysical plasmas.


\begin{acknowledgements}

This work was supported by  PPARC (HEM, GDZ, NRB) and by the 
  NSERC (MCC) (Natural Sciences and 
Engineering Research Council of Canada). One of us (MCC) wish to acknowledge the welcoming 
environment during her sabbatical visit to DAMTP, The Centre for Mathematical Sciences, 
Cambridge, funded via a PPARC fellowship.  
MCC is also sincerely grateful to the Master and Fellows of St Edmund's College for the award of a 
Visiting Fellowship, and their wonderful hospitality.

\end{acknowledgements}


\bibliographystyle{aa}

% 1- run latex paper-FeXXIII_3may  with \bibliography{biblio}
%      and coment \bibliography{paper-FeXXIII_13may}
% 2- run bibtex  paper-FeXXIII_3may with \bibliography{biblio}
%      and coment \bibliography{paper-FeXXIII_13may}
% 3- run latex paper-FeXXIII_3may.tex  with \bibliography{paper-FeXXIII_13may}
%      and coment \bibliography{biblio}

%\bibliography{biblio}

\bibliography{paper-FeXXIII_5aug}


%\vfil\eject




\end{document}



%\end{document}


%Figure 1
\begin{figure}
{\centerline{\psfig{figure=log-omega_1-15.ps,height=8.0cm,width=8.8cm,angle=0}}}
\caption{
An optically allowed transition}
\end{figure}

%Figure 2
\begin{figure}
{\centerline{\psfig{figure=0105-b.ps,height=8.0cm,width=8.8cm,angle=0}}}
\caption{
Optically allowed: 
full, present; dotted, Bhatia \& Mason (1986); 
dashed, Zhang \& Sampson (1992) }
\end{figure}

%Figure 3
\begin{figure}
{\centerline{\psfig{figure=0103-a.ps,height=8.0cm,width=8.8cm,angle=0}}}
\caption{An optically allowed intersystem transition}
\end{figure}

%Figure 4 
\begin{figure}
{\centerline{\psfig{figure=0103-b.ps,height=8.0cm,width=8.8cm,angle=0}}}
\caption{Optically allowed intersystem: 
full, present; dotted, Bhatia \& Mason (1986); 
dashed, Zhang \& Sampson~(1992)}

\end{figure}

%Figure 5 
\begin{figure}
{\centerline{\psfig{figure=0109-a.ps,height=8.0cm,width=8.8cm,angle=0}}}
\caption{An optically forbidden (electric quadrupole) transition}
\end{figure}


%Figure 6 
\begin{figure}
{\centerline{\psfig{figure=0109-b.ps,height=8.0cm,width=8.8cm,angle=0}}}
\caption{Optically forbidden (electric quadrupole): full, 
present; dotted, 
Bhatia \& Mason (1986); dashed, Zhang \& Sampson (1992); 
dot-dashed, Chen \& Ong (1998)}
\end{figure}


%Figure 7
\begin{figure}
{\centerline{\psfig{figure=0102-a.ps,height=8.0cm,width=8.8cm,angle=0}}}
\caption{An optically forbidden intersystem transition}
\end{figure}

%Figure 8
\begin{figure}
{\centerline{\psfig{figure=0102-b.ps,height=8.0cm,width=8.8cm,angle=0}}}
\caption{Optically forbidden intersystem: full, 
present; dotted, Bhatia \& Mason (1986); 
dashed, Zhang \& Sampson~(1992)}
\end{figure}

%Figure 9
\begin{figure}
{\centerline{\psfig{figure=0102-c.ps,height=8.0cm,width=8.8cm,angle=0}}}
\caption{Optically forbidden intersystem: full, 
present; dotted, Bhatia \& Mason (1986); 
dashed, Zhang \& Sampson~(1992)}

\end{figure}


%Figure 10
\begin{figure}
{\centerline{\psfig{figure=0102-d.ps,height=8.0cm,width=8.8cm,angle=0}}}
\caption{Optically forbidden intersystem: 
full, present; dotted, Bhatia \& Mason (1986); 
dashed, Zhang \& Sampson~(1992)}

\end{figure}

%Figure 11
\begin{figure}
{\centerline{\psfig{figure=0102-e.ps,height=8.0cm,width=8.8cm,angle=0}}}
\caption{Optically forbidden intersystem: 
full, present; dotted, Bhatia \& Mason (1986); 
dashed, Zhang \& Sampson~(1992)}
\end{figure}

%Figure 12
\begin{figure}
{\centerline{\psfig{figure=0102-f.ps,height=8.0cm,width=8.8cm,angle=0}}}
\caption{Optically forbidden intersystem: 
full, present; dotted, Bhatia \& Mason (1986); 
dashed, Zhang \& Sampson~(1992)}

\end{figure}


\end{document}

%----------------------------------------------------------------------
\begin{table}[!htbp] %add [h] placement to break table across pages
\caption{Wavelength, $\lambda$(\AA); weighted oscillator strength, $gf$; 
transition probability, $A_{ji}(s^{-1})$. Theoretical results from 
AUTOSTRUCTURE.}
\label{as_rad}
\begin{tabular}{@{}lrrcc@{}}
\hline\noalign{\smallskip}
 $i$-$j$ & $\lambda$(\AA) & $gf$ & $A_{ji}(s^{-1})$ & Type of transition  \\ 
\hline\noalign{\smallskip}  
1-11 & 11.2115 &   & 5.4 & M1 \\
1-13 & 11.0063 & 0.25 & 4.6 10$^{12}$ & E1 \\ 
1-15 & 10.9697 & 0.41 & 7.6 10$^{12}$ & E1 \\ 
1-17 & 10.8585 &   & 1.9 & M1 \\ 
1-18 & 10.8528 &   & 8.8 10$^{7}$ & E2 \\ 
1-20 & 10.7687 &   & 2.1 10$^{10}$ & E2 \\ 
1-22 & 10.6645 & 1.3 10$^{-2}$ & 2.5 10$^{11}$ & E1 \\ 
1-23 & 10.5630 &   & 1.5 & M1 \\ 
1-25 & 10.4960 & 2.3 10$^{-2}$ & 4.7 10$^{11}$ & E1 \\ 
1-26 & 10.4910 &   & 2.9 10$^{7}$ & E2 \\ 
1-27 & 10.4896 &   & 4.8 10$^{3}$ & M1 \\ 
1-30 & 10.3933 &   & 5.3 10$^{3}$ & M1 \\ 
1-33 & 10.3668 &   & 3.2 10$^{3}$ & M1 \\ 
1-35 & 10.3627 &   & 3.3 10$^{7}$ & E2 \\ 
1-36 & 10.3448 & 5.0 10$^{-3}$ & 1.0 10$^{11}$ & E1 \\ 
1-37 & 10.2957 &   & 6.4 10$^{7}$ & E2 \\ 
1-42 & 10.2272 & 2.0 10$^{-4}$ & 4.3 10$^{9}$ & E1 \\ 
1-46 & 10.1520 & 1.7 10$^{-2}$ & 3.8 10$^{11}$ & E1 \\ 
1-47 & 8.3520 &   & 1.1 & M1 \\ 
1-50 & 8.3072 & 2.7 10$^{-2}$ & 8.6 10$^{11}$ & E1 \\ 
1-52 & 8.2956 & 0.13 & 4.3 10$^{12}$ & E1 \\ 
1-53 & 8.2733 &   & 0.18 & M1 \\ 
1-54 & 8.2722 &   & 2.0 10$^{7}$ & E2 \\ 
1-56 & 8.2565 &   & 2.8 10$^{9}$ & E2 \\ 
1-62 & 8.0753 & 3.1 10$^{-4}$ & 1.1 10$^{10}$ & E1 \\ 
1-63 & 8.0483 &   & 5.8 10$^{-3}$ & M1 \\ 
1-64 & 8.0317 &   & 4.0 10$^{3}$ & M1 \\ 
1-65 & 8.0306 &   & 8.0 10$^{5}$ & E2 \\ 
1-70 & 7.9994 & 2.3 10$^{-3}$ & 8.2 10$^{10}$ & E1 \\ 
1-73 & 7.9937 &   & 1.2 10$^{8}$ & E2 \\ 
1-75 & 7.9931 & 3.0 10$^{-4}$ & 1.0 10$^{10}$ & E1 \\ 
1-77 & 7.9647 &   & 2.0 10$^{3}$ & M1 \\ 
1-79 & 7.9602 &   & 2.0 10$^{6}$ & E2 \\ 
1-80 & 7.9570 &   & 8.6 10$^{2}$ & M1 \\ 
1-81 & 7.9447 &   & 4.0 10$^{6}$ & E2 \\ 
1-87 & 7.9258 & 6.3 10$^{-5}$ & 2.2 10$^{9}$ & E1 \\ 
1-91 & 7.9167 &   & 4.4 10$^{7}$ & E2 \\ 
1-96 & 7.9124 &   & 5.7 10$^{-3}$ & M1 \\ 
1-97 & 7.9102 & 4.6 10$^{-3}$ & 1.6 10$^{11}$ & E1 \\ 
1-98 & 7.9097 &   & 1.3 10$^{8}$ & E2 \\ 
%
5-12 & 12.144 &   2.9 10$^{-2}$ & 1.3 10$^{12}$ & E1 \\
5-20 & 11.724 &  1.64 & 1.6 10$^{13}$ & E1 \\ 
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\end{tabular}
\end{table}