3 Results

We calculated collision strengths $\Omega$ for fine-structure transitions 3s²3p⁵-3s3p⁶, 3s²3p⁵-3s²3p⁴3d and 3s3p⁶-3s²3p⁴3d. In Table 4 we show a limited comparison with earlier work, the DW calculation of Bhatia & Doschek (1995), at energies above all thresholds. (Note that they also give data at 9 Ryd, and make a comparison with Mason (1975) at 5.5 Ry, but we cannot meaningfully compare at these low energies because of resonances, see Fig. 1.) Generally the agreement is reasonably good for excitations to level 4 and above (3s²3p⁴3d levels), but rather poorer for the 3s²3p⁵-3s3p⁶ doublet, the latter may be explained by the difference in calculated oscillator strength between the two calculations, as summarised in Table 3.

Figures 2 to 4 show some illustrative plots of the calculated collision strength at low scattering energies. All these transitions are affected by resonances in the first 2 or 3 Rydbergs above threshold: these resonances arise primarily from the 3s²3p⁴3d states, with resonances arising from the 3s3p⁵3d and higher states having negligible effect on transitions from the ground state (Figs. 2, 3). However, Fig. 4 show significant resonance structure up to 6 Ryd above threshold for excitation out of the 3s3p⁶ initial state, and this justifies the inclusion of the 3s3p⁵3d and higher levels in the 180-level BPRM calculation in order to obtain accurate data for these transitions.

Figure 2: Collision strength for 3s23p5 2P $^{\rm o}_{3/2,1/2}$-3s3p6 2S $^{\rm e}_{1/2}$ (1-3 and 2-3) in Fe X, as a function of electron energy (Ryds) relative to threshold: --, $\Omega$ from the present 180-level BPRM calculation; - - - -, the resulting thermal average ($\Upsilon$) plotted against kT in Ryds; . . . . ., $\Upsilon$ from Mohan et al. (1994); + + +, $\Omega$ from Bhatia & Doschek (1995)

Figure 3: Collision strength for 3s23p5 2P $^{\rm o}_{3/2}$-3s23p43d 4D $^{\rm e}_{5/2,7/2}$(1-4 and 1-5) in Fe X, as a function of electron energy relative to threshold: notation as in Fig. 2

Figure 4: Collision strength for 3s3p6 2S $^{\rm e}_{1/2}$-3s23p43d 4D $^{\rm e}_{5/2,7/2}$(3-4 and 3-5) in Fe X, as a function of electron energy relative to threshold: notation as in Fig. 2

Collision strengths are computed for the required fine-structure transitions over a sufficiently wide and fine energy mesh in order to be able to integrate over a Maxwellian distribution to obtain the effective collision strength $\Upsilon$, from which the excitation and de-excitation rate coefficients can easily be obtained (Hummer et al. 1993). Our energy mesh was determined by increasing the number of points until the integration converged: resulting in an energy spacing of 0.001-0.002 Ryd in the resonance regions, a total of 7460 energy points. The range of temperatures chosen was $\pm$ 0.8 dex of the temperature of maximum ionic abundance given by Shull & Van Steenberg (1982), and our final results are tabulated in Table 6.

 

 

Table 5: Effect of resonances on the derived level populations for Fe X. "BD'' is from Table IV A of Bhatia & Doschek (1995) for electron density 10¹⁰ cm^-3 and 10⁶ K. "Present''substitutes our rates from Tables 2 & 6 for those of BD

i	BD	Present	i	BD	Present
1	7.23E-01	6.19E-01	17	1.16E-08	1.31E-08
2	1.36E-01	1.73E-01	18	1.97E-02	1.55E-02
3	1.86E-09	1.39E-09	19	5.36E-09	1.31E-08
4	1.46E-06	3.70E-07	20	1.31E-02	1.28E-02
5	2.92E-02	4.10E-02	21	1.33E-02	1.05E-02
6	1.03E-07	1.19E-07	22	5.67E-08	4.48E-08
7	2.30E-08	4.43E-08	23	1.02E-08	6.11E-09
8	4.24E-02	8.68E-02	24	4.81E-03	3.88E-03
9	1.14E-09	1.66E-09	25	5.37E-10	5.73E-10
10	1.86E-02	3.69E-02	26	3.81E-08	7.29E-09
11	4.11E-08	3.09E-08	27	8.40E-11	7.83E-11
12	2.18E-08	1.90E-09	28	1.65E-10	3.25E-12
13	1.72E-09	6.33E-09	29	3.88E-11	4.92E-11
14	1.11E-09	1.85E-09	30	2.23E-10	2.08E-10
15	2.30E-09	2.16E-09	31	6.26E-11	7.48E-11
16	7.91E-09	3.32E-08

Our effective collision strengths $\Upsilon$ are also plotted in Figs. 2-4 as a function of kT Rydbergs, alongside the collision strength $\Omega$: the figures illustrate that the enhancement of $\Upsilon$due to low-energy resonances extends to surprisingly large temperatures ( $\sim 10^{6}$ K). Typical enhancements are factors of two or three for transitions from the ground state (Figs. 2-3) and up to an order of magnitude for the optically forbidden transitions out of the excited 3s3p⁶ level (Fig. 4).

For comparison we also plot the DW $\Omega$ from Bhatia & Doshek (1995) and the early R-matrix calculation of $\Upsilon$ of Mohan et al. (1994), showing that although our present results agree well with these at higher energies (see also Table 4), these other calculations appear to underestimate or ignore the resonance contribution at low temperatures.

To see the effect of the resonance enhancement more clearly, we recalculate in Table 5 the level populations given by Bhatia & Doschek (1995) for electron density 10¹⁰ cm^-3 and 10⁶ K, with no proton excitation or blackbody radiative excitation. We use all our radiative and collisonal rates from Tables 2 and 6 (i.e. for transitions involving levels 1, 2 and 3), and complete the dataset up to level 31 using Bhatia and Doschek's data: the rate equations are then solved as in their Eq. (3) for the level populations. Our resonance-enhanced $\Upsilon$ for 1-2 (²P _3/2-1/2), which we published in an earlier IP paper (Pelan & Berrington 1995), gives some redistribution of population between level 1 and 2. But the total ²P ground population drops 7% when the $\Upsilon$ from the ground state to higher levels also includes resonances as in the present work, and the population of ⁴F_9/2 and ⁴F_7/2 (levels 8 and 10 in Table 5) doubles.

   

Table 6: Effective collision strengths for Fe X fine structure transitions as a function of $\log T$ (Kelvin). The level indexing (i,i') is defined in Table 1. (The 1-2 data is from Pelan & Berrington 1995)

i-i'	5.4	5.6	5.8	6.0	6.2	6.4	6.6	6.8	7.0
1-2	2.97	2.69	2.27	1.79	1.35	0.99	0.73	0.54	0.40
1-3	0.7939	0.6812	0.5868	0.5133	0.4602	0.4241	0.4008	0.3867	0.3792
1-4	0.1742	0.1496	0.1262	0.1045	0.0849	0.0674	0.0522	0.0394	0.0291
1-5	0.2389	0.2063	0.1755	0.1468	0.1203	0.0963	0.0749	0.0566	0.0415
1-6	0.0993	0.0856	0.0722	0.0597	0.0484	0.0384	0.0298	0.0226	0.0168
1-7	0.0418	0.0360	0.0304	0.0250	0.0201	0.0158	0.0121	0.0091	0.0067
1-8	0.1702	0.1480	0.1252	0.1036	0.0839	0.0664	0.0514	0.0390	0.0291
1-9	0.0715	0.0625	0.0523	0.0428	0.0349	0.0284	0.0232	0.0189	0.0154
1-10	0.1281	0.1097	0.0913	0.0744	0.0595	0.0469	0.0368	0.0291	0.0234
1-11	0.0846	0.0722	0.0599	0.0488	0.0391	0.0308	0.0239	0.0184	0.0141
1-12	0.1132	0.1001	0.0863	0.0729	0.0606	0.0500	0.0411	0.0340	0.0286
1-13	0.0697	0.0603	0.0507	0.0419	0.0338	0.0264	0.0199	0.0146	0.0105
1-14	0.0944	0.0814	0.0682	0.0573	0.0480	0.0392	0.0311	0.0241	0.0185
1-15	0.0792	0.0729	0.0661	0.0574	0.0476	0.0380	0.0295	0.0225	0.0169
1-16	0.0543	0.0469	0.0394	0.0329	0.0283	0.0255	0.0240	0.0232	0.0228
1-17	0.1343	0.1209	0.1065	0.0930	0.0816	0.0726	0.0659	0.0611	0.0580
1-18	0.1351	0.1206	0.1054	0.0907	0.0785	0.0699	0.0648	0.0625	0.0621
1-19	0.1619	0.1375	0.1140	0.0953	0.0820	0.0732	0.0676	0.0644	0.0629
1-20	0.1627	0.1438	0.1257	0.1100	0.0974	0.0882	0.0823	0.0793	0.0787
1-21	0.1177	0.0995	0.0815	0.0650	0.0507	0.0387	0.0289	0.0211	0.0150
1-22	0.1101	0.0933	0.0768	0.0622	0.0500	0.0403	0.0330	0.0277	0.0241
1-23	0.1045	0.0891	0.0758	0.0648	0.0560	0.0493	0.0447	0.0420	0.0409
1-24	0.1288	0.1096	0.0943	0.0822	0.0728	0.0661	0.0622	0.0608	0.0614
1-25	0.0580	0.0530	0.0489	0.0453	0.0425	0.0406	0.0394	0.0388	0.0386
1-26	0.0594	0.0526	0.0473	0.0435	0.0416	0.0415	0.0424	0.0437	0.0450
1-27	1.908	1.917	1.925	1.933	1.939	1.943	1.950	1.963	1.982
1-28	3.856	3.870	3.898	3.934	3.966	3.992	4.018	4.053	4.101
1-29	0.4783	0.4762	0.4802	0.4924	0.5122	0.5362	0.5608	0.5839	0.6047
1-30	6.805	6.883	6.963	7.032	7.083	7.124	7.171	7.236	7.323
1-31	0.2161	0.2216	0.2240	0.2274	0.2363	0.2512	0.2690	0.2863	0.3018
2-3	0.3508	0.3111	0.2762	0.2481	0.2274	0.2131	0.2039	0.1985	0.1958
2-4	0.0596	0.0504	0.0418	0.0342	0.0274	0.0216	0.0166	0.0125	0.0092
2-5	0.0539	0.0444	0.0358	0.0284	0.0222	0.0170	0.0127	0.0093	0.0067
2-6	0.0532	0.0460	0.0389	0.0324	0.0264	0.0210	0.0163	0.0122	0.0090
2-7	0.0321	0.0283	0.0244	0.0206	0.0170	0.0137	0.0108	0.0082	0.0061
2-8	0.0292	0.0240	0.0192	0.0151	0.0116	0.0088	0.0066	0.0048	0.0034
2-9	0.0312	0.0282	0.0249	0.0216	0.0183	0.0150	0.0118	0.0090	0.0067
2-10	0.0498	0.0424	0.0352	0.0286	0.0227	0.0177	0.0136	0.0105	0.0082
2-11	0.0539	0.0461	0.0385	0.0314	0.0251	0.0196	0.0151	0.0114	0.0085
2-12	0.0787	0.0692	0.0588	0.0484	0.0386	0.0299	0.0226	0.0169	0.0127
2-13	0.0468	0.0399	0.0329	0.0271	0.0228	0.0200	0.0184	0.0173	0.0166
2-14	0.0360	0.0313	0.0262	0.0216	0.0175	0.0138	0.0106	0.0079	0.0057
2-15	0.0308	0.0270	0.0232	0.0194	0.0156	0.0122	0.0093	0.0070	0.0051
2-16	0.0516	0.0453	0.0386	0.0327	0.0285	0.0258	0.0240	0.0227	0.0216

 

Table 6: continued

i-i'	5.4	5.6	5.8	6.0	6.2	6.4	6.6	6.8	7.0
2-17	0.0471	0.0399	0.0329	0.0266	0.0212	0.0166	0.0128	0.0096	0.0070
2-18	0.0866	0.0710	0.0568	0.0452	0.0357	0.0275	0.0205	0.0149	0.0106
2-19	0.0757	0.0677	0.0592	0.0503	0.0424	0.0362	0.0317	0.0284	0.0260
2-20	0.0924	0.0798	0.0677	0.0563	0.0458	0.0364	0.0281	0.0211	0.0154
2-21	0.1118	0.0979	0.0852	0.0745	0.0661	0.0600	0.0561	0.0542	0.0539
2-22	0.0845	0.0726	0.0608	0.0501	0.0411	0.0339	0.0285	0.0247	0.0222
2-23	0.0642	0.0544	0.0458	0.0392	0.0348	0.0328	0.0325	0.0334	0.0352
2-24	0.0611	0.0532	0.0467	0.0414	0.0370	0.0337	0.0316	0.0308	0.0309
2-25	0.0381	0.0343	0.0304	0.0261	0.0215	0.0168	0.0125	0.0090	0.0062
2-26	0.0447	0.0399	0.0356	0.0316	0.0276	0.0238	0.0207	0.0183	0.0167
2-27	0.6641	0.6596	0.6610	0.6714	0.6911	0.7169	0.7446	0.7714	0.7961
2-28	0.1400	0.1386	0.1371	0.1393	0.1488	0.1646	0.1828	0.1999	0.2147
2-29	1.520	1.530	1.540	1.548	1.551	1.550	1.549	1.553	1.564
2-30	0.0292	0.0267	0.0242	0.0219	0.0197	0.0178	0.0162	0.0150	0.0142
2-31	4.289	4.335	4.386	4.431	4.462	4.481	4.500	4.533	4.583
3-4	0.0732	0.0606	0.0474	0.0351	0.0250	0.0173	0.0118	0.0079	0.0053
3-5	0.0749	0.0621	0.0486	0.0361	0.0259	0.0180	0.0123	0.0083	0.0055
3-6	0.0572	0.0480	0.0380	0.0285	0.0204	0.0142	0.0096	0.0064	0.0043
3-7	0.0337	0.0296	0.0244	0.0188	0.0137	0.0095	0.0065	0.0043	0.0028
3-8	0.0452	0.0356	0.0268	0.0193	0.0134	0.0090	0.0060	0.0039	0.0025
3-9	0.1418	0.1254	0.1036	0.0795	0.0575	0.0397	0.0266	0.0175	0.0113
3-10	0.0435	0.0347	0.0263	0.0189	0.0131	0.0088	0.0058	0.0038	0.0025
3-11	0.0400	0.0320	0.0243	0.0176	0.0123	0.0084	0.0057	0.0038	0.0026
3-12	0.1864	0.1659	0.1386	0.1074	0.0783	0.0545	0.0368	0.0245	0.0161
3-13	0.0519	0.0426	0.0332	0.0245	0.0174	0.0119	0.0080	0.0053	0.0034
3-14	0.0768	0.0657	0.0523	0.0395	0.0287	0.0203	0.0141	0.0098	0.0068
3-15	0.0655	0.0568	0.0471	0.0367	0.0270	0.0191	0.0132	0.0091	0.0063
3-16	0.0578	0.0521	0.0435	0.0338	0.0250	0.0181	0.0132	0.0099	0.0077
3-17	0.0858	0.0739	0.0599	0.0463	0.0347	0.0257	0.0193	0.0148	0.0118
3-18	0.0880	0.0722	0.0561	0.0417	0.0299	0.0209	0.0143	0.0097	0.0065
3-19	0.0956	0.0817	0.0659	0.0506	0.0383	0.0296	0.0240	0.0206	0.0186
3-20	0.0776	0.0654	0.0523	0.0398	0.0292	0.0209	0.0147	0.0103	0.0073
3-21	0.0742	0.0599	0.0463	0.0343	0.0245	0.0171	0.0117	0.0080	0.0055
3-22	0.0670	0.0558	0.0437	0.0322	0.0227	0.0154	0.0103	0.0067	0.0044
3-23	0.0485	0.0409	0.0325	0.0245	0.0177	0.0123	0.0084	0.0056	0.0037
3-24	0.0481	0.0401	0.0328	0.0259	0.0196	0.0142	0.0099	0.0067	0.0045
3-25	0.0415	0.0376	0.0328	0.0282	0.0241	0.0208	0.0184	0.0166	0.0154
3-26	0.0521	0.0470	0.0419	0.0372	0.0331	0.0298	0.0273	0.0256	0.0244
3-27	0.2772	0.2710	0.2328	0.1819	0.1331	0.0930	0.0632	0.0422	0.0280
3-28	0.1967	0.1746	0.1408	0.1057	0.0754	0.0520	0.0351	0.0234	0.0154
3-29	0.1179	0.1042	0.0839	0.0629	0.0449	0.0309	0.0208	0.0138	0.0091
3-30	0.1712	0.1370	0.1046	0.0772	0.0558	0.0398	0.0283	0.0203	0.0146
3-31	0.1199	0.0960	0.0736	0.0545	0.0396	0.0285	0.0205	0.0149	0.0110

Thus, we conclude that it is not safe to calculate rates from earlier tabulations of the collision strength without taking into account resonance enhancement. We believe that, by including resonance structure associated with 180 levels, we have included the most significant resonance effects on transitions to the 31 lowest levels.

Acknowledgements

This work was done with the support of a PPARC grant GR/K97608. We would like to thank Drs. P. Young and H. Mason for providing the level populations code.