2. Atomic structure and radiative data

The atomic structure problem has been solved by employing the program SUPERSTRUCTURE (Eissner et al. 1974; Nussbaumer & Storey 1978), which has been developed over the years at University College London. This program uses multi-configuration expansions for the atomic eigenfunctions (CI). The structure and radiative calculations can be carried out in both LS coupling and intermediate coupling schemes. In the intermediate coupling case we diagonalise the Breit Pauli Hamiltonian, which includes the one-body and two-body relativistic operators, as specified in Eissner et al. (1974). The radial parts of the one electron wave-functions are calculated in scaled Thomas-Fermi-Dirac potentials. The scaling parameters, , are chosen to minimise selected sets of LS coupled term energies.

The ion has a core and five spectroscopically active electrons which occupy orbitals nl with and . We will be mostly interested in radiative data between the ground odd-parity configuration and the first two excited , even-parity configurations. However, it is necessary to include a greater number of more highly excited configurations, of both parities, in the wavefunctions basis set, in order to get accurate results. In Table 1 (click here) we list the configuration sets employed in different calculations to explore the accuracy of the wavefunction. For sets 1 to 3 we optimised the potential by minimising the energies of all the terms included in the expansion, therefore obtaining orbitals all of spectroscopic type. For set 3A the minimisation was done on 20 LS coupling terms only, those belonging to the , and configurations. In such a way we could study the effects of using 4s, 4p, 4d and 4f correlation orbitals, all taken of hydrogenic nature (a similar calculation carried out with a 4f Thomas-Fermi correlation orbital led to identical conclusions). These correlation orbitals are approximations to the real orbitals, introduced in order to allow for correlation with the missing configurations in the basis set, and can be highly contracted compared to the spectroscopic ones. The scaling parameters for the potential employed in the set 3A calculation are , , , , , , , , , . The wavefunctions obtained in the different computations have been used to calculate the length and velocity forms of the oscillator strength for transitions between LS coupling terms or intermediate coupling fine-structure levels. The agreement between the two forms and their stability as more configurations are added to the basis set are used as an indicator of the quality of the computation. In Table 2 (click here) we present results for the two forms of the oscillator strength, weighted with the statistical weight of the initial level, for selected optically allowed transitions in LS coupling between the ground and the first two excited configurations in FeXII, for the basis sets illustrated above.

The first point of interest is the clear improvement in going from the coarse 4 configuration model (set 1) to the much more comprehensive set including n = 4 electrons (set 3), as testified by the closer agreement of the and values. Correlation effects introduced by the n = 4 correlation orbitals (set 3A) do not consistently improve the results and any improvement is often only marginal.

   

set configurations

1 3s²3p³, 3s3p⁴, 3s²3p²3d, 3p⁵

2 set 1 + 3s3p³3d, 3p⁴3d, 3s3p²3d², 3p³3d², 3s²3p3d², 3s²3d³, 3s3p3d³, 3p²3d³

3 set 2 + (3s²3p²)4s, 4p, 4d, 4f; (3s3p³)4s, 4p, 4d, 4f; (3p⁴)4s, 4p, 4d, 4f

3A same as set 3 but with 4s, 4p, 4d, 4f correlation orbitals (hydrogenic)

   

Transition set 1 set 2 set 3 set 3A

gf(L) gf(V) gf(L) gf(V) gf(L) gf(V) gf(L) gf(V)

8.048 4.459 6.203 5.723 6.076 5.794 6.116 5.702

0.707 0.353 0.679 0.459 0.669 0.536 0.670 0.696

3.225 1.563 2.414 2.210 2.396 2.193 2.373 2.238

5.184 2.665 3.298 3.264 3.176 3.089 3.086 2.851

10.431 6.108 7.978 7.637 7.805 7.769 7.870 7.291

0.336 0.086 0.264 0.213 0.266 0.230 0.255 0.280

3.200 2.107 2.532 2.084 2.498 2.229 2.465 2.119

6.633 3.848 5.032 4.721 4.949 4.817 4.948 4.455

   

Configuration Level

(set 3) (set 3A)

3s²3p³ 0 0 0

44477 42789 41555

49049 46907 46088

77755 76895 74108

84059 82587 80515

3s3p^{4 4}P_5/2 274093 274389 274373

⁴P_3/2 283799 283883 284005

⁴P_1/2 288194 288235 288307

²D_3/2 343277 341900 339761

²D_5/2 345171 343595 341703

²P_3/2 396678 394281 389668

²P_1/2 401278 399595 394352

²S_1/2 417216 415694 410401

3s²3p²3d (³P)⁴F_3/2 435809 431849 420258

(³P)⁴F_5/2 439573 435616 424022

(³P)⁴F_7/2 445081 441131 429530

(³P)⁴F_9/2 451936 447979 436385

(¹D)²F_5/2 452611 449292 437194

(³P)⁴D_1/2 455867 452638 440316

(³P)⁴D_7/2 456492 453159 440941

(³P)⁴D_3/2 456966 453713 441414

(³P)⁴D_5/2 461606 458342 446055

(¹D)²F_7/2 471304 468104 455887

(¹D)²G_7/2 507320 503510 491903

(¹D)²G_9/2 510296 506378 494879

(³P)²P_3/2 513685 511268 501800

(³P)⁴P_5/2 524880 521908 512510

(³P)²P_1/2 525739 523255 513850

(³P)⁴P_3/2 528938 526123 516740

(³P)⁴P_1/2 531540 528826 519770

(¹S)²D_3/2 539623 537312 526120

(¹S)²D_5/2 550427 547826 538040

(¹D)²D_3/2 567870 565364 554030

(¹D)²D_5/2 569025 566648 554610

(¹D)²P_1/2 587307 586413 568940

(³P)²F_5/2 593418 590771 576740

(¹D)²P_3/2 595306 594486 577740

(¹D)²S_1/2 593956 592965 579630

(³P)²F_7/2 597798 595133 581180

(³P)²D_5/2 621026 618968 603930

(³P)²D_3/2 622632 620649 605480

There is however considerably better agreement between theoretical and observed energy levels when correlation orbitals are used, as can be seen in Table 3 (click here). We therefore regard the set 3A as our best approximation, as far as the provision of structure and radiative data is concerned. Another important point is apparent in the magnitude of several values of for transitions from terms of the excited configuration down to the configuration. Clearly radiative cascades from those levels are extremely important in populating the levels of the ground configuration. The availability of accurate collisional data for the population mechanisms of levels is therefore a key issue, which will be stressed in the next section. In Table 3 (click here) we present the list of fine-structure levels for the three energetically lowest configurations of FeXII, along with calculated and observed energy values. Theoretical energies are those obtained with our set 3 and set 3A models. Experimental values have been taken from the compilation by Corliss & Sugar (1982) and, where available, from the updated list by Jupen et al. (1993). The average difference between theoretical (set 3A) and observed values was found to be for the ground configuration, for the configuration and for the configuration. The improvement of our computation over previous works is shown by the comparison with corresponding results by Flower (1977) (, and respectively) and by Tayal & Henry (1986) (, and ). For the lowest twelve levels of the configuration experimental energies are not yet available. In order to fill this gap we used theoretical energies that were empirically corrected by adding the weighted average of the difference for other levels in the same configuration having the same parent term, whose energies are experimentally known. These estimates are distinguished by italic type in the column listing the experimental values.

   

Transition gf(L)

present work (set 3A) Bromage et al. (1978) Tayal & Henry (1986)

0.190 0.190 0.212

0.128 0.130 0.144

0.065 0.065 0.072

3.012 3.190

2.035 2.13

0.995 1.01

0.081 0.082 0.142

0.225 0.230 0.256

0.177 0.170 0.188

0.535 0.54 0.484

0.591 0.61 0.628

1.330 1.520 1.428

0.079 0.080 0.040

3.22 3.39

0.285 0.280 0.324

0.388 0.420 0.438

1.336 1.380 1.374

0.162 0.160 0.078

1.166 1.120 1.038

4.581 4.820

0.135 0.130

0.095 0.098 0.116

0.131 0.100 0.150

0.176 0.150 0.198

0.810 0.890

0.450 0.500

0.073 0.072 0.088

0.202 0.180 0.224

0.158 0.130 0.176

0.231 0.200 0.248

0.764 0.780

3.036 3.310

0.495 0.500

Finally in Table 4 (click here) we show weighted oscillator strengths, calculated with our set 3A model, for selected electric dipole transitions between fine-structure levels in intermediate coupling. There is generally a good agreement between these data, the corresponding values from Tayal & Henry (1986) and, in particular, those from Bromage et al. (1978). As in the latter work the authors performed empirical adjustments on the Slater parameters in order to minimise discrepancies between computed and measured energy levels, we regard the particularly good agreement with their oscillator strengths as highly satisfactory.