2 The calculation

Paper I in this series gives the basic atomic theory, approximations and computer codes used in the IRON Project. For electrons incident with kinetic energies relative to the ground state of the target less than or equal to 350 Ry we make use of 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. Relativistic effects are allowed for, either by using the Breit-Pauli version of the R-matrix code, or by running the non-relativistic version and then carrying out an algebraic transformation of the appropriate collision matrix elements by means of Saraph's (1978) code JAJOM. We use the former procedure for energies below 116 Ry and the latter for energies from 116 to 350 Ry. We are able to extend our R-matrix collision strengths beyond 350 Ry with some confidence by using A. Burgess's graphics program OmeUps with the appropriate high energy limits (see Burgess & Tully 1992). For the optically allowed transitions the limits are determined by the oscillator strengths (see Table 7), while for the non-exchange optically forbidden transitions we have calculated the Born limits using the program discussed by Burgess et al. (1997) and results are given in Table 8.

The radial orbitals for the Be-like target are as follows: $P_{\rm 1s}$, $P_{\rm 2s}\,$ are from Clementi & Roetti (1974). $P_{\rm 2p}\,$ is the 2 exponent function

$$\begin{eqnarray} P_{\rm 2p}(r)&=&544.2971\;r^{2}\exp(-11.83976\;r)\nonumber\\ & &+63.0511\;r^{2}\exp(-20.83824\;r) \end{eqnarray}$$ (4)

which we obtained by using Hibbert's (1975) variational program CIV3 to minimise the sum of the energies of the $\rm 1s^2\,2s 2p\,\, ^1P^o \, and \, ^3P^o$ terms with trial exponents and coefficients taken from the Opacity Project. For the remaining orbitals P_nl, with $nl = {\rm 3s, 3p, 3d, 4s, 4p, 4d, 4f}$, we used the minimum number of exponents dictated by nl and calculated their values, as given in Table 2, by minimising the sum of the energies of the ${\rm 1s^2\,2s}nl\,\,{\rm ^1}l\, {\rm and\, ^3}l$ terms. We used the minimization routine VA04A by setting IDAVID = 0 and IVA04A = 1. (Note that the same procedure was adopted by Berrington et al. (1998) in spite of the unintentionally contradictory statements they make.) Furthermore, by setting MAXIT1 = 5 we limited the maximum number of iterations to 5. Additional work by one of us (JAT) has shown that some improvement can be obtained by using values of MAXIT1 larger than 5.

Our collision calculation makes use of theoretical target energies produced by the Breit-Pauli R-matrix code. They are given in Table 3 along with the observed energies for some of the levels taken from Corliss & Sugar (1982). Most of these observed values, with some small differences, can also be found on the Web page of the National Institute of Standards and Technology, http://physics.nist.gov. As can be seen there is very good agreement between theory and experiment, where we assume 1 Ry = 109737.32 cm^-1. Finally, we find that the data in Table 3 are in excellent agreement with results we have obtained using Hibbert's (1975) atomic structure code CIV3.

In order to delineate the multitude of resonance peaks we ran the Breit-Pauli code at 7704 values of the collision energy starting at 3.15210 Ry, relative to the ground state, and going up to 103.05816 Ry. Originally we meant to go as far as 115 Ry but finally abandoned this goal because of the excessive amount of computer time required. We therefore covered the interval between 103.05816 and 116 Ry by making a linear extrapolation backwards using the values of the collision strength at 116 and 127.5 Ry. In a few cases, especially for optically allowed transitions, a noticeable step up occurs in the collision strength when the energy increases beyond 103.05816 Ry, see Fig. 2.

  

Table 2: Exponents for the $\rm Fe^{+22}$ radial orbitals using analytic forms similar to that shown in (4). The coefficients are fixed by orthonormality conditions

  

Table 3: Fe XXIII level energies in rydberg units relative to the ground state. Theoretical results from the Breit-Pauli R-matrix program. Observed results from Corliss & Sugar (1982) assuming 1 Ry = 109737.32 $\rm cm^{-1}$. % diff is the percentage difference between the theoretical and observed energies

  

Table 4: Labels chosen to identify levels dominated by the same SLJ term

2.1 Energy levels

The 98 energy levels given in Table 3, together with the way of indexing them, are from the Breit-Pauli branch of the R-matrix program. Using the CIV3 code we have been able to make the identifications given in the columns with the heading "Label''$\!$. We use the customary procedure of letting the label be the configuration with the largest absolute mixing coefficient. However in four cases we abandoned this procedure in order to avoid the confusing situation of having the same label for more than one level. The last column in Table 4 shows SLJ terms which make a significant contribution to each of 4 pairs of levels where this problem arises. The mixing coefficients given in Table 4 are from CIV3 with ICSTAS = 1.