4 Results and discussion

In Table 5 we compare our total collision strengths between the the two levels of the ground term and the levels of the two even parity configurations 3s3p² and 3s²3d, with the work of BK93 and the values given by M 94. At the energies given in Table 5, (10 and 30 Ryd) none of the calculations contain any resonance features. The results of BK93 were obtained using the distorted wave method supplemented by the Coulomb-Bethe method for high partial waves. Their target basis contained only the five electron configurations given in Table 1 and their calculations were made at 10, 20 and 30 Ryd, above all thresholds. The agreement is generally good, as one would expect at these relatively high energies, with an average difference at 10 Ryd of 16.9%. The values quoted in M 94 at 10 Ryd are data computed by Dufton and Kingston in 1982 and deposited in the Belfast atomic databank. The calculations were made in the close-coupling approximation (see DK91 for more details). Again the agreement with the present work is good, with an average difference of 13.5%. Most of the difference between the present work and the other two calculations is attributable to the large differences for the strong allowed transitions to levels 8 and 9 (3s3p^2 2S_1/2 and ²P_1/2). If these are excluded, the mean difference between our results at 10 Ryd and those of BK93 is only 6.1% and with the values given by M 94, 4.1%. In our calculation, these two levels interact strongly, mainly through the spin-orbit interaction and the strength of the interaction is, to a first approximation, inversely proportional to the energy separation between them. In the calculation of BK93, this separation is 34715 cm^-1, compared to the experimental value of 23832 cm^-1 and our calculated value from Table 3 of 24267 cm^-1. We are therefore confident that the present work represents this interaction much more accurately than the results of BK93, and that consequently the distribution of collision strength between the 3s3p^2 2S_1/2 and ²P_1/2 levels is also more accurately represented. In DK91, target level energies are not given, but the collision strengths they obtain for the transitions between 3s²3p ²P$^{\rm o}$ and 3s3p^2 2S_1/2 and ²P_1/2 are very similar to those obtained by BK93 at 10 Ryd and are therefore also significantly different to those presented here.

In Table 6 we compare collision strengths at 6 Ryd from an early distorted wave calculation by Mason (1975) with the results of DK91 and with the present work. At this energy, the calculations of Mason (1975) and DK91 can be compared directly, since the distorted wave method does not include resonance effects and the highest target threshold of DK91 lies lower at 4.198 Ryd. There are significant differences between the two calculations caused by the limited target expansion used in the earlier work (Mason 1975, 1994). In the present work, the highest target threshold lies at 7.681 Ryd, so there are resonance features present at 6 Ryd. The values given in Table 6 were derived from the calculated collision strengths by averaging over the energy range 5.5-6.5 Ryd. These average values are significantly larger than the results of DK91, showing that resonances converging on the terms of the 3s3p3d electron configuration are important in this energy region.

   

Table 5: Comparison of collision strengths above all thresholds

i	j	10 Ryd			30 Ryd
		BK93 $^\dagger$	M 94$^{\ast}$	Present	BK93	Present
1	3	0.0170	0.016	0.0176	0.0140	0.0165
	4	0.0138	0.014	0.0127	0.0061	0.0062
	5	0.0105	0.010	0.0100	0.0045	0.0047
	6	0.716	0.717	0.735	0.859	0.931
	7	0.0257	0.025	0.0240	0.0141	0.0141
	8	0.986	0.991	1.316	1.208	1.701
	9	1.238	1.282	0.910	1.528	1.166
	10	0.924	0.954	0.906	1.135	1.162
	11	2.240	2.243	2.279	2.789	2.918
	12	0.0349	0.038	0.0392	0.0271	0.0293
2	3	0.0091	0.008	0.0086	0.0078	0.0080
	4	0.0202	0.020	0.0199	0.0123	0.0141
	5	0.0554	0.054	0.0579	0.0421	0.0512
	6	0.0843	0.082	0.0714	0.0732	0.0617
	7	1.146	1.133	1.131	1.357	1.395
	8	0.358	0.320	0.141	0.426	0.170
	9	1.368	1.414	1.550	1.677	1.954
	10	4.499	4.678	4.592	5.260	5.835
	11	0.626	0.630	0.619	0.762	0.764
	12	4.230	4.238	4.276	5.291	5.449
$^\dagger$ Bhatia & Kastner (1993).
$^{\ast}$ Mason (1994).

  

 

Table 6: Comparison of collision strengths at 6 Ryd

i	j	Mason	DK91 $^\dagger$	Present
		(1975)		(average)
1	3	0.0178	0.0200	0.0249
	4	0.0136	0.0167	0.0265
	5	0.0103	0.0127	0.0272
	6	0.930	0.671	0.704
	7	0.0257	0.0237	0.0915
	8	1.103	0.9326	1.278
	9	1.667	1.208	0.888
	10	1.196	0.900	0.914
	11	2.800	2.117	2.485
	12	0.0411	0.0358	0.190
2	3	0.0085	0.0101	0.0190
	4	0.0211	0.0243	0.0425
	5	0.0616	0.0665	0.0944
	6	0.0986	0.0767	0.163
	7	1.483	1.062	1.157
	8	0.325	0.302	0.198
	9	1.690	1.334	1.527
	10	5.840	4.412	4.468
	11	0.783	0.594	0.827
	12	5.296	3.997	4.754
$^\dagger$ Dufton & Kingston (1991).

In Fig. 1, we show the collision strength for the 3s²3p(²P $^{\rm o}_{1/2})~-$ 3s3p²(²D_5/2) transition, with the results in the resonance region averaged over 0.5 Ryd intervals. As described above, there is good agreement between the present work and that of DK91 in the non-resonant region above 8 Ryd, while at 6 Ryd, our results are significantly higher. The resonant enhancement seen in the collision strength in Fig. 1 between 4.3 and 7.7 Ryd is entirely due to the presence of the 3s3p3d electron configuration in the target. This enhancement is not accounted for in DK91 or BK93. It is evident from the figure that in this case, the contributions from the resonance region will be important in determining the thermally averaged collision strength as long as the mean thermal energy of the electrons is less than about 20 Ryd. This conclusion is confirmed by Fig. 2, which shows the thermally averaged collision strength for the 3s²3p(²P $^{\rm o}_{1/2})~-$ 3s3p²(²D_5/2) transition. We compare our current results with those obtained by DK91 and with values derived from the distorted wave collision strength data of BK93. Although the work of DK91 does include some resonance effects, it is clear that, for this transition, the most important resonance series are those converging on the 3s3p3d configuration which are absent in that calculation. Consequently, the results of DK91 are very similar to those of BK93 in which resonance effects are completely absent. We find similar resonance enhancements in all transitions between the 3s²3p and 3s3p²configurations, when comparing to both the work of BK93 and that of DK91, with the largest enhancements being in those transitions which are intrinsically weak. The effective collision strength in the 3s²3p(²P $^{\rm o}_{1/2})~-$ 3s3p²(⁴P_5/2) transition, for example, is increased by a factor of 2.9 at log T=6.2. The strong allowed transitions are also enhanced, but by much smaller factors.

The increases in the collision strengths between the 3s²3p and 3s3p² configurations will lead to corresponding increases in the populations of the levels of the 3s3p² configuration. The populations of these levels will also be increased by excitation from the ground 3s²3p to the levels of the 3s3p3d configuration followed by radiative cascading to 3s3p² and 3s²3d. Again, the most strongly affected levels will be those whose excitation from the ground configuration is intrinsically weak. We return to this point is Sect. 6.

In Table 7, the final thermally averaged collision strengths for the strongest EUV transitions between the 3s²3p and the 3s3p² and 3s²3d electron configurations are given as a function of electron temperature. The complete set of effective collision strengths among all of the forty levels listed in Table 3 are available in the electronic version of this paper.