3 Target representation

As shown in Table 1, the Fe XV target contains all the 35 fine-structure levels within the n=3 complex. Configuration interaction (CI) wavefunctions are obtained with the structure code SUPERSTRUCTURE, originally developed by Eissner et al. (1974) with extensions by Nussbaumer & Storey (1978). CI arising from configurations containing orbitals with n>3 is neglected. The one-electron orbitals are generated in a statistical Thomas-Fermi-Dirac-Amaldi model potential $V(\lambda_l)$ described by Eissner & Nussbaumer (1969). The scaling parameters $\lambda_{l}$ are computed variationally so as to minimize the sum of the non-relativistic term energies. The optimized parameters for the present calculation are: $\lambda_0=1.1256$;$\lambda_1=1.0429$; $\lambda_2=1.0696$.

In Table 1 we compare the present level energies with experiment (Churilov et al. 1985, 1989; Litzén & Redfors 1987; Redfors 1988) and with the computed results of BM, BMB and CNP. A relevant feature in this level structure has to do with the assignments of the $3{\rm d}^2\ ^1{\rm D}_2$ and $^3{\rm P}_2$ levels (i=30 and i=33, respectively, in Table 1). These two levels are strongly mixed by relativistic couplings up to the point of making unambiguous assignments almost meaningless. The listed assignments are those given by experiment whereas SUPERSTRUCTURE usually inverts them (see, for instance, BM). Furthermore, energy positions for the $3{\rm d}^2\ ^3{\rm P}_{\rm J}$ levels have not been actually measured; the values listed in Table 1 have been obtained by Churilov et al. (1989) by fitting to spectroscopic data, making the order of levels 33 and 34 somewhat uncertain. For this reason they are treated in the present computations as degenerate.

  

Table 1: Comparison of experimental and theoretical energy levels (Rydberg units) for the Fe XV target. Expt: Churilov et al. (1985, 1989), Litzén & Redfors (1987) and Redfors (1988). Pres: present results. BM: Bhatia & Mason (1997). BMB: Bhatia et al. (1997). CNP: Christensen et al. (1985)

The agreement between present level energies and the measured values is better than 1% except for the 3p3d$\,^1$F$^{\rm o}_3$ and ¹P$^{\rm o}_1$ where it deteriorates to $\sim 2$%. The target used by BM (see Table 1) is very similar to the present one thus leading to very close level energies. The target selected by BMB includes the 78 levels from the following configurations: 3s², 3s3p, 3p², 3s3d, 3p3d, $3{\rm s}4l$ and $3{\rm p}4l$. A notable exclusion in this ansatz is the important $3{\rm d}^2$ configuration which thus results in a poorly represented 3p$^2\,^1$S₀ level, e.g. incorrect energy position above the 3s3d$\,^3$D₂ level (see Table 1). The target by CNP contains the 14 levels that arise from the 3s², 3s3p, 3p², 3s3d configurations and two additional levels from 3s4s; they also take into account extensive CI with configurations including orbitals with n=4 and n=5. All their level energies agree to better than 1% with experiment.

  

Table 2: Comparison of theoretical radiative rates A_ij (s^-1) for transitions to the lowest five levels of the Fexv target. Pres: present results. BM: Bhatia & Mason (1997). BMB: Bhatia et al. (1997). CNP: Christensen et al. (1985). ($a+b\equiv a\times 10^b$)

Following BM, computed A-values for transitions to the lowest five levels of Fexv are compared in Table 2. The agreement between present data and those by BM is as expected very good (within 2%). The comparison with CNP is also satisfactory: 89% of the A-values agree to within 10%, only finding differences of $\sim$13% for the 8-5, 6-5 and 11-5 transitions. The comparison with BMB, on the other hand, is less favourable as only 74% agree to 10%; large discrepancies (up to 37%) are found for transitions involving the 3p$^2\,^1$S₀ level (10-3, 10-5) and transitions involving the 3s3p configuration (7-3, 7-4, 14-4, 6-5, 9-5). A poor level of agreement is also found with BMB in a more extensive comparison with their listed gf-values (see Table 3) where only 70% agree to within 10%. Moreover, by running a structure calculation with the same target as BMB (same configurations and $\lambda_{l}$ parameters) but now including the $3{\rm d}^2$ configuration, the numbers of gf-values within the 10% accord goes up to 94% (see Table 3); larger differences are now only found for transitions with very small gf-values, e.g. 19-1, 23-1, 26-1, 23-6, 23-10 and 23-14. This finding has two important implications; firstly, by excluding the 3d² configuration BMB have weakened the general reliability of their target and, secondly, the neglect of CI from the n=4 complex in our target does not seem to lead to major consequences.

  

Table 3: Theoretical weighted oscillator strengths g_if_ij (length formulation) for the Fe XV target. Pres1: present results. BMB: Bhatia et al. (1997). Pres2: structure calculation with the same target as BMB but including the $3{\rm d}^2$configuration. ($a\pm b\equiv a\times 10^{\pm b}$)

  

Table 4: Comparison of collision strengths $\Omega_{ij}$ for Fe XV for transitions with j<15 at an incident electron energy of 50 Ryd. Some transitions with small collision strengths have been excluded. Pres: present results. BM: Bhatia & Mason (1997). BMB: Bhatia et al. (1997). CNP: Christensen et al. (1985). (Note: the data for the latter were computed at E=50.38 Ryd. $a\pm b\equiv a\times 10^{\pm b}$)