3 The results

In Figs. 2 and 3 we compare calculated fine-structure collision strengths from the present 80-term R-matrix plus JAJOM model and the 9-level Dirac R-matrix model (Norrington & Grant 1987) in the scattering energy region up to the 3d4s thresholds, and we also compare with distorted-wave calculations of Nussbaumer & Storey (1982).

 

 

Table 3: Effective collision strengths for Fe VII 3d² fine structure transitions (i-i' using the indexation of Table 2) as a function of logT. A = Calculated E2 and M1 transition probability of the upper state (s^-1, with a superscript indicating a power of ten factor)

i-i'	A	4.3	4.5	4.75	5.0	5.25	5.5	5.75	6.0
1-2	.325-1	3.71	3.35	2.98	2.69	2.48	2.20	1.79	1.35
1-3	.167-8	1.30	1.17	1.02	.926	.862	.776	.634	.469
1-4	3.25	1.01	.959	.918	.873	.813	.721	.593	.455
1-5	.135	.295	.299	.299	.302	.305	.292	.259	.217
1-6	.502-1	.619	.633	.628	.623	.619	.585	.504	.403
1-7	.174	.516	.549	.548	.534	.518	.473	.387	.288
1-8	.959-3	1.14	1.24	1.26	1.20	1.11	.971	.786	.595
1-9	.181	.134	.115	.097	.087	.080	.071	.057	.043
2-3	.466-1	4.55	4.11	3.64	3.30	3.04	2.71	2.21	1.66
2-4	.603	1.38	1.29	1.23	1.16	1.08	.952	.780	.596
2-5		.228	.235	.231	.226	.222	.207	.174	.133
2-6	.762-1	.807	.833	.837	.837	.837	.794	.693	.564
2-7	.697-1	1.02	1.06	1.06	1.05	1.03	.964	.818	.637
2-8	.343	1.59	1.74	1.76	1.68	1.56	1.36	1.11	.837
2-9		.190	.162	.138	.123	.113	.100	.081	.061
3-4	.139-2	1.72	1.60	1.51	1.42	1.32	1.16	.943	.716
3-5		.171	.187	.191	.186	.181	.167	.139	.104
3-6		.649	.690	.692	.677	.663	.616	.516	.390
3-7	.735-1	1.89	1.94	1.95	1.94	1.93	1.84	1.60	1.30
3-8	.503	2.06	2.25	2.28	2.18	2.02	1.77	1.43	1.09
3-9		.249	.213	.181	.162	.149	.132	.107	.080
4-5	.472-6	.159	.172	.179	.178	.170	.154	.128	.099
4-6	.572-1	.490	.531	.553	.544	.515	.462	.382	.292
4-7	.191	1.022	1.06	1.07	1.03	.952	.833	.668	.495
4-8	.414-2	2.01	2.02	1.91	1.76	1.62	1.43	1.16	.889
4-9	26.7	.509	.473	.468	.483	.492	.480	.439	.384
5-6	.115-2	.350	.370	.380	.380	.389	.402	.375	.306
5-7	.139-7	.315	.324	.324	.329	.347	.363	.342	.287
5-8		.155	.164	.178	.186	.183	.166	.138	.106
5-9		.030	.035	.040	.041	.039	.034	.028	.021
6-7	.743-2	1.13	1.17	1.18	1.20	1.25	1.31	1.23	1.03
6-8		.466	.495	.536	.558	.549	.500	.416	.320
6-9	6.88	.088	.105	.120	.123	.116	.103	.084	.064
7-8	.454-4	.857	.903	.961	.988	.965	.875	.726	.558
7-9	1.11	.173	.197	.220	.225	.215	.192	.159	.122
8-9		.255	.310	.346	.345	.320	.273	.209	.146

Figure 3: Collision strength for electron excitation from the ground state $^{3}{\rm F}_{2}$ to $^{1}{\rm G}_{4}$ (upper plot) and $^{1}{\rm S}_{0}$ (lower) of Fe VII: -- present 80-term R-matrix calculation; - - - 9-level Dirac R-matrix (Norrington & Grant 1987); - - - - - - distorted-wave (Nussbaumer & Storey 1982)

These figures illustrate for the first time the effect of resonances, not included in previous calculations, in the collision strength over this energy range. These resonances are converging to 3d4l and 3p⁵3d³ excited states, and mask any differences in the prediction of the two earlier calculations of a straight-line background, indicating instead a much more complex energy dependency.

The large low-lying resonance near 0.5 Ryd was identified by an analysis of the Hamiltonian eigenvalues and vectors in the vicinity, and is due to 3p⁵3d³4p N+1-electron configurations, which are seen to dominate the low energy collision rate particularly for $^{3}{\rm F}_{2}-^{1}{\rm G}_{4}$. This confirms the importance of including 3p-hole correlation in the scattering calculation for this ion.

Because of the importance of such near-threshold resonances, we repeated the calculation using experimental energies to correct our target energies, with no appreciable difference in the final rates.

As a further test, we used the Breit-Pauli R-matrix approach in the 3d² threshold energy region, thus including fine-structure splitting explicitly. Only 5 terms (i.e. 9 levels) were included in this test because of computational constraints, though the same N+1 electron configurations were included as in our main 80-term R-matrix plus JAJOM run. Replacing the low energy "JAJOM'' collision strengths with those from the Breit-Pauli test run, and recalculating the effective collision strengths, gives an estimate of the error due to numerical difficulties in establishing the relative position of thresholds and resonances in the "JAJOM'' results presented in Table 3 as 20%.

Table 3 tabulates the resulting effective collision strength for Fe VII 3d² transitions in the electron temperature range log T(K) = 4.3-6. This includes the temperature of maximum ionic abundance which is given by Shull & Van Steenberg (1982) as log T = 5.6. We also calculated, using our wavefunctions in CIV3, the E2 and M1 transition probability of these forbidden transitions, and we show these also in Table 3. In conclusion, we have used an 80-term R-matrix model to calculate collision strengths for 3d² transitions which show in some cases strong enhancement of background due to large and broad open-shell resonances, and therefore show big differences with earlier compilations based on the 9-level Dirac R-matrix approach (Keenan & Norrington 1987, 1991). We have discussed the difficulties in modelling such cases, in a computer-tractable way, and estimate our results at 20% at best. Further improvements would require larger basis sets with more configuration interaction and relativistic effects to accurately represent the resonance structure near the thresholds.

Acknowledgements

This work was done with the support of a PPARC grant GR/K97608.