3 Results

In this section we present a small sample of the fine structure collision strengths in three different approximations: (a) 19BP, (b) 8CC, and (c) 34CC, and compare their Maxwellian-averaged collision strengths (the term coupling results, 8CC+TCC, are not considered as they are found to be nearly identical with the NR 8CC results).

The extensive resonance structures are displayed in Figs. 1 for the collision strengths for the first transition $\Omega \rm (^4F_{3/2}~-~^4F_{5/2})$.As mentioned in the previous section, the mesh is considerably more refined in the low energy region E < 0.25 Rydberg, than at higher energies, so as to resolve accurately the near threshold resonances. The 8CC and the 34CC calculations assume degenerate fine structure level energies belonging to a given LS term, whereas the 19BP calculation employs observed energies. The 19BP results in Fig. 1a differ in detail from the 8CC and the 34CC results (Figs. 1b,c) , both of which are very nearly the same, as expected for low-lying transitions; the differences are only expected to manifest themselves at higher energies owing to additional target states in the larger 34CC calculation. Figures 1b and 1c are nearly identical up to about 0.4 Rydberg. The 19BP calculations show significantly different positions and shapes for the individual features. The dense resonance structures and the background collision strengths appear consistent, so except at the low temperatures where there is no abundance of Fe VI in astronomical objects and transitions between very high-lying levels the averaged collision strengths are almost not influenced by relativistic effects, as seen from the comparisons in Tables 2 and 3 (described below).

  

Figure 1: Collision strengths for the first transition $\Omega(^4{\rm F}_{3/2}~-~^4{\rm F}_{5/2})$:a) 19BP Breit-Pauli R-matrix calculation; b) 8CC nonrelativistic algebraic recoupling; c) 34CC nonrelativistic algebraic recoupling. The "$\bullet$'' and the "*'' near the threshold energy were calculated by Nussbaumer & Storey (1978) and Garstang et al. (1978), respectively

Figures 2 present further examples of resonances in other transition $\rm ^4F_{3/2}~-~^4F_{7/2}$ within the ground term. One main feature is highlighted. In Figs. 2a-c the near-threshold region is full of dense resonances as in Figures 1, but the elevated background at low energies E < 0.25 Rydbergs is more pronounced relative to higher energies. The reason for the high density of resonances in this low energy range is the presence of a number of closely spaced thresholds (Table 1) that give rise to several overlapping Rydberg series of resonances.

  

Figure 2: $\Omega(^4{\rm F}_{3/2}~-~^4{\rm F}_{7/2})$ for fine structure levels within the ground term

The dominant role of resonances is clear, particularly compared to the two earlier calculations by Nussbaumer & Storey (1978) and Garstang et al. (1978), which correspond only to the non-resonant background. The Nussbaumer and Story calculations were in DW approximation that does not enable a treatment of resonances. However, the Garstang et al. calculations were in the close coupling approximation using a small basis set (Garstang et al. 1978), but they did not obtain the resonance structures and presented only averaged values (these are denoted as asterisks near the threshold energy). The computed rate coefficients are higher by several factors in comparision with the earlier collision strengths. For example, the rate coeffcient for the transition $\rm ^4F_{3/2} ~-~^4\!_{5/2}$ (Table 1, first transition) is approximately a factor of 5 higher (at 20 000 K) than the previous collision strengths shown in Fig. 1c. For the $\rm ^4F_{3/2}~-~^4F_{7/2}$ transition the differences with previous data are about a factor of 7.

Because of the larger target expansion in the 34CC case (as shown in Fig. 7 of CP), there are more extensive resonance structures in the 34CC case compared to the 19BP or the 8CC case at high impact energies. Consequently, as seen below, the 34CC rate coefficients should show a larger resonance enhancement especially at high electron temperatures.

The procedure for obtaining the Maxwellian-averaged collision strength, or the effective collision strength, can be found in earlier publications in this series (e.g. see Papers I, VI or XVIII). This quantity is defined as

$$\Upsilon _{ij}=\int _{0}^{\infty}\Omega _{ij}{\rm e}^{-\epsilon _j/kT}{\rm d} (\epsilon_j/kT).$$ (1)

Fe VI is abundant in H II regions at around 20 000 K (Hyung & Aller 1997), and in collisionally ionized coronal type sources at around 200 000 K (Arnaud & Rothenflug 1992). Tables 2 and 3 correspond to these rather disparate temperature ranges. We compare the Maxwellian-averaged collision strengths, in the three approximations, for a few of the low-lying transitions at three representative temperatures for Fe VI. Surprisingly, we find that the differences between the three sets of calculations is large only for transitions between very high-lying levels or at the lowest electron temperature, 10 000 K, when the Maxwellian samples a small energy range above threshold, typically about 1 eV or 0.1 Rydberg. At higher temperatures, considered in Tables 2 and 3, the differences are small; for example at 100 000 K the differences are no more than a few percent (Table 3). It follows that the relativistic effects are not important as far as the calculation of rate coefficient is concerned. This fact is of considerable practical significance since it implies that we can accurately compute the effective collision strengths for the low-lying transitions from among the 80 fine structure levels using the large 34CC dataset. However the fine structure effects are expected to manifest themselves more strongly for transitions among the closely spaced higher lying levels, and the uncertainties (discussed later) will be larger than for the transitions among the 19 levels explicitly studied herein. In the practical calculations of $\Upsilon _{ij}$ in Eq. (1), the impact energies $\epsilon _j$ go up to 100 Rydbergs instead of $\infty$. According to the temperatures of abundance for Fe VI in astronomical objects under coronal equilibrium, the maximum temperature is 10⁶ K (Table 3). Therefore the Maxwellian factor in Eq. (1) ${\rm e}^{-\epsilon _j/kT}={\rm e}^{-100*157885/10^6}\simeq {\rm e}^{-16}$, which ensures the convergence for the calculation of Eq. (1) over the full temperature range covered. However, for more highly ionized systems such as the ongoing calculation of Fe XVII, the upper bound of $\epsilon _j$ used should be at least a few hundred Rydbergs as the temperature of the maximum abundance of Fe XVII is 10⁶ K to 10⁷ K.

  

Figure 3: The collision strengths enhanced by CBe top-up for the optically allowed transitions a) $\rm 3{\rm d}^3\ ^4{\rm F}_{3/2} - 3{\rm d}^24p\ ^4G^{\rm o}_{5/2}$,b) $3\rm {\rm d}^3\ ^4{\rm F}_{3/2} - 3{\rm d}^24p\ ^4F^{\rm o}_{3/2}$,c) $\rm 3{\rm d}^3\ ^4{\rm F}_{5/2} - 3{\rm d}^24p\ ^4G^{\rm o}_{7/2}$,and d) $\rm 3{\rm d}^3\ ^4{\rm F}_{7/2} - 3{\rm d}^24p\ ^4G^{\rm o}_{9/2}$,from the 34CC calculation

Figures 3 show the collision strengths for the optically allowed transitions (a) $\rm 3{\rm d}^3\ ^4F_{3/2} - 3{\rm d}^24p\ ^4G^{\rm o}_{5/2}$,(b) $\rm 3{\rm d}^3\ ^4F_{3/2} - 3{\rm d}^24p\ ^4F^{\rm o}_{3/2}$,(c) $\rm 3{\rm d}^3\ ^4F_{5/2} - 3{\rm d}^24p\ ^4G^{\rm o}_{7/2}$,and (d) $\rm 3{\rm d}^3\ ^4F_{7/2} - 3{\rm d}^24p\ ^4G^{\rm o}_{9/2}$.As mentioned earlier, the Coulomb-Bethe approximation was employed to estimate the contributions from high partial waves. It is clear from the figures that relative resonance contributions are not as strong as in the forbidden transitions and the rate coefficients are dominated by the high energy region where the collision strength has the Bethe asymptotic behavior, $\Omega \sim \ln(\epsilon)$.

As discussed in detail in the earlier work by CP on the bound channel expansion in the close coupling calculations for Fe VI, these (N+1)-electron functions are found to appear as resonances in the collision strengths. Often in close coupling calculations the bound channel configurations are not well represented from the point of view of a reasonably complete configuration-interaction expansion. This is particularly the case when the continuum channel expansion over the target states does not include the same configurations as the bound channels, which can then lead to large pseudoresonances in the cross sections. However, if care is exercised in the choice of the continuum and the bound channel expansions, the latter set corresponds well to physical resonances. For example, CP find that one particular bound channel configuration $\rm 3p^53{\rm d}^5$ leads to a considerable but diffuse rise in the near-threshold background of the collision strength $\Omega(^4{\rm F}_{3/2}~-~^4{\rm F}_{5/2})$and $\Omega (^4{\rm F}_{5/2}~-~^4{\rm F}_{9/2})$ (Figs. 6a,b of CP are plotted with and without this configuration). We also note that such bound channel resonances were also found to have an appreciable effect in the near-threshold region in an earlier work on the photoionization of Fe IV (Bautista & Pradhan 1997).

The Maxwellian-averaged collision strengths were calculated for all 3610 non-vanishing transitions between 80 energy levels shown in Table 4 (available in electronic form) for 21 temperatures ranging from 10 000 to 100 0000 K.