3 Target representation

The Fe XVI target is represented by a 12-level approximation: 3s$\,^2$S_1/2, 3p$\,^2$P$^{\rm o}_{1/2}$, 3p$\,^2$P$^{\rm o}_{3/2}$,$3{\rm d}\,^2{\rm D}_{3/2}$,$3{\rm d}\,^2{\rm D}_{5/2}$, $4{\rm s}\,^2{\rm S}_{1/2}$,4p$\,^2$P$^{\rm o}_{1/2}$, 4p$\,^2$P$^{\rm o}_{3/2}$,4d$\,^2$D_3/2, 4d$\,^2$D_5/2, 4f$\,^2$F$^{\rm o}_{5/2}$ and $4{\rm f}\ ^2{\rm F}^{\rm o}_{7/2}$.The wavefunctions are obtained with the structure code SUPERSTRUCTURE, originally developed by Eissner et al. (1974) and generalized by Nussbaumer & Storey (1978). A summary of the code's main features is given by Eissner (1991). In this approach the wavefunctions are expressed in a configuration expansion of the type

$$\Psi=\sum_i\phi_ic_i \ ,$$ (23)

where the basis functions $\phi_i$ are constructed from one-electron orbitals computed in a Thomas-Fermi-Dirac- Amaldi statistical model potential $V(\lambda)$ as described by Eissner & Nussbaumer (1975) for $V(\lambda_l)$ and Nussbaumer & Storey (1978) for $V(\lambda_{nl})$.The scaling parameters $\lambda_{nl}$ are computed variationally so as to minimize the weighted sum of the non-relativistic term energies. The $\lambda_{nl}$ parameters for the present calculation are listed in Table 1 along with other properties of the target orbitals. Of particular relevance to the topping-up procedure are the last point of inflection and the mean radius of each orbital function. Tabulated radial functions are input to Stage 1 of the R-matrix code. Beyond the radial distance $r_{\rm cut}$ a Whittaker expansion can be employed up to the R-matrix radius of 4.0 Bohr radii a₀; there the most diffuse orbital (4d) has decayed to relative magnitude 0.002.

For the present system it is relatively easy to obtain an accurate target representation. Internal consistency and comparison with previous calculations and experiment (see Tables 2 and 3) suggest an accuracy of the target level energies and f-values within the 2% uncertainty range. Note that levels 9 and 10 are labeled incorrectly in Table 2 of Tayal (1994), whereas they are tabulated correctly in the present Table 2. The reported computed oscillator strengths for the 4-8 transition differ by up to a factor of 100. Varying the present $\lambda_{nl}$by as much as 10% would change our f-value of 0.0063 by not more than 20% (while unbalancing the excellent agreement between length and velocity results). Besides, the present oscillator strength for this transition agrees closely with the measurement listed in Fuhr et al. (1981) and the theoretical value obtained by Sampson et al. (1990). Other differences with the calculation by Cornille et al. (1997), namely for transitions 1-2, 2-6, 3-6, 1-7 and 1-8 in Table 3, are small, and they can be explained as the result of alternative optimization procedures in SUPERSTRUCTURE such as different weighting in the variational functional or fewer variational parameters (perhaps $\lambda_l$ rather than $\lambda_{nl}$).

  

Table 1: Properties of the Fe XVI target orbitals: binding energy in the potential $V(\lambda_{nl})$, mean radius, last point of inflection (both after Schmidt orthogonalization) and cut-off radius of SUPERSTRUCTURE output

  

Table: Comparison of experimental and theoretical energy levels (Rydberg units) for the Fe XVI target. Expt: Corliss & Sugar (1982). Pres: present results. T: Tayal (1994). CDMBB: Cornille et al. (1997)

  

Table: Comparison of computed absorption oscillator strengths f_ij (length formulation) for the Fe XVI target. Pres: present results. T: Tayal (1994); CDMBB: Cornille et al. (1997); FMWY: Fuhr et al. (1981); SZF: Sampson et al. (1990). Hutton et al. (1988) give experimental values for the first two entries: 0.115 $\pm$ 0.007 and 0.244 $\pm$ 0.015