2. The calculation

Paper I gives the basic atomic theory, approximations and computer codes employed in the IRON Project. The CC approximation known as the R-matrix method is used. In the present case we have taken account of channel coupling up to the n=4 levels. Relativistic effects are allowed for as explained later.

The radial orbitals for the Li-like target are as follows: , are from Clementi & Roetti (1974). is the 2 exponent function

which we obtained by using Hibbert's (1975) variational program CIV3 to minimise the energy of the term. and are from Tully et al. (1990). Each remaining orbital P_nl, with , has the minimum number of exponents dictated by nl. The values of these exponents were calculated using CIV3 to minimise the appropriate term energy. The orbital exponents for n=3, 4 are given in Table 1 (click here).

 

 

. . . .

P_nl r¹ r² r³ r⁴

. . . .

14 . 437 7 . 9574 8 . 0058

9 . 600263 7 . 847094

8 . 0050

13 . 360597 6 . 031843 6 . 055029 6 . 021776

8 . 209951 5 . 974792 5 . 957608

6 . 367924 5 . 971795

6 . 000065

. . . .

Although configuration interaction (CI) wavefunctions are used to describe the target terms, in practice each term is dominated by a single configuration. The target energies used in the collision calculation were, with one important exception, adjusted to match the accurate experimental levels of Reader et al. (1992). These are given in Table 2 (click here) after being converted to Ry (1 Ry = 109737.32 cm^-1). The exception concerns each pair of 4l fine-structure levels which we forced to be degenerate with the corresponding LS term.

 

 

i Term j uncertainty

2 1/2 3 . 57201 0.00007 0.004

3 3/2 4 . 74549 0.00013 -0.014

4 1/2 84 . 497 0.015 0.034

5 1/2 85 . 461 0.024 0.048

6 3/2 85 . 815 0.025 0.024

7 3/2 86 . 197 0.015 0.034

8 5/2 86 . 321 0.022 0.019

9 1/2 113 . 584 0.006 0.035

10 1/2 113 . 990 0.015 0.030

11 3/2 114 . 136 0.015 0.018

12 3/2 114 . 266 0.020 0.050

13 5/2 114 . 321 0.025 0.040

14 5/2 114 . 342 0.015 0.028

15 7/2 114 . 379 0.022 0.014

Theoretical LS-coupling oscillator strengths calculated using the present wavefunctions and observed transition energies (see Table 2 (click here)) are compared in Table 3 (click here) with those which Peach et al. (1988) obtained in the Opacity Project. Our length (L) and velocity (V) forms agree well and there is also fairly good overall agreement between our length oscillator strengths and those of Peach et al. (1988). This suggests that our choice of wavefunctions is satisfactory for computing reliable collision data.

 

Transition L V OP

0.0525 0.0606 0.052

0.386 0.383 0.386

0.0956 0.0944 0.0955

0.0881 0.0947 0.0875

0.425 0.421 0.424

0.122 0.126 0.121

0.0170 0.0174 0.017

0.00365 0.00376 0.0037

0.0399 0.0395 0.0395

0.684 0.683 0.683

0.122 0.122 0.157

0.0149 0.0171 0.0151

0.599 0.597 0.598

0.0123 0.0124 0.0123

0.0263 0.0274 0.0268

1.016 1.016

0.00076 0.00078

 

Collision strengths for fine structure transitions are obtained from two R-matrix calculations which we now describe.

BP

is a 15-level fine-structure calculation using the Breit-Pauli Hamiltonian version of the R-matrix program (Berrington et al. 1995). This is the most accurate approach when the collision energy is comparable to the level splitting in the target. This method was used in the scattering energy range up to the highest threshold.

JAJOM

is a 9-term LS-coupling calculation plus a transformation to intermediate coupling which we apply to the reactance matrix using the JAJOM program written by Saraph (1978). This was used from just above the highest threshold up to a scattering energy of 265 Ry. For allowed transitions we use a "top-up" procedure which consists in assuming that beyond J=80 the partial collision strengths form a geometrical series. In this way we are able to complete the sum to infinity of partial waves analytically.

For the purpose of comparing our results numerically with those of Zhang et al. (1990) we choose an energy that lies above the highest target term included in our calculation, namely 4f. The comparison is shown in table 4 (click here) where it can be seen that differences are less than 10% for all transitions out of . For transitions from the 2p levels the differences are greater with some of them as high as 14%.

 

 

Transition BP DW

0.254 0.253

0.472 0.475

0.0137 0.0151

0.0053 0.0054

0.0101 0.0099

0.0118 0.0118

0.0176 0.0177

0.0025 0.0029

0.0011 0.0010

0.0022 0.0020

0.0021 0.0020

0.0031 0.0030

0.0012 0.0011

0.0016 0.0014

0.0196 0.0191

0.0011 0.0010

0.0142 0.0155

0.0034 0.0034

0.0609 0.0613

0.0065 0.0056

The approximations in each method are comparable at higher energies and any differences here are presumably caused by different "topping" up procedures and possibly by the fact that Zhang et al. (1990) use orbitals obtained by solving the Dirac equation with a Dirac-Fock-Slater potential. We expect really important differences to occur only at lower energies where resonance structures such as those shown in Figs. 1 (click here)-2 (click here) and 4 (click here)-7 (click here) occur. Resonances can have a big effect on effective collision strengths, as seen in Figs. 8 (click here) and 9 (click here).

Thermal averaging of the collision strengths is done using the "linear interpolation" method described by Burgess & Tully (1992). The resulting effective collision strengths are given in Table 5 (click here) for the astrophysically important temperature range where Fe⁺²³ is abundant under conditions of coronal ionization equilibrium (see Arnaud & Rothenflug 1985). For temperatures below two million degrees the abundance will be negligible. For this reason we begin our tabulation of in Table 5 (click here) at log. Astrophysical situations may exist where Fe⁺²³ is abundant at temperatures lower than this; in these cases one would need to extend the temperature range below K. This should pose no problem since our collision strengths will be preserved for posterity at the CDS (Centre de données astronomiques de Strasbourg) and some other databanks.

  
Figure 1: collision strength shown over the range (i.e. from to ). Full line: present Breit-Pauli calculation; broken line: DW calculation by Zhang et al. (1990)

  
Figure 2: collision strength shown over the range (i.e. from to ). Full line: present Breit-Pauli calculation; broken line: DW calculation by Zhang et al. (1990)

 

Transition 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0

1.661^-1 1.747^-1 1.853^-1 1.976^-1 2.117^-1 2.278^-1 2.459^-1 2.659^-1 2.873^-1 3.095^-1

3.172^-1 3.331^-1 3.523^-1 3.744^-1 4.000^-1 4.298^-1 4.642^-1 5.027^-1 5.445^-1 5.884^-1

1.603^-2 1.562^-2 1.522^-2 1.493^-2 1.484^-2 1.499^-2 1.534^-2 1.581^-2 1.631^-2 1.677^-2

5.573^-3 5.533^-3 5.566^-3 5.792^-3 6.328^-3 7.284^-3 8.747^-3 1.077^-2 1.339^-2 1.658^-2

1.038^-2 1.041^-2 1.052^-2 1.097^-2 1.198^-2 1.378^-2 1.650^-2 2.027^-2 2.511^-2 3.101^-2

1.280^-2 1.274^-2 1.263^-2 1.262^-2 1.284^-2 1.333^-2 1.408^-2 1.503^-2 1.608^-2 1.714^-2

1.960^-2 1.954^-2 1.930^-2 1.920^-2 1.943^-2 2.009^-2 2.118^-2 2.264^-2 2.444^-2 2.644^-2

2.497^-3 2.507^-3 2.526^-3 2.559^-3 2.621^-3 2.719^-3 2.842^-3 2.976^-3 3.105^-3 3.224^-3

1.100^-3 1.128^-3 1.174^-3 1.251^-3 1.374^-3 1.562^-3 1.834^-3 2.204^-3 2.681^-3 3.262^-3

2.176^-3 2.231^-3 2.323^-3 2.472^-3 2.705^-3 3.056^-3 3.564^-3 4.267^-3 5.184^-3 6.315^-3

2.095^-3 2.090^-3 2.086^-3 2.086^-3 2.098^-3 2.129^-3 2.188^-3 2.277^-3 2.397^-3 2.537^-3

3.168^-3 3.160^-3 3.154^-3 3.158^-3 3.177^-3 3.224^-3 3.311^-3 3.438^-3 3.598^-3 3.776^-3

1.233^-3 1.212^-3 1.189^-3 1.166^-3 1.146^-3 1.133^-3 1.130^-3 1.136^-3 1.148^-3 1.164^-3

1.646^-3 1.618^-3 1.587^-3 1.556^-3 1.529^-3 1.511^-3 1.506^-3 1.515^-3 1.538^-3 1.568^-3

3.734^-2 4.120^-2 4.291^-2 4.166^-2 3.807^-2 3.346^-2 2.896^-2 2.516^-2 2.225^-2 2.017^-2

4.738^-3 3.722^-3 2.950^-3 2.383^-3 2.004^-3 1.796^-3 1.748^-3 1.846^-3 2.076^-3 2.420^-3

1.650^-2 1.604^-2 1.562^-2 1.529^-2 1.513^-2 1.517^-2 1.538^-2 1.569^-2 1.602^-2 1.634^-2

7.156^-3 6.702^-3 5.999^-3 5.245^-3 4.568^-3 4.024^-3 3.626^-3 3.363^-3 3.214^-3 3.154^-3

5.389^-2 5.534^-2 5.734^-2 6.046^-2 6.524^-2 7.223^-2 8.188^-2 9.456^-2 1.104^-1 1.293^-1

1.267^-2 1.189^-2 1.063^-2 9.202^-3 7.834^-3 6.629^-3 5.633^-3 4.859^-3 4.296^-3 3.913^-3

4.298^-4 4.128^-4 3.933^-4 3.730^-4 3.549^-4 3.437^-4 3.448^-4 3.626^-4 3.989^-4 4.533^-4

2.745^-3 2.734^-3 2.723^-3 2.721^-3 2.741^-3 2.794^-3 2.873^-3 2.958^-3 3.038^-3 3.104^-3

1.402^-3 1.343^-3 1.268^-3 1.179^-3 1.081^-3 9.818^-4 8.916^-4 8.177^-4 7.634^-4 7.283^-4

9.901^-3 1.011^-2 1.043^-2 1.091^-2 1.161^-2 1.261^-2 1.396^-2 1.573^-2 1.793^-2 2.054^-2

2.571^-3 2.432^-3 2.253^-3 2.035^-3 1.790^-3 1.534^-3 1.290^-3 1.070^-3 8.847^-4 7.351^-4

1.698^-3 1.711^-3 1.742^-3 1.802^-3 1.900^-3 2.046^-3 2.245^-3 2.492^-3 2.772^-3 3.063^-3

1.051^-3 9.836^-4 8.993^-4 8.017^-4 6.971^-4 5.947^-4 5.035^-4 4.299^-4 3.760^-4 3.402^-4

1.044^-2 8.312^-3 6.629^-3 5.355^-3 4.483^-3 3.996^-3 3.872^-3 4.084^-3 4.604^-3 5.392^-3

8.007^-3 7.342^-3 6.490^-3 5.632^-3 4.886^-3 4.303^-3 3.888^-3 3.624^-3 3.487^-3 3.444^-3

4.098^-2 4.002^-2 3.863^-2 3.726^-2 3.624^-2 3.568^-2 3.556^-2 3.577^-2 3.616^-2 3.664^-2

2.691^-2 2.621^-2 2.503^-2 2.390^-2 2.319^-2 2.314^-2 2.389^-2 2.552^-2 2.803^-2 3.134^-2

1.108^-1 1.129^-1 1.156^-1 1.202^-1 1.279^-1 1.399^-1 1.569^-1 1.795^-1 2.080^-1 2.419^-1

8.680^-4 8.342^-4 7.957^-4 7.566^-4 7.232^-4 7.053^-4 7.139^-4 7.576^-4 8.400^-4 9.600^-4

1.376^-3 1.319^-3 1.246^-3 1.161^-3 1.068^-3 9.774^-4 8.972^-4 8.360^-4 7.998^-4 7.902^-4

7.294^-3 7.216^-3 7.123^-3 7.040^-3 7.016^-3 7.078^-3 7.204^-3 7.349^-3 7.479^-3 7.580^-3

5.102^-3 4.974^-3 4.820^-3 4.653^-3 4.499^-3 4.397^-3 4.387^-3 4.503^-3 4.759^-3 5.152^-3

2.029^-2 2.056^-2 2.099^-2 2.169^-2 2.278^-2 2.440^-2 2.671^-2 2.981^-2 3.374^-2 3.846^-2

1.908^-3 1.823^-3 1.721^-3 1.610^-3 1.502^-3 1.411^-3 1.350^-3 1.326^-3 1.338^-3 1.376^-3

3.834^-3 3.812^-3 3.809^-3 3.850^-3 3.954^-3 4.142^-3 4.432^-3 4.821^-3 5.285^-3 5.787^-3