2. Calculations

In the following sections we only sketch the atomic structure and scattering calculations since detailed descriptions of the various methods are to be found elsewhere (IP93, Norrington & Grant 1987).

2.1. Semi-relativistic and Breit-Pauli R-matrix method

To begin with we recall that a consistent treatment of relativistic effects in both the target and scattered electron wavefunction such as in the BP\ method tends to lead to a problem of prohibitively large dimensions. This is due to the number of scattering channels in the R-matrix which are associated with the target fine-structure levels. They increase rapidly with the number of target terms included. Thus we employ close-coupling representations for the total wavefunctions target + electron based on a reasonably small number of target terms. We have generated configuration-interaction target wavefunction expansions using 7 (XeIII ), 10 (XeIV ), 6 (XeVI ), 5 (BaII ) and 3 (BaIV ) LS terms. A compilation of the spectroscopic and correlation target configurations is to be found in Table 1 (click here). The one-electron orbitals nl have been optimized using the SUPERSTRUCTURE package (Eissner et al. 1974; Nussbaumer & Storey 1978). Table 2 (click here) shows the corresponding adjustable scaling parameters in the statistical-model potential. From Table 3 (click here) it is obvious that the agreement between the calculated fine-structure energy levels and the experimental measurements generally is better than 10%. Nevertheless in the scattering calculations we have adjusted the theoretical target threshold energies to those measured except for a few levels where experimental energies are not available. We note that in Table 3 (click here) we have reordered the designation of some calculated levels, e.g. ²S _1/2 and ²P _1/2 in XeVI. These levels are strongly mixed by spin-orbit coupling and it is well known that SUPERSTRUCTURE could yield incorrect term designations. However, this does not imply that the wavefunctions of these target states are inaccurate.

 

XeIII XeIV XeVI BaII BaIV

4f5s²5p³ 4f5s²5p² 4f²5s 5s²5p⁶5d 4f5s²5p⁴

5s²5p⁴ 5s²5p³ 4f5s² 5s²5p⁶6s 4f5s5p⁵

5s²5p³5d 5s²5p²5d 4f5s5p 5s²5p⁶6p 5s²5p⁵

5s²5p³5f 5s²5p5d² 4f5p5d 5s²5p⁶6d 5s²5p⁴5d

5s²5p³6s 5s5p⁴ 5s²5p 5s²5p⁶7s 5s²5p⁴6d

5s²5p²5d² 5s5p³5d 5s²5d 5s²5p⁵5d6p 5s²5p⁴7d

5s²5p5d³ 5s5p²5d² 5s²5f 5s²5p⁵6s6p 5s5p⁶

5s5p⁵ 5p⁵ 5s5p² 5s²5p⁵6p² 5s5p⁵5d

5s5p⁴5d 5p⁴5d 5s5p5d 5s²5p⁵6p6d

5s5p³5d² 5p³5d² 5s5p5f 5s²5p⁵6p7s

5p⁶ 5s5d²

5p⁵5d 5p³

5p⁴5d² 5p²5d 5p5d²

Table 1: Configurations included in the target representations. All configurations include 1s²2s²2p⁶3s²3p⁶3d¹⁰4s²4p⁶4d¹⁰

 

 

XeIII XeIV XeVI BaII BaIV

1.4065 1.4067 1.4066 1.4053 1.4048

1.1252 1.1250 1.1249 1.1240 1.1236

1.0776 1.0774 1.0769 1.0769 1.0764

1.0559 1.0556 1.0553 1.0571 1.0567

1.0393 1.0391 1.0386 1.0404 1.0399

1.0242 1.0239 1.0235 1.0252 1.0248

1.0499 1.0494 1.0478 1.0564 1.0558

1.0461 1.0461 1.0449 1.0518 1.0515

1.0467 1.0470 1.0453 1.0533 1.0530

1.0610 1.0634 0.9918 1.0474

1.0594 1.0606 1.0560 1.0707 1.0752

1.0676 1.0664 1.0566 1.0728 1.0811

1.1116 1.0980 1.0707 1.0871 1.0612

1.0853 1.0134

1.2491 1.0837 1.1268

1.0910 1.1573 1.0861 1.2776

Table 2: Values for the adjustable scaling parameters in the statistical-model potential used to calculate the one-electron orbitals

 

 

XeIII BP XeIV BP XeVI BP

5s²5p^{4 3}P ₂ 0.0 0.0 5s²5p³ 0.0 0.0 5s²5p 0.0 0.0

³P ₁ 0.08925 0.09002 0.12090 0.13221 0.14215 0.13104

³P ₀ 0.07409 0.07829 0.15957 0.17024 5s5p^{2 4}P _1/2 -- 0.78493

¹D ₂ 0.15582 0.16465 0.25549 0.27424 ⁴P _3/2 -- 0.84614

¹S ₀ 0.32900 0.34103 0.32486 0.33678 ⁴P _5/2 -- 0.90621

5s5p⁵ 0.89544 0.89138 5s5p^{4 4}P _1/2 0.99560 0.97653 ²D _3/2 1.13790 1.10225

0.94379 0.94508 ⁴P _3/2 0.97436 0.95373 ²D _5/2 1.17763 1.12862

0.98721 0.98809 ⁴P _5/2 0.90820 0.88848 ²P _1/2 1.29252 1.32147

1.08465 1.13247 ²D _3/2 1.11109 1.13357 ²P _3/2 1.44994 1.49561

5s²5p³5d 1.02310 1.08250 ²D _5/2 1.14341 1.15894 ²S _1/2 1.43977 1.45668

1.01703 1.07244 ¹S _1/2 1.37362 1.41206 5s²5d ²D _3/2 1.64256 1.68784

1.01931 1.07194 5s²5p²5d ²P _1/2 1.24657 1.29287 ²D _5/2 1.66132 1.70721

1.02472 1.07405 ²P _3/2 1.21223 1.25997

1.02695 1.07325 ⁴F _3/2 1.23003 1.28973

1.10473 1.17872 ⁴F _5/2 1.24384 1.31168

1.06837 1.14450 ⁴F _7/2 1.29061 1.35729

1.11105 1.18585 ⁴F _9/2 1.33062 1.41806

²F _5/2 1.29240 1.36066

²F _7/2 1.32143 1.39262

⁴D _1/2 1.32232 1.37085

⁴D _3/2 1.33232 1.38798

⁴D _5/2 1.35491 1.41955

⁴D _7/2 1.42032 1.49284

BaII BP BaIV BP

5s²5p⁶6s ²S _1/2 0.0 0.0 5s²5p⁵ 0.0 0.0

5s²5p⁶5d ²D _3/2 0.04441 0.04779 0.15992 0.16672

²D _5/2 0.05171 0.05856 5s5p^{6 2}S _1/2 1.14643 1.14358

5s²5p⁶6p 0.18464 0.19546 5s²5p⁴5d ⁴D _7/2 -- 1.36803

0.20005 0.21806 ⁴D _5/2 1.31188 1.35983

5s²5p⁶7s ²S _1/2 0.38597 0.38870 ⁴D _3/2 1.31757 1.36843

5s²5p⁶6d ²D _3/2 0.41872 0.41758 ⁴D _1/2 1.33520 1.39080

²D _5/2 0.42060 0.42032

Table 3: XeIII, IV, VI and BaII, IV fine-structure energy levels in Ryd. Columns BP contain the calculated SUPERSTRUCTURE results. The experimental values are also given

 

As in Paper I we apply the following approaches to the solution of the electron scattering problem with the inclusion of relativistic effects: (a) the semi-relativistic TCC and (b) the full intermediate coupling BP R-matrix method for the large-scale calculations. In the TCC method the collisional problem is efficiently solved in LS-coupling and collision strengths for transitions between fine-structure target levels are obtained through algebraic recoupling of the transmission matrices (T=1-S) to intermediate coupling. Additionally J-J coupling between the target terms is included using a perturbation treatment with term-coupling coefficients (Saraph 1978). Clearly the applicability of this approach must be carefully investigated since it requires the term splittings to be small compared to the term separations. On the other hand the BP Hamiltonian contains the spin-orbit interaction and algebraic recoupling of the Hamiltonian matrices from LS to a pair coupling scheme includes all scattering channels explicitly including fine-structure.

The numerical solution of the scattering problem is achieved by means of the RAL version of the Iron Project R-matrix package (Eissner, priv.\ commun.) which can be run both in (a) LS and (b) intermediate coupling mode. Subsequently we employ the asymptotic region codes STGFJ (IP93) and STGFJJ (Eissner, priv. commun.) to calculate the collision strengths for case (a) and (b) respectively. For simplicity channel coupling has been neglected and Coulomb wavefunctions have been used in the asymptotic region. Test calculations have revealed that this approximation affects the collision strengths by at most for ions with residual charge z=1 but the effect of channel coupling decreases rapidly with increasing z.

Partial wave contributions are included for all total angular momenta and parity symmetries with for XeIII , for XeIV and XeVI , for BaIV and for BaII . With this choice, convergence for all the collision strengths could be achieved except for the dipole allowed transitions in BaII\ where a new "top-up'' procedure in STGFJJ was employed to estimate the contribution of high partial waves. The energy mesh was determined in terms of the effective quantum number relative to the next higher target threshold. Particularly narrow resonance structures of the collision strengths could be resolved with a small step width . However, close to thresholds where exceeds a value of we have used a constant interval length in (z-scaled) energy and the resonances have been averaged according to Gailitis' method (Gailitis 1963).

2.2. Dirac R-matrix method

Since we have applied the BP approach to the computation of collision strengths for heavy ions of xenon and barium it is desirable to estimate the accuracy of our results by comparison with the DC method. We consider XeIV as a test case and employ the multiconfiguration Dirac-Fock code GRASP² of Parpia et al. (1996) to obtain wavefunctions and energies for the 10 lowest jj-coupled configuration-state functions (CSF) formed from 5s²5p³ and 5s5p⁴ (Table 4 (click here)). Apart from 4f5s²5p² and 5s5p²5d² all correlation configurations from Table 1 (click here) have been included so that the computational dimensions could be kept to a reasonably small level. The self-consistent orbital calculation was performed in the extended average level (EAL) mode which involves an optimisation of the Hamiltonian trace weighted by the degeneracies of the particular CSF. We note that in our calculation only the Coulomb electron-electron interaction is included in the Hamiltonian and higher-order terms such as full transverse Breit and quantum electrodynamic contributions have been neglected.

 

XeIV Obs. DC BP

5s²5p³ 0.0 0.0 0.0

0.12090 0.15401 0.15164

0.15957 0.19469 0.20206

0.25549 0.29059 0.29641

0.32486 0.35324 0.38215

5s5p^{4 4}P _1/2 0.99560 0.99367 1.04365

⁴P _3/2 0.97436 0.97011 1.01678

⁴P _5/2 0.90820 0.90842 0.93688

²D _3/2 1.11109 1.14294 1.18177

²D _5/2 1.14341 1.17645 1.22011

Table 4: XeIV fine-structure energy levels (Ryd) used in the Dirac R-matrix method. The results in column DC are from a ten-state multiconfiguration Dirac-Fock EAL calculation. For comparison column BP contains the corresponding SUPERSTRUCTURE results. The experimental values from Tauheed et al. (1993) are also given

 

We have applied the DARC package (Norrington, priv. commun.) to the numerical solution of the collisional problem. Because the residual charge of the XeIV ion is relatively small and the channel energies of less than 10 Ryd are low, the asymptotic equations could be solved using non-relativistic Coulomb wavefunctions. Accordingly collision strengths were obtained from the asymptotic region code DSTGF (Norrington, priv. commun.) which has a structure similar to the BP code STGFJJ. The choice of computational parameters such as the maximum total angular momentum of the partial wave contribution tallies with the corresponding BP calculation.