3. The scattering problem

For the solution of the inelastic scattering problem we employed the R-matrix technique (Burke et al. 1971; Berrington et al. 1987; Burke & Berrington 1993; Hummer et al. 1993). This method has been implemented in the Queen's University Belfast R-matrix suite of programs, the latest and most updated version of which is thoroughly described in Berrington et al. (1995).

The scattering calculation is carried out in LS coupling, but the mass and Darwin relativistic energy corrections are included (Saraph & Storey 1996). The transformation to pair coupling including the effects of intermediate coupling in the target are included as discussed in the first IRON Project paper (Hummer et al. 1993).

Two sets of R-matrix calculations have been performed in order to analyse critically previous computations and to provide new electron scattering data for FeXII. These will be described in the two following subsections.

3.1. 7 term R-matrix computation

Our initial set of R-matrix calculations was aimed at studying the effect of varying target representations on collisional data and, in particular, on the resonance structure of the collision strengths below the highest excitation threshold in the target. As in Tayal et al. (1987) the lowest seven LS coupling terms, , , , , , , , were included in the expansion of the total wavefunction for the target. The energies for these excitation thresholds were calculated as averages of observed values, weighted over the fine-structure levels, and are tabulated as the first seven entries in Table 5 (click here). Experimental energies were preferred to theoretical values in order to determine as accurately as possible the positions of the convergence limits for Rydberg series of resonances. The seven target terms were represented by CI expansions constructed with a common set of radial functions, describing the radial charge distribution of the target. Three different combinations of seven orthogonal one-electron orbitals 1s, 2s, 2p, 3s, 3p, 3d and 4f were selected, in order to match as closely as possible the details of Tayal et al. (1987) calculation. In the first computation radial waves were obtained in the way described in Sect. 2 (click here) by including nine configurations in the basis set, , , , , , , , , , where the correlation orbital was chosen of Thomas-Fermi type (). The bar over the principal quantum number n indicates the correlation nature of the orbital. In the second calculation we added the two extra configurations and to the basis set, hence investigating the effect of a correlation orbital of hydrogenic type (). In these two calculations the size of the R-matrix "box'', within which exchange and correlation interactions are treated explicitly by performing a CI expansion of the collision complex wavefunction, was set at a = 3.22 a.u. and 16 continuum orbitals were included to ensure convergence for electron energies spanning the range 0.4 to 100 Ry. In our third calculation the same set of eleven configurations was kept in the basis expansion, but this time with 3d and 4f orbitals of spectroscopic type. Due to the presence of a diffuse 4f spectroscopic orbital, compared to the more contracted correlation type equivalent (see Table 6 (click here)), it was necessary to increase a to 5.34 a.u. and to include a total of 24 continuum orbitals in the problem. The wavefunction for the collision complex was, in all three cases, expanded on a basis set of 72 intermediate states, including partial waves of singlet, triplet and quintet spin multiplicities, both odd and even parities and with total orbital angular momenta L from 0 to 12, which corresponds to an expansion in intermediate coupling with values of J=L+S from 0 to 10. The variation of the collision strength, as a function of electron energy, with target structure is presented in the three plots of Fig. 1 (click here), Fig. 2 (click here) and Fig. 3 (click here), corresponding to the three calculations described above. Here collision strengths for the fine-structure forbidden transition are plotted in the energy region between the and target thresholds, revealing the complicated pattern of different series of resonances. A very fine energy mesh of was chosen to delineate this complex resonance structure. Clearly it is the correlation nature of the orbital which introduces the broad resonant features visible in Fig. 2 (click here). This important point will be further discussed in Sect. 4 (click here).

  
Figure 1: Collision strength for FeXII forbidden transition . R-matrix calculation including 7 LS coupling target terms with correlation orbital

  
Figure 2: Collision strength for FeXII forbidden transition . R-matrix calculation including 7 LS coupling target terms with , correlation orbitals

  
Figure 3: Collision strength for FeXII forbidden transition . R-matrix calculation including 7 LS coupling target terms with 3d, 4f spectroscopic orbitals

   

Configuration Term E(Ry)

3s²3p³ 0.0 0.40346 0.71424

3s3p^{4 4}P 2.55069

²D 3.10674

²P 3.56514

²S 3.73984

3s²3p²3d (³P)⁴F 3.91364

(³P)⁴D 4.03242

(¹D)²F 4.08134

(¹D)²G 4.49761

(³P)²P 4.60933

(³P)⁴P 4.69420

(¹S)²D 4.85952

(¹D)²D 5.05186

(¹D)²S 5.22953

(¹D)²P 5.23802

(³P)²F 5.27875

(³P)²D 5.50906

Table 5: LS coupling target terms included in our R-matrix calculations

   

Orbital 3d, 4f spectroscopic

3d 0.68862 0.69790 0.69789 0.67926 0.67929

4f 1.42519 0.83966 0.84152 0.83838 0.83985

3.2. 19 term R-matrix computation

A higher quality set of collisional data for FeXII was obtained by increasing the size of the expansion for both the target and the system total wavefunctions. In a new, ab initio R-matrix calculation we extended the target representation by including in the target the 12 LS coupling terms of the configuration, (, (, (, (, (, (, ( (, (, (, (, (. The list of energies for the total set of 19 LS coupling target terms employed in this calculation is given in Table 5 (click here). The lowest four energy values for the configuration (italic type) have been calculated from the corrected level energies presented in Table 3 (click here), due to lack of observed energies for the corresponding fine-structure levels, as discussed in Sect. 2 (click here). It should be noted that the energy value for the ( term in Table 5 (click here) does not correspond to the experimental energy given in Table 3 (click here). This observed value was modified in order to preserve the theoretical ordering of the term energies, as determined by the R-matrix computer code, where the ( term falls in between the ( and ( terms. Failing to preserve the theoretical ordering can lead to indexing problems in the R-matrix program. The correction was taken as the difference between the calculated and experimental energy values for the ( term. The radial waves describing the distribution of the bound electrons in the target were obtained with the twelve configuration basis of set 2 in Table 1 (click here). This represented a good compromise between target quality and computational speed. The scaling parameters for the potential employed in the set 2 structure calculation are , , , , , . A total of 16 continuum orbitals was included and the inner region boundary was set at 3.09 a.u. An initial expansion of the system total wavefunction on a basis set including partial waves with total angular momenta turned out to be insufficient to ensure convergence of the collision strengths for some of the forbidden transitions at high electron energies. The final 19 state calculation was therefore carried out for all partial waves with , again for both odd and even parities and singlet, triplet and quintet spin states, including a total of 102 intermediate states. In Fig. 4 (click here) we present results for the same transition illustrated in Fig. 1 (click here), this time spanning the whole closed channel energy range, going from the lowest ( ) to the highest ( ) excitation threshold.

  
Figure 4: Collision strength for FeXII forbidden transition . R-matrix calculation including 19 LS coupling target terms

The rapidly varying behaviour of the collision strength as a function of electron energy is again noticeable, this time enriched by the additional series of resonances converging to the extra thresholds belonging to the configuration. A top-up procedure to estimate the contributions to the collision strengths from partial waves with was adopted in the open channel energy region, above the highest excitation threshold. By assuming that the partial collision strengths form a geometric series with a geometric scaling factor equal to the ratio of the last two adjacent terms explicitly included, we obtain for the contribution, S, from
 
This top-up procedure was found to be appropriate for the ten optically forbidden transitions within the ground configuration, with a maximum correction for of at the highest electron energy considered (100 Ry). At the lowest energy in the open channel region (6 Ry) the maximum correction due to is found to be . Below 6 Ry, no top-up is considered necessary.

However for the optically allowed transitions the contribution from predicted by Eq. (1) is much larger (). Alternative top-up techniques are needed and results for these transitions will be presented elsewhere. Final collision strength values for the forbidden transitions between the five levels of the ground configuration are presented in Table 7 (click here), on a grid of energy points above the highest excitation threshold.

   

Transition E(Ry)

6.5 10 15 20 30 50 100

0.055 0.047 0.038 0.032 0.024 0.016 0.011

0.086 0.073 0.061 0.052 0.041 0.030 0.021

0.009 0.007 0.005 0.004 0.003 0.001 0.001

0.017 0.013 0.009 0.007 0.004 0.002 0.001

0.170 0.153 0.136 0.125 0.111 0.098 0.088

0.295 0.291 0.286 0.288 0.286 0.290 0.290

0.203 0.197 0.191 0.190 0.187 0.186 0.186

0.223 0.219 0.214 0.215 0.213 0.215 0.216

0.675 0.665 0.652 0.655 0.650 0.656 0.658

0.068 0.062 0.055 0.050 0.045 0.040 0.036

Electron excitation rates, obtained by averaging collision cross sections over a Maxwellian distribution of electron energies, are better represented in terms of effective (or thermally averaged) collision strengths, given by
 
where E_j is the colliding electron kinetic energy relative to the upper level j of the transition.

   

4 6 8 10 12 14 16 18 20 25 30

0.268 0.227 0.196 0.174 0.157 0.143 0.132 0.123 0.115 0.100 0.088

0.268 0.233 0.207 0.187 0.171 0.158 0.148 0.139 0.131 0.117 0.106

0.074 0.063 0.055 0.048 0.043 0.039 0.035 0.032 0.030 0.025 0.022

0.304 0.257 0.219 0.190 0.168 0.150 0.136 0.124 0.114 0.096 0.082

2.375 1.989 1.699 1.482 1.315 1.184 1.078 0.991 0.918 0.778 0.680

0.763 0.679 0.617 0.570 0.535 0.507 0.485 0.466 0.451 0.423 0.403

1.418 1.236 1.084 0.966 0.874 0.800 0.740 0.691 0.650 0.570 0.514

0.556 0.490 0.444 0.411 0.385 0.366 0.350 0.337 0.327 0.307 0.293

1.779 1.572 1.421 1.311 1.227 1.162 1.110 1.068 1.032 0.966 0.919

1.241 1.082 0.940 0.826 0.736 0.663 0.604 0.555 0.514 0.434 0.378

   

4 5 6 7 8 9 10

0.0732 0.0631 0.0559 0.0505 0.0463 0.0429 0.0401

0.0901 0.0796 0.0720 0.0663 0.0617 0.0580 0.0549

0.0175 0.0147 0.0126 0.0111 0.0100 0.0090 0.0083

0.0642 0.0529 0.0450 0.0392 0.0347 0.0312 0.0283

0.5487 0.4655 0.4080 0.3658 0.3335 0.3079 0.2872

0.3764 0.3600 0.3488 0.3407 0.3345 0.3297 0.3258

0.4399 0.3929 0.3606 0.3370 0.3190 0.3048 0.2934

0.2751 0.2638 0.2561 0.2505 0.2463 0.2430 0.2403

0.8580 0.8199 0.7940 0.7752 0.7609 0.7497 0.7407

0.3020 0.2538 0.2205 0.1960 0.1773 0.1625 0.1505

We have integrated our collision strengths using the linear interpolation technique described in Burgess & Tully (1992) and results are tabulated in Table 8 (click here) and Table 9 (click here) for two different temperature ranges. In integrating the collision strengths we made the assumption , i.e. constant at energies above the last calculated point. The contribution coming from this energy region to the total increases with and is generally . When it is we used italic type in Table 9 (click here). For the transitions , in particular, this contribution can be as high as at the highest temperatures. However, from Table 7 (click here) we see that the assumption of constant at high energies, particularly for the transitions , is excellent.