2. The calculation

The R-matrix computer packages utilized in the present computations have been described in detail by Berrington et al. (1987). The twelve target eigenstates which are included in the wavefunction representation of the MgVI parent ion are 2s²2pS, ²D, ²P; 2s2pP, ²D, ²S, ²P; 2pP; 2s²2p²3sP, ²P, ²D and ²S. These target eigenstates are represented by configuration-interaction type wave-expansions constructed from seven orthogonal one-electron basis orbitals, consisting of four "spectroscopic" (1s, 2s, 2p, 3s) and three "pseudo" type orbitals (, , ), the latter being included to allow explicitly for additional correlation effects. The radial part of each of these orbitals may be expanded in the form:

Since the R-matrix program requires all the target states to be represented in terms of a single-orbital basis, our choice of orbital parameters (, , ) were as follows. The 1s, 2s and 2p orbital parameters were taken directly from the analysis of Clementi & Roetti (1974). The parameters for the 3s spectroscopic orbital were obtained by optimizing on the energy of the 2s²2p²3sP state using the configuration-interaction computer package CIV3 as described by Hibbert (1975). The configurations 2s2p⁴, 2s²2p²3s and 2s2p³[³P] were included in this optimization procedure. The parameters for the pseudo-orbital were similarly obtained by optimizing on the energy of the 2pP state using the configurations 2s²2p³, 2p⁵ and 2p⁴[³P], while those for the pseudo-orbital were found by optimizing on the 2s2pP level with configurations 2s2p⁴, 2s²2p²[³P] and 2s2p³[³P]. Finally the parameters for the only n=4 pseudo-orbital, , were obtained by optimizing on the energy of the 2s2pD state with configurations 2s2p⁴, 2p⁴3s, 2p and 2s2p³[¹D]. The parameters adopted for all seven orbitals are presented in Table 1 (click here). Table 2 (click here) displays the configurations, formed from this set of basis orbitals, which are retained in the wavefunction expansion for both even and odd parity target states.

   

The twelve target eigenstates included in the present calculation correspond to 23 fine-structure levels, each of which is assigned an index number in Table 3 (click here). Also shown in this table are the evaluated target state energies in atomic units relative to the 2s²2pS ground state of MgVI. An examination of these energies can give a reasonable measure of the accuracy of the ion representation utilized in the present calculation. On comparing with the most recent experimental values of Kaufman & Martin (1991) we see that most of the theoretical thresholds differ by less than 1 from the experimental values, the greatest disparity occuring for the 2s²2pS - 2s²2pD separation with a difference of less than 6%. Excellent agreement is also evident between the present thresholds and those obtained by the International Opacity Project (Burke & Lennon, unpublished). Following the normal R-matrix procedures the present calculation has been carried out with the energies shifted to their experimental values. Also shown in Table 3 (click here) are the total number of configurations retained to describe each state.

A further indication of the reliability of the present wavefunction representation can be found by examining the close conformity between the length and velocity oscillator strength values for transitions between the target states. In Table 4 (click here) we present oscillator strengths, obtained using the configuration-interaction computer code CIV3 (Hibbert 1975), for optically allowed transitions among the lowest twelve LS states of MgVI, in both the length and velocity formulations. Transitions for which the oscillator strength is less than 10^-5 are not included. Excellent agreement is evident between the present length and velocity values with differences better than 9. There is at present a severe lack of other available oscillator strengths for MgVI. Hence, our comparison is limited to those obtained by the Opacity Project. Good agreement is, however, evident between their length values and the length values obtained by the present calculation. This would suggest that the present length values can be assigned an accuracy by using the difference between the present results and those from the Opacity Project. With a few exceptions, this would suggest an accuracy of better than 5. We note that the results of Bhatia & Mason (1980) for the 2s²2p³ - 2s2p⁴ transitions are always larger than both the present data and the Opacity Project values by typically 30%.

The theory describing the expansion of the MgVI ion plus electron system has been discussed in detail by Burke & Robb (1975). The R-matrix boundary was set at 5.6 a.u., twenty continuum orbitals were included for each channel angular momentum and the electron-impact energy range of interest lies between 0 and 50 Rydbergs. All incident partial waves with total angular momentum , for both even and odd parities, and singlet, triplet and quintet multiplicities, were included in the wavefunction expansion. These seventy-eight partial waves were found to be sufficient to ensure convergence of the collision strengths for the optically forbidden transitions. Contributions from the partial waves with total angular momentum L>12 become much more important for the dipole allowed transitions and need to be taken into account. It has been conclusively shown by Ramsbottom et al. (1994, 1995, 1996a, 1997), when investigating electron-impact excitation of NIV, NeVII, SII and ArIV respectively, that it is possible to estimate quite accurately these additional contributions for allowed transitions by assuming the partial collision strengths form a geometric series with a geometric scaling factor equal to the ratio of two adjacent terms. Contributions from the high partial waves L>12 can be as much as 15-20% for some of the slowest converging transitions.

The R-matrix computer packages produce collision strengths for transitions between LS states only. However, for many astrophysical and plasma applications it is often the collision strengths among the fine-structure levels which are required. Transforming to a jj-coupling scheme can easily be achieved by utilizing the program developed by Saraph (1978). Finally a thermally averaged effective collision strength () may be derived from the collision strength () by

where is the electron temperature in K, the energy of the final electron and k is Boltzmann's constant.

   

Index J Level Mg VI Present Opacity No.

State LS Energy Configs.

1 3/2 0.0000 0.0000 0.0000 0.0000 31

2 3/2 0.2668 0.2523 0.2592 0.2652 70

3 5/2

4 1/2 0.3971 0.3827 0.3915 0.3762 86

5 3/2

6 1/2 1.1259 1.1341 1.1219 1.1283 70

7 3/2

8 5/2

9 3/2 1.5657 1.5572 1.5560 1.6029 97

10 5/2

11 1/2 1.8455 1.8308 1.8372 1.8680 47

12 1/2 1.9601 1.9403 1.9495 2.0214 85

13 3/2

14 1/2 2.9704 2.9741 2.9650 86

15 3/2

16 1/2 4.0865 4.0802 4.0985 70

17 3/2

18 5/2

19 1/2 4.1527 4.1448 4.1648 85

20 3/2

21 3/2 4.2984 4.2776 4.3031 97

22 5/2

23 1/2 4.5254 4.4808 4.5164 47

   

- f_L f_V f_L f_V

- 0.1967 0.2158 0.2008 0.2168 0.261

- 0.0985 0.0996 0.1255 0.1285

- 0.1168 0.1264 0.1180 0.1280 0.155

- 0.1679 0.1802 0.1680 0.1840 0.202

- 0.0442 0.0458 0.0465 0.0466

- 0.0481 0.0490 0.0582 0.0591

- 0.0348 0.0368 0.0347 0.0375 0.046

- 0.0708 0.0777 0.0718 0.0775 0.092

- 0.0913 0.0987 0.0893 0.0975 0.113

- 0.0705 0.0741 0.0635 0.0655

- 0.0371 0.0382 0.0365 0.0378

- 0.0168 0.0169 0.0248 0.0247

- 0.1012 0.1073 0.1020 0.1120

- 0.0380 0.0419 0.0373 0.0415

- 0.2025 0.2181 0.2067 0.2250 -