4. Discussion

We have used various R-matrix techniques for the calculation of collision strengths: the semi-relativistic TCC and the full intermediate coupling BP method. It is interesting to compare the collision strengths as a function of colliding electron energy obtained with the different approaches. This comparison demonstrates the influence of the consistent treatment of relativistic effects in the system target + electron on the effective collision strengths. Here we focus on selected KrIII, IV, V transitions from the ground state to excited states within the 4p ground configurations.

The collision strength of the fine-structure transition in KrIII is dominated by complex resonance structures at low electron energies (Fig. 1 (click here)a). The position and shape of the resonances close to threshold clearly affect the effective collision strengths especially at low temperatures. In the BP calculation the threshold is not at zero energy due to fine-structure splitting whereas in the TCC method the fine-structure thresholds are assumed to be degenerate. The effect of the full relativistic approach is well reflected in the effective collision strength (Fig. 2 (click here)a). In particular, for a temperature of 2000 K BP yields an effective collision strength which is 30% lower compared to TCC. The difference reduces with increasing temperature (10% at 10000 K) and both methods agree in the high temperature limit as expected.

  
Figure 1: Collision strength as a function of the incoming electron energy (Ryd) in the near threshold region of the a) KrIII\ PP, b) KrIV\ SD and c) KrV PS transitions (full curve: BP, dotted: TCC method). The arrows on the energy axis mark the target fine-structure threshold energies

  
Figure 2: Effective collision strength as a function of for the transitions a) KrIII PP, b) KrIV SD and c) KrV PS (full curve: BP, dotted: TCC method)

As to the SD transition in KrIV\ the BP calculation exhibits a remarkable Rydberg series of resonances in the collision strength at low energies (Fig. 1 (click here)b). The series is converging to the P threshold and superimposed on a broad resonance associated with bound channels in the J=4 partial wave of even symmetry. Again the effect of fine-structure splitting can be clearly seen at the excitation threshold. Hence it is not surprising that the BP effective collision strength exceeds the TCC result by almost a factor of 2 at a temperature of 2000 K (Fig. 2 (click here)b). However, the deviations rapidly diminish with increasing temperature such that the TCC results approach the BP values for .

Finally, Fig. 1 (click here)c demonstrates the collision strength for the PS transition in KrV where similarly to the case of KrIV the lowering of the collisional background close to threshold is prominent in the BP calculation. Moreover the comparison exhibits an important feature of the BP calculations, the shift of resonances due to relativistic spin-orbit effects on collisional complex terms. Accordingly the BP effective collision strength is much lower (35% at 2000 K) as compared to TCC (Fig. 2 (click here)c).

To our knowledge experimentally measured collision strengths are not available for any krypton ions. Were the krypton abundances in NGC 7027 known we could infer the effective collision strengths from the parameters in Tables 10-13 of PB94 which have been determined from the observed line intensities. It has been shown in PB94 that

with being the actual nebular abundance of the ion emitting the line and the solar system elemental abundance relative to hydrogen. is the effective collision strength for the transition from the ground state to the upper level j of the line . Since abundances are not known a priori we consider proper ratios of parameters, e.g.\

for transitions in the KrIV\ S-D multiplet at 5868.0 Å (1-2) and 5346.1 Å (1-3). The labelling of the transitions in braces refers to the indices of the fine structure energy levels in Table 5 (click here). For a nebular temperature of 13500 K (see Table 2 (click here) in PB94) the TCC and BP calculations yield which is well within the error limit of the measurements ( of Péquignot & Baluteau. We note that agreement could not be achieved with a non-relativistic approach where the collision strengths for excitation from the S ground state to the individual D fine-structure levels are obtained from LS coupling data using statistical weights and thus satisfy the relation . This result clearly demonstrates the importance of relativistic effects on the electron excitation of krypton ions.

On the other hand the parameters prove to be useful for determining the krypton abundance in NGC 7027. In their analysis (PB94) Péquignot & Baluteau tentatively presumed that collision strengths are similar for homologous forbidden transitions in Ar and Kr ions. However, by comparing our collision strengths for KrIII-V with the results of previous calculations for ArIII-V (Johnson & Kingston 1990; Zeippen et al. 1987; Mendoza 1983) we found that some effective collision strengths for Kr are larger by more than a factor of 2 compared to Ar. Consequently it is interesting to reanalyse the spectral data of PB94 and apply our new collision strengths for krypton ions.

We follow the method of PB94 except that we do not neglect the individual behaviour of the complex ions in collisional processes. Then for homologous transitions in argon and krypton ions we consider the ratio

and using (6) we rewrite (8) as

We emphasize that denotes the ratio of ionic abundances in NGC 7027 and is the ratio of solar system elemental abundances. In Table 6 (click here) derived from the values in Tables 10-13 of PB94 are listed for selected homologous transitions in Ar and Kr (n=2,3,4) ions. The transitions have been chosen such that for a specific ion number the uncertainty of quoted in PB94 is minimal.

  
Table 6: Homologous transitions in Ar and Kr ions used for determining the krypton abundance in NGC 7027

The nebular abundance of krypton in units of the solar system abundance (see Table 3 (click here) in PB94) is then given as

where is the ionization fraction of argon obtained from recent photoionization models of NGC 7027 (see Table 17 in PB94). The ratios of effective collision strengths have been calculated for electron temperatures according to Table 2 (click here) in PB94. The ``ionization correction factor'' icf (see Table 17 in PB94) accounts for contributions of unobserved highly ionized Ar and Kr ions to the sum in (10). No data are available for Ar and Kr, thus is set arbitrarily to unity as these ions are negligible.

Using (10) and the values of , and listed in Table 6 (click here) we find that krypton is overabundant in NGC 7027 by a factor of relative to the solar system value. We note that our result is lower by a factor of compared to the estimate of PB94. This is mainly due to the use of our calculated collision strength for the KrIV SD transition which exceeds that of the homologous transition in ArIV\ by more than a factor of 2.