Resolution of the forbidden (J = 0 → 0) excitation puzzle in Mglike ions
^{1} Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK
email: luis.fernandezmenchero@strath.ac.uk
^{2} Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
Received: 21 January 2015
Accepted: 8 April 2015
We investigate the source of the discrepancy between Rmatrix and distortedwave (DW) collision strengths for J − J′ = 0–0 transitions in Mglike ions, for example 3s^{2}^{1}S_{0}–3p^{2}^{1}S_{0}, as reported previously. We find it to be due to the neglect of coupling, for example via 3s3p ^{1}P_{1}, as done by most DW codes. We have implemented an option to account for such coupling as a perturbation within the autostructure DW code. This removes the discrepancy of a factor ~10 and ~100 for Fe^{14 +} and S^{4 +}, respectively, for such transitions. The neglect of coupling would have affected (to some degree) the atomic data for a few weak optically forbidden transitions in other isoelectronic sequences if they were calculated with DW codes such as FAC and HULLAC. In addition, we compare the Fe^{14 +} line intensities predicted with the Rmatrix collision strengths against observations of solar active regions and flares; they agree well. For Fe^{14 +}, we suggest that the best density diagnostic ratio is 327.0/321.8 Å.
Key words: atomic data / techniques: spectroscopic
© ESO, 2015
1. Introduction
In a recent paper (FernándezMenchero et al. 2014b), we used the intermediate coupling frame transformation Rmatrix method to calculate electronimpact excitation data for all ions of the Mglike isoelectronic sequence from Al^{+} to Zn^{18 +} for all transitions involving levels up to principal quantum number n = 5. We also carried out distortedwave (DW) calculations using the code autostructure (AS; Badnell 2011) and exactly the same atomic structure as we used for the Rmatrix calculations. We compared our ASDW results, as well as those from other DW codes viz. the Flexible Atomic Code (FAC; Gu 2003) developed by Landi (2011) and the code UCLDW (Eissner 1998; Saraph 1972) developed by Christensen et al. (1985), with our Rmatrix results for Fe^{14 +}. We found large differences between our Rmatrix background collision strengths and those from ASDW and FAC by Landi (2011) for some weak J − J′ = 0–0 transitions, for example, by an order of magnitude for the 3s^{2}^{1}S_{0}–3p^{2}^{1}S_{0}. Conversely, the UCLDW results of Christensen et al. (1985) broadly agreed with the Rmatrix results. Clearly, the source of the difference must lie in the treatment of the (DW) scattering problem, not in the description of the atomic structure.
It is important to understand the origin of this large disagreement in the collision strengths calculated with different methods. Even today, the DW method is still relied upon extensively for calculations of electronimpact data for excitation and ionization of atoms, ions, and molecules. It is stored in databases such as CHIANTI^{1} (Landi et al. 2013) and OPEN ADAS^{2}. These data is then used in turn by plasma modellers to determine spectral diagnostics for both astrophysical and magnetic fusion plasmas.
2. Methodologies
The Rmatrix closecoupling and DW methods solve the formal scattering equations for the colliding electron in their respective approximations (Eissner & Seaton 1972). Thence, they both calculate the elements of the reactance matrix K, which is related to the transmission matrix T by (1)and the resulting scattering matrix, S = 1 − T, is unitary. The collision strength (Ω_{ij}) and cross section (Q_{ij}) for any transition i − j is then easily determined since (2)The Rmatrix method solves the closely coupled scattering equations and so naturally determines all elements of the Kmatrix for a given set of target levels. A significant advantage of the DW method is that it does not need to calculate the entire Kmatrix since it solves uncoupled radial scattering equations. Formally, it can make use of (3)for small K , which is usually the case for atoms that are ionized several times (Hayes & Seaton 1977).
In most astrophysical and magnetic fusion plasmas the main population of any given ion lies in its ground and metastable levels M. The main radiating properties of the full set of excited levels N are then determined by collisional excitation from levels M to N, followed by radiative cascade. The DW method then only needs to solve an M × N problem, for M ≪ N, as opposed to the N^{2} problem for the Rmatrix method. Of course, the DW method normally neglects resonances, but their contribution rapidly diminishes for more highly excited levels.
The codes ASDW, FAC and, indeed, HULLAC (BarShalom et al. 1988), all make use of Eq. (3), sometimes referred to as the weak coupling DW approximation, and are said to be nonunitarized DW methods. However, the UCLDW code has the option of using Eq. (1) and is said to be a unitarized DW method when it does so, this is sometimes referred to as the strong coupling DW approximation. We speculated (FernándezMenchero et al. 2014b) that this might be the source of the differences in weak collision strengths for forbidden J − J′ = 0–0 transitions since the use of Eq. (1) treats the close coupling as a perturbation within the DW method (Seaton 1961), while the use of Eq. (3) neglects it completely. Although ASDW has the ability of calculating all elements of the Kmatrix (M = N), it did so on the fly, the full matrix was never held. We have now implemented an option to retain the full Kmatrix and so utilize Eq. (1) to give a unitarized method (ASUDW).
3. Results
3.1. Atomic structure
The atomic structure was calculated with the program autostructure (Badnell 2011). We included the configurations { (1s^{2} 2s^{2} 2p^{6}) 3s^{2},3s 3p,3s 3d,3p^{2},3p 3d,3d^{2},3 { s,p,d } nl } with n = 4,5 and for l = 0–4. This yields 283 intermediate coupling levels. The energies calculated for the 15 lowest levels of Fe^{14 +} and S^{4 +} are shown in Tables 1 and 2. Further details of the atomic structure calculation can be found in FernándezMenchero et al. (2014b).
Fe^{14 +} target levels.
S^{4 +} target levels.
3.2. Collisions
In Figs. 1 and 2 we show the electronimpact excitation collision strength Ω for the transition 3s^{2}^{1}S_{0} − 3p^{2}^{1}S_{0} of the ions Fe^{14 +} and S^{4 +}. We compare the results of Rmatrix and (nonunitarized) ASDW calculations by FernándezMenchero et al. (2014b) with the UCLDW calculations of Christensen et al. (1985) and FACDW of Landi (2011) for Fe^{14 +} and UCLDW of Christensen et al. (1986) for S^{4 +} and compare them with present (unitarized) ASUDW results. The AS(U)DW results were obtained using exactly the same target atomic structures as the Rmatrix results.
Fig. 1 Electronimpact excitation collision strength versus the impact energy for the transition 3s^{2}^{1}S_{0}–3p^{2}^{1}S_{0} of Fe^{14 +} . Full line: Rmatrix (FernándezMenchero et al. 2014b); ×: ASDW (FernándezMenchero et al. 2014b); +: ASUDW (present work); □: FACDW (Landi 2011); ◇: UCLDW (Christensen et al. 1985). 

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Fig. 2 Electronimpact excitation collision strength versus the impact energy for the transition 3s^{2}^{1}S_{0}–3p^{2}^{1}S_{0} of S^{4 +}. Full line: Rmatrix (FernándezMenchero et al. 2014b); ×: ASDW (FernándezMenchero et al. 2014b); +: ASUDW (present work); ◇: UCLDW (Christensen et al. 1986). 

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The background collision strengths of Christensen et al. (1985, 1986) obtained with the UCLDW code are quite close to the Rmatrix strengths; the differences can be attributed to the different atomic structures used. However, the FACDW results of Landi (2011) (Fe^{14 +} only) and the ASDW results of FernándezMenchero et al. (2014b) differ by large factors from the Rmatrix results for this transition: ~10 for Fe^{14 +} and ~100 for the S^{4 +}. However, the ASUDW results show a dramatic increase over the nonunitarized ones by very similar factors. This demonstrates that this transition is dominated (~90% and ~99%) by coupling. It also shows that the original speculation of FernándezMenchero et al. (2014b) was correct: the results of Christensen et al. (1985, 1986) were obtained using the unitarized option of the UCLDW code.
We carried out a series of calculations for which we progressively reduced the number of target configurations included in the scattering calculation to determine the source of the coupling for the 3s^{2}^{1}S_{0}–3p^{2}^{1}S_{0} transition. It was still present with a threeconfiguration (3s^{2},3s3p,3p^{2}) target expansion, but disappeared when we omitted the 3s3p. As expected, perhaps, the dominant coupling mechanism is thus the double dipole mechanism: 3s^{2}^{1}S_{0} → 3s3p ^{1}P_{1} → 3p^{2}^{1}S_{0}.
Given that the Rmatrix and AS(U)DW calculations used the same atomic structure, one might wonder about the consistency of the highenergy behaviour of the collision strengths: Rmatrix and ASUDW should tend to the same Born limit as the nonunitarized ASDW. As demonstrated by FernándezMenchero et al. (2014b), a reduced Burgess–Tully diagram (Burgess & Tully 1992) shows that the Rmatrix collision strength turns over at high energy and attains the same limit as the ASDW collision strength. The present ASUDW results follow the same behaviour at high energy as those of the Rmatrix.
In Table 3 we compare Maxwellianintegrated effective collision strengths Υ for the 3s^{2}^{1}S_{0}–3p^{2}^{1}S_{0} transition in Fe^{14 +} at a temperature of 2.5 × 10^{6} K. The present ASUDW effective collision strength still differs by a factor 2 from that of the Rmatrix (Eissner et al. 1999; FernándezMenchero et al. 2014b). We recall that the collision strengths of FernándezMenchero et al. (2014b) used exactly the same atomic structure for the target, which means that the difference is mainly due to the contribution from resonances. The difference in background collision strengths is no more than ≈15% – see also Fig. 1. The (unitarized) UCLDW result of Christensen et al. (1985) closely agrees with the present ASUDW result. The nonunitarized ASDW and FACDW results also closely agree with each other, but are more than a factor of 10 smaller than the unitarized DW results. The UCLDW result of Bhatia & Mason (1997) is nearly a factor of 10 smaller again than the nonunitarized results. This indicates that Bhatia & Mason (1997) probably used the nonunitarized UCLDW option and that their atomic structure is somewhat different for this transition.
Electronimpact excitation effective collision strengths for the transition 3s^{2}^{1}S_{0}–3p^{2}^{1}S_{0} of Fe^{14 +} at an electron temperature of T = 2.25 × 10^{6} K.
3.3. Comparison to observations for Fe^{14+}
We have seen the large differences in the collision strengths of the 1–10 3s^{2}^{1}S_{0}–3p^{2}^{1}S_{0} transition in Fe^{14 +}. It is therefore useful to validate our Rmatrix results against observations. As briefly discussed in FernándezMenchero et al. (2014b), a significant fraction of the population of the upper level 103p^{2}^{1}S is due to the above transition, the direct excitation from the ground state. This upper level mainly decays with an allowed transition to level 5, .
This means that the intensity of the 5–10 transition is directly affected by the collision strength of the above 1–10 3s^{2}^{1}S_{0}–3p^{2}^{1}S_{0} transition, therefore we calculated the level populations for this ion using our atomic data (FernándezMenchero et al. 2014b) to compare the relative intensity of the 5–10 transition to those of other lines. We only considered lines close in wavelength to avoid possible issues in terms of instrument calibration.
The 5–10 transition is in itself a troublesome line in terms of its identifications. This transition (together with other lines) was identified by Churilov et al. (1985) using laboratory spectra with a line observed at 324.98 Å. The identifications were mainly based on wavelength coincidences (the line intensities were not calibrated). There were previous suggestions that a solar line at 323.57 Å was instead due to this transition (Cowan & Widing 1973), therefore Keenan et al. (1993) considered Skylab S082A intensities of several solar flares to assess whether the identification was correct. The Skylab observations confirmed the identification made by Churilov et al. (1985), although there is a large scatter in the intensity of this line, which is always weak in the solar spectra. The 324.98 Å line was invisible in the active region SERTS89 spectra of Thomas & Neupert (1994), which led Young et al. (1998) to suggest that the identification of the 324.98 Å line as the 5–10 transition was probably not correct.
One way to compare at once the observed intensities of several lines with the predicted ones is to plot the emissivity ratios R_{ji} (Del Zanna et al. 2004), which are basically the ratios of the observed (I_{ob}, energy units) and the calculated line emissivities as a function of the electron density N_{e}: (4)where N_{j}(N_{e},T_{e}) is the population of the upper level j relative to the total number density of the ion, calculated at a fixed temperature T_{e} (the ratios of the lines considered here have little temperature sensitivity, and we have taken as T_{e} the value of peak ion abundance in ionization equilibrium); λ_{ji} is the wavelength of the transition, A_{ji} is the spontaneous radiative transition probability, and C is a scaling constant that is the same for all the lines within one observation. If experimental and theoretical intensities agree, all lines should be closely spaced or intersect for a near isodensity plasma. The value of C is chosen so that the emissivity ratios R_{ji} are near unity where they intersect.
Fig. 3 Emissivity ratio plots for some Fe^{14 +} EUV lines observed by Skylab (above) and SERTS97 (below). 

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If we consider the first of the flares considered by Keenan et al. (1993) and plot the emissivity ratios as a function of density, we obtain the results shown in Fig. 3 (top). There is excellent (to within a relative few percent) agreement between observed and predicted intensities for the 327.0, 292.3, 321.8, and 325.0 Å lines at a density of 10^{10.3} cm^{3}, which in turn agrees excellently well with the densities obtained from other ions. This is an improvement over the atomic data used at the time by Keenan et al. (1993). The 312.6 Å line is known to be blended (probably with Co XVII), and the 317.6 Å has been known to be severely blended (possibly with Na VI).
There are various other extrem UV (EUV) observations of the Fe XV lines, but often spectra have not been properly calibrated or did not have enough resolution or sensitivity (the 325.0 Å line is weak). However, there is a wellcalibrated SERTS97 spectrum (Brosius et al. 2000) where the 325.0 Å line was visible, however. Most of the lines listed as due to Fe XV are severely blended, but good agreement is found in the intensities of the 327.0, 321.8, and 325.0 Å lines, as shown in Fig. 3 (bottom). As already mentioned, the 312.6 Å is known to be blended with Co XVII, while the 304.9 Å line is possibly blended with Mn XIV. Similar results were obtained (using different atomic data) by Keenan et al. (2005). We also considered the SERTS89 spectrum, where more lines were observed (but not the 5–10 transition), finding an overall good agreement between theory and observation.
In conclusion, the few solar observations of the 5–10 line show very good agreement between the observed and predicted intensity of this line, which confirms the reliability of the Rmatrix calculations. We note that Kastner & Bhatia (2001) built an ion population model using the Iron Project Rmatrix calculations of Eissner et al. (1999). Their predicted intensity of the 5–10 transition, relative to the 4–7 327.03 Å line, is 0.085, relatively close to the two observed values, 0.105 and 0.115, and our predicted value of 0.115 (at a density of 10^{10} cm^{3}). Indeed, the effective collision strengths of the calculations reported by Eissner et al. (1999) are close to ours, as shown in Table 3.
Finally, we note that Keenan et al. (1993) suggested that the 325.0 Å transition would be an excellent density diagnostic for the solar corona, but the 327.0 Å line has a similar sensitivity at typical active region/flare densities, so it is to be preferred because it is much stronger. We suggest that the best diagnostic ratio for Fe XV is the 327.0/321.8 Å. The lines are both strong, unblended, and are close in wavelength.
3.4. Other sequences
Fig. 4 Electronimpact excitation collision strength versus the impact energy for the transition 2s^{2}^{1}S_{0} − 2p^{2}^{1}S_{0} of O^{4 +}. Full line: Rmatrix (FernándezMenchero et al. 2014a); ×: ASDW (present work); +: ASUDW (present work). 

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What about systems other than Mglike? The obvious one is Belike – the n = 2 analogue: 2s^{2},2s2p,2p^{2}. In Fig. 4 we show the Rmatrix collision strengths derived by FernándezMenchero et al. (2014a) for O^{4 +} (the same residual charge as Mglike S) and compare them with the present nonunitarized and unitarized AS(U)DW results, calculated with the same atomic structure. We see that that Rmatrix background collision strengths and the ASUDW are similar in magnitude to the S^{4 +} results shown in Fig. 2. In contrast, however, the nonunitarized ASDW results are much larger than in the corresponding S case – neglect of coupling only reduces them by a factor of ~2 instead of ~100. The reason for this, it turns out, is that the 2s^{2} is more strongly mixed with the 2p^{2}^{1}S_{0} than in the corresponding n = 3 case. There is enough admixture of 2s^{2} in the 2p^{2}^{1}S_{0} state for it to proceed directly through the target mixing. In contrast, the 3p^{2}^{1}S_{0} state is pure enough that very little collision strength arises directly.
Since the transition J − J′ = 0–0 takes place through target state mixing, which is small in general, the total collision strength is expected to be strongly sensitive to small changes in the atomic structure. We have found that making small changes in the atomic structure, so that the mixing coefficients change, can change the collision strength calculated with ASDW by a factor 10, while the collision strength calculated with ASUDW remains much more stable (to within ~20%).
We have detected this coupling effect only in very weak optically forbidden transitions (J − J′ = 0–0). In the coronal approximation, where the population of an ion is concentrated in the ground state, the population of such upper states in general does not come from a direct excitation from the ground state, but from radiative cascading from more excited states. However, the 3p^{2} case, being a double electron jump from the ground, is only populated weakly by cascade. In higher density plasmas (≳10^{14} cm^{3}), such as magnetic fusion, the 3s3p ^{3}P_{2} population can be expected to drive the 3p^{2}^{1}S_{0} population by direct excitation.
4. Conclusion
We have implemented an option in the ASDW code to convert the reactance Kmatrices to the transmission Tmatrices that gives rise to unitary scattering Smatrices – ASUDW. Physically, this corresponds to treating all coupling of the scattering equations as a perturbation. The effect of coupling is very large for select transitions: J = 0 → 0 in Mglike ions.
The neglect of coupling is the reason for the large differences found in FernándezMenchero et al. (2014b) for these transitions between Rmatrix and distorted wave results, including those that used exactly the same atomic structure. The implementation of ASUDW corrects for this difference.
We compared the theoretical line intensities obtained using the Rmatrix results for Mglike iron with solar observations and found good agreement, confirming the reliability of the calculations.
Finally, we point out that the neglect of coupling would have affected (to some degree) the atomic data for a few weak optically forbidden transitions in other isoelectronic sequences, calculated with nonunitarized DW codes such as FAC and HULLAC.
Acknowledgments
The present work was funded by STFC (UK) through the University of Strathclyde UK APAP network grant ST/J000892/1 and the University of Cambridge DAMTP astrophysics grant.
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All Tables
Electronimpact excitation effective collision strengths for the transition 3s^{2}^{1}S_{0}–3p^{2}^{1}S_{0} of Fe^{14 +} at an electron temperature of T = 2.25 × 10^{6} K.
All Figures
Fig. 1 Electronimpact excitation collision strength versus the impact energy for the transition 3s^{2}^{1}S_{0}–3p^{2}^{1}S_{0} of Fe^{14 +} . Full line: Rmatrix (FernándezMenchero et al. 2014b); ×: ASDW (FernándezMenchero et al. 2014b); +: ASUDW (present work); □: FACDW (Landi 2011); ◇: UCLDW (Christensen et al. 1985). 

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In the text 
Fig. 2 Electronimpact excitation collision strength versus the impact energy for the transition 3s^{2}^{1}S_{0}–3p^{2}^{1}S_{0} of S^{4 +}. Full line: Rmatrix (FernándezMenchero et al. 2014b); ×: ASDW (FernándezMenchero et al. 2014b); +: ASUDW (present work); ◇: UCLDW (Christensen et al. 1986). 

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In the text 
Fig. 3 Emissivity ratio plots for some Fe^{14 +} EUV lines observed by Skylab (above) and SERTS97 (below). 

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In the text 
Fig. 4 Electronimpact excitation collision strength versus the impact energy for the transition 2s^{2}^{1}S_{0} − 2p^{2}^{1}S_{0} of O^{4 +}. Full line: Rmatrix (FernándezMenchero et al. 2014a); ×: ASDW (present work); +: ASUDW (present work). 

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In the text 