3 The scattering calculation

The close-coupling method in the R-matrix formulation was employed, using the programs developed for the IRON Project (Hummer et al. 1993). The mass and Darwin terms of the Breit-Pauli operator were included explicitly. Calculated target energies were replaced by the energies $E_{\rm target}$ listed in Col. 4 of Tables 2 to 4, to ensure that the series limits of resonances were as accurately positioned as possible. Orbitals of high valence states extend far out and can cause numerical problems in the scattering calculations. Therefore, the radius of the "inner region'' of the R-matrix method was relatively large, 19.2au for Cl V, 24.3au for S IV and 32.0au for P III. Up to 20 basis functions had to be used in the R-matrix method to span these extended inner regions. We considered scattering electron energies up to 1.5 Ryd; this range being sufficient for the determination of collision rates in plasmas with temperatures up to 30000 K. The expansion over total angular momenta converged for these energies by J=8.

The collision strengths are completely dominated by resonances. Since the aim of this calculation is to provide collision rates the delineation of the resonances has to be detailed enough so as to avoid errors when the collision strengths are sampled by the Maxwell velocity distribution function. This is particularly critical near the excitation threshold where we used small steps $\delta \nu =0.0002$ with $\nu$ the effective quantum number relative to the nearest higher threshold. When $\nu\gt 25$ resonances due to the corresponding threshold were averaged using the Gailitis method (Seaton 1983) and $\nu$ was taken relative to the next higher threshold. At energies between about 1 Ryd and 1.5 Ryd the closely-packed target states made this approach impractical and instead we used a small constant step in energy. In order to ensure that no essential information was lost the calculation was repeated at points shifted by half that steplength. This process was repeated with the steplength halved until it was found that the results obtained from two sets of non-overlapping data points differed by less than 1%. The final results were obtained by integrating over all datapoints.

  

Figure 2: Low energy collision strength for excitation of 3s23p(2P$^{\rm o}$1/2 - 2P$^{\rm o}$3/2) in P III, obtained using a 22 term CC expansion. $\times \quad \times \quad \times$  DW calculations by Krüger & Czyzak (1970)

These scattering calculations were performed in LS coupling and in order to obtain collision strengths for the fine-structure transition the T-matrix elements were transformed algebraically to pair coupling. The spin-orbit interaction between the target terms was included as a perturbation by a second transformation that incorporated the so-called term coupling coefficients (Saraph 1978). In practice, at the low energies considered here the collision strengths are hardly affected by term coupling.

  

Figure 3: Low energy collision strength for excitation of 3s23p(2P$^{\rm o}$1/2 - 2P$^{\rm o}$3/2) in S IV, obtained using a 21 term CC expansion. $\times \quad \times \quad \times$  DW calculations by Krüger & Czyzak (1970)

The fine-structure splitting of the target terms was neglected. This leads to some inaccuracy in the collision rate at the lowest temperature, but is not as serious for these light ions as for the ions discussed in IP XI due to the relatively smaller fine-structure splittings. The low energy collision strengths are shown in Figs. 2 to 4 where the results of the distorted wave calculations of Krüger & Czyzak (1970) are included for comparison.

  

Figure 4: Low energy collision strength for excitation of 3s23p(2P$^{\rm o}$1/2 - 2P$^{\rm o}$3/2) in Cl V, obtained using a 21 term CC expansion. $\times \quad \times \quad \times$  DW calculations by Krüger & Czyzak (1970), the point at 0.4 Ryd was obtained by interpolation