Free Access
Volume 566, June 2014
Article Number A123
Number of page(s) 7
Section Atomic, molecular, and nuclear data
Published online 23 June 2014

© ESO, 2014

1. Introduction

The resonance transition of Ni xi at 148.4 Å is one of the strongest lines in the extreme ultraviolet (EUV) region of the spectrum (see, e.g. Behring et al. 1972). This ion also produces several transitions in the soft X-rays and EUV. Ni xi lines are also observed in laboratory plasmas, such as those of the JET tokamak described by Mattioli et al. (2004).

Ni xi is isoelectronic with Fe ix, and lines from this ion are in principle useful to measure electron densities and temperatures. In particular, the two Ni xi lines at 207.9 and 211.4 Å are analogues of the Fe ix 241.7, 244.9 Å lines, which are an excellent density diagnostic (Storey et al. 2002), and are observed by the Hinode EUV Imaging Spectrometer (EIS, see Culhane et al. 2007), as discussed by Landi & Young (2009).

To date, the best scattering calculations for this ion are those of Aggarwal & Keenan (2007) and Bhatia & Landi (2011). Aggarwal & Keenan (2007) carried out, with the Dirac Atomic R-matrix Code (DARC, originally written by Norrington and Grant), R-matrix calculations involving the energetically lowest 17 levels, from the 3s23p6, 3s23p53d, and 3s3p63d configurations. Bhatia & Landi (2011) carried out a scattering calculation with an extended target, including the main n = 3, 4 configurations. However, they used the distorted-wave (DW) approximation within the Flexible Atomic Code (FAC, see Gu 2004).

Both calculations have considerable limitations. As pointed out by Bhatia & Landi (2011), the small target employed by Aggarwal & Keenan (2007) limited the accuracy of their results. That an R-matrix calculation with a larger target was needed, to include all the main resonances, was already pointed out by Aggarwal & Keenan (2007) themselves. Indeed, our earlier calculations for Fe ix (Storey et al. 2002) have clearly shown the need for a larger n = 3 target, to include all the main resonances contributing to the lowest three configurations, for ions of this sequence.

The DW calculations are inherently limited because the resonance enhancements are normally not included, and as a result collision strengths of the lowest configurations can easily be underestimated. Until recently, it was commonly thought that DW calculations were quite accurate for the higher configurations and this is often the case. However, there are many instances within an ion where DW calculations underestimate collision strengths by factors of two or more. For example, in O’Dwyer et al. (2012) we pointed out that collision strengths to the Fe ix 3s23p54s levels can be significantly underestimated if the resonances due to the 3s23p54p levels are not included. The effect is notable for the 3s23p54s levels because they have small collision strengths from the ground configuration. However, similar issues have been found for other Fe ix configurations in our recent large-scale R-matrix calculation (Del Zanna et al. 2014), but also in general with all other coronal iron ions: Fe x (Del Zanna et al. 2012b), Fe xi (Del Zanna & Storey 2013), Fe xii (Del Zanna et al. 2012a), and Fe xiii (Del Zanna & Storey 2013).

A new R-matrix calculation for all the main n = 3, 4 levels of Ni xi was therefore needed. The paper is organised as follows. In Sect. 2 we outline the methods we adopted for the scattering calculations. In Sect. 3 we present our results and in Sect. 4 we reach our conclusions.

thumbnail Fig. 1

LS terms of the target configuration basis. The lowest 273 terms which produce levels having energies below the dashed line have been retained for the close-coupling expansion.

2. Methods

The atomic structure calculations were carried out using the autostructure program (Badnell 2011) which originated from the superstructure program (Eissner et al. 1974). autostructure constructs target wavefunctions using radial wavefunctions calculated in a scaled Thomas-Fermi-Dirac-Amaldi statistical model potential with a set of scaling parameters. The program also provides radiative rates and infinite energy Born limits. These limits are particularly important for two reasons. First, they allow a consistency check of the collision strengths in the scaled Burgess & Tully (1992) domain (see also Burgess et al. 1997). Second, they are used in the interpolation of the collision strengths at high energies.

The R-matrix method used in the scattering calculation is described in Hummer et al. (1993) and Berrington et al. (1995). We performed the calculation in the inner region in LS coupling and included mass and Darwin relativistic energy shifts.

The outer region calculation used the intermediate-coupling frame transformation (ICFT) method (Griffin et al. 1998), in which the transformation of the multi-channel quantum defect theory unphysical K-matrix to intermediate coupling uses the so-called term-coupling coefficients (TCCs) in conjunction with level energies.

Dipole-allowed transitions were topped-up to infinite partial wave using an intermediate coupling version of the Coulomb-Bethe method as described by Burgess (1974) while non-dipole allowed transitions were topped-up assuming that the collision strengths form a geometric progression in J (see Badnell & Griffin 2001).

The collision strengths were extended to high energies by interpolation using the appropriate high-energy limits in the Burgess & Tully (1992) scaled domain. The high-energy limits were calculated with autostructure for both optically-allowed (see Burgess et al. 1997) and non-dipole allowed transitions (see Chidichimo et al. 2003).

We have also carried out Breit-Pauli DW calculations using the recent development of the autostructure code, described in detail in Badnell (2011). We have used the same target for the DW and the ICFT calculations.

The temperature-dependent effective collisions strength Υ(ij) were calculated by assuming a Maxwellian electron distribution and linear integration with the final energy of the colliding electron.

3. Results

Since we are primarily interested in the main diagnostics within the n = 3 levels, we have chosen a target which includes all the main n = 4 levels. In our recent calculation for Fe ix (Del Zanna et al. 2014) we even included all the main n = 5 configurations, but for the n = 3 levels we did not find any significant differences in the collision strengths from the previous calculations (Storey et al. 2002), which included only n = 3 levels and the 3s23p54s and 3s23p54p configurations. This means that the n = 5 levels produce negligible resonance enhancement for the n = 3 levels. We also did not find significant differences for the 3s23p54s and 3s23p54p configurations, which again means that the n = 5 levels produce negligible resonance enhancement for these configurations.

We have therefore chosen as our configuration basis the complete set of 29 n = 3, 4 configurations, shown in Fig. 1 and listed in Table 1. They give rise to 1089 LS terms and 2635 fine-structure levels. The scaling parameters λnl for the potentials in which the orbital functions are calculated are also given in Table 1. The 599 fine-structure levels arising from the (energetically) lowest 273 LS terms were retained for the scattering calculation. They include all the spectroscopically important n = 4 levels.

Table 1

Target electron configuration basis and orbital scaling parameters λnl.

Table 2

Level energies for Ni xi.

Table 3

List of the strongest Ni xi lines.

Table 2 presents a selection of fine-structure target level energies Et, compared to experimental energies Eexp. The experimental energies are mostly taken from the NIST compilation (Kramida et al. 2013), although we have revised some experimental energies (R in the Table) and suggested new tentative ones (TN in the Table), as described below in Sect. 3.1.

A set of “best” energies Eb was obtained with a linear fit between the Eexp and Et values. The Eb values were used (whenever the experimental energies Eexp were not available) within the R-matrix calculation to obtain a better position of the resonance thresholds. The experimental energies Eexp and the “best” energies Eb were also used to calculate the transition probabilities, which was done with a separate autostructure calculation.

The expansion of each scattered electron partial wave was done over a basis of 21 functions within the R-matrix boundary. The partial wave expansion within the exchange calculations was extended to a maximum total orbital angular momentum quantum number of L = 16. This produces reliable collision strengths up to about 70 Ryd.

The outer region calculation includes exchange up to a total angular momentum quantum number J = 25 / 2. We have added to the exchange contributions those from a non-exchange calculation, which extended from J = 27 / 2 to J = 73 / 2. The outer region exchange calculation was performed in a number of stages. A coarse energy mesh (with 96 points) was chosen above all resonances. The resonance region itself was calculated with an increasing number of points, until 9600, so we could check the convergence, as in Del Zanna et al. (2010).

For diagnostic application purposes, it is useful to just discuss the atomic data for the strongest transitions. We have constructed an ion population model with the new R-matrix collision strengths, complemented with a set of A-values calculated separately with exactly the same configuration basis, but with the experimental and best energies. We then calculated line intensities and looked at how levels are populated at log Ne [cm-3] = 9 (a typical solar active region density) and log Te [K] = 6.1, the temperature of maximum ion abundance in ionization equilibrium. The brightest lines are listed in Table 3. We also list, within the table, the intensities calculated with the CHIANTI model, which uses the Aggarwal & Keenan (2007) collision strengths for the first 17 levels and those of Bhatia & Landi (2011) for the rest. Table 3 also lists our A-values, and those calculated by Bhatia & Landi (2011) with the FAC.

The intensities in the lines are relative to the strongest, resonance line (1–13, at 148.374 Å). The oscillator strength for this transition as calculated by Aggarwal & Keenan (2007) is 2.62 with FAC and 2.66 with GRASP, i.e. almost the same as our value (2.72), however, there is a considerable (about 30%) difference in the effective collision strength for this transition, as shown in Fig. 2. This difference results in a decrease of the predicted intensity of this line, hence affects all the ratios listed in Table 3. The reason for this difference is puzzling.

thumbnail Fig. 2

Thermally-averaged collision strengths for a selection of transitions (see text).

The oscillator strength for the resonance line as calculated with FAC by Bhatia & Landi (2011) is 2.67, i.e. it is also very close to ours. However, in that case their DW collision strengths are in good agreement with ours, as shown in Fig. 3. Our DW results are also in good agreement with the background collision strength as calculated with the R-matrix suite of codes, and the collision strength correctly converges toward the high-energy limit, so the effective collision strength for this line calculated by Aggarwal & Keenan (2007) is the odd one out.

A quick look at Table 3 clearly shows that many lines from the 3s23p53d levels are significantly increased in intensity, compared to the CHIANTI model. There is good agreement in the A-values, hence the differences are due to increased populations of the upper levels. For example, the populations of levels 3 and 4 are significantly increased. The reason is not just an increase in direct excitation to these levels, but a combined effect which includes increased cascading, in turn due the larger model and also increased excitation, similar to what we found for the coronal iron ions. Direct excitation contributes 25% to the population of level 3, while decays from levels 4 and 11 contribute 16%. The collision strengths to levels 3, 4, and 11 are all increased compared to the Aggarwal & Keenan (2007) results, as shown in Fig. 2. About 43% of the population of level 4 is due to decays from levels 8 and 11 and 15% to direct excitation from the ground state. Again, collision strengths to the 4, 8, and 11 levels are all increased compared to the Aggarwal & Keenan (2007) results.

On the other hand, there is good agreement with the CHIANTI model for the decays from the 3s23p43d2 levels, indicating that little resonance enhancement is present (at peak temperatures) in the present larger target, since in the CHIANTI model the excitation rates for these levels are derived from the DW collision strengths of Bhatia & Landi (2011).

thumbnail Fig. 3

Collision strength for the resonance 148.374 Å transition, averaged over 0.05 Rydbergs. Boxes: the results of our DW calculation. Stars: the results of the DW FAC calculation by Bhatia & Landi (2011).

Finally, for the n = 3 − 4 transitions, we find a similar situation as we saw for the iron coronal ions, i.e. significant increases in the intensities of many among the strongest transitions compared to the CHIANTI model. The A-values for the transitions generally agree well, so the main difference is due to the resonance enhancement that is present in the R-matrix calculations, but not in the DW values as calculated by Bhatia & Landi (2011) (note that our DW values are normally in good agreement with theirs). The effect is particularly pronounced for the 3s23p54s levels, where differences are factors of two or so. Figure 2 shows as an example the effective collision strength for the 3s23p61S0–3s23p54s1P1 transition, the strongest of the n = 3 − 4 array. Large enhancements also occur for the 3s23p54d levels.

3.1. Discussion on level identifications

The strongest resonance transition at 148.374 Å was identified by Alexander et al. (1965) which allowed the energy of the 3s23p53d3P1 level to be established. Svensson et al. (1974) applied some extrapolations along the isoelectronic sequence and used tentative identifications of some forbidden lines (at 3302.8, 3020.1, 2000.3, and 1715.3 Å) to suggest the identification of the three EUV solar lines observed by Behring et al. (1976) at 211.44, 207.935, and 186.983 Å with the decays from the 3s23p53d3P2, 3P1, and 3D1. These identifications, although tentative at the time, turned out to be important, because they allow the energies of these levels (relative to the ground state) to be established.

Surprisingly, Edlen & Smitt (1978) later revised some of the identifications of a few forbidden lines establishing the relative energies of some among the 3s23p53d levels. By combining these with the energies of the 3P2, 3P1, and 3D1 levels suggested by Svensson et al. (1974), it was therefore possible to establish also the energies of the 3F4, 3D2, and 1F3 levels. The energies of the other 3s23p53d levels were either estimated or known in terms of differences.

The main decays from the 3s23p54s and 3s23p54d configurations (from the 1P1 and 3P1) were identified using laboratory spectra by Even-Zohar & Fraenkel (1968). A few of the lines from the 3s23p53d–3s23p54f transition array were identified for the first time by Fawcett et al. (1972) using laboratory spectra. These lines were later re-measured by Swartz et al. (1976) with greater accuracy. With the measurements of the 81.378 and 81.138 Å lines, decays from the 3s23p54f3D1 (No. 525) to the 3s23p53d3P1,0 levels, we obtain the energy of the 3s23p53d3P0 level (No. 2).

We note that the energy of the 3s23p53d3F2 level adopted by NIST (504 070 cm-1) is the value suggested by Edlen & Smitt (1978), presumably based on the predicted energy difference with the known 3F4 level. This value turns out to be incorrect, because it predicts a wavelength of 198.38 Å for the decay to the ground state, which is not observed. Our theoretical splitting between the 3F2 and 3F4 levels suggests an energy of 503 971 cm-1 for the 3F2, implying a wavelength of 198.42 Å for the decay to the ground state. There is indeed a weak line observed by Hinode EIS at exactly the right wavelength, at 198.424 (Del Zanna 2012b), and with about the right intensity.

Again using the theoretical 3F splittings, the estimated energy of the 3F3 should be 497 364 cm-1 and not 497 520 as listed by NIST. The energy of the 3D3 is only known relative to the 3F3 via the 3338.5 Å line (wavelength in air) as proposed by Edlen & Smitt (1978), hence the energy of the 3D3 should be 527 309 cm-1 and not 527 470 cm-1 as listed by NIST. The energy difference between this estimate and our target is 9950 cm-1, in excellent agreement with the energy differences of the 3D1 (9834 cm-1) and 3D2 (10 039 cm-1). A similar argument applies to the 1D2, for which we estimate the energy to be 530 687, and not 530 830.0 cm-1 as listed by NIST. We obtain this number by adopting the observed wavelength (in air) of the 3F31D2, 3000.0 Å (Edlen & Smitt 1978), and our revised energy for the 3F3.

The energies of the 3s3p63d levels are still unknown. Landi & Young (2009) proposed the identification of the 3s23p53d3F4–3s3p63d3D3 (5–16) transition with a line observed at 278.73 Å, blended with Al v in their on-disk solar observation of a cool loop. The identification was presumably based on matching the ab-initio theoretical wavelength. We are unable to confirm this identification. If this line was partly due to Ni xi, it should be visible in off-limb spectra of the quiet Sun, where the Al v becomes invisible. In reality, there is no line in off-limb quiet-Sun spectra at this wavelength (Del Zanna 2012b). Also, the energy difference with our target energy would differ significantly from the energy differences of the other 3s23p53d levels. Applying the same energy difference of nearby levels results in a predicted wavelength of 288.5 Å, very close to a line observed at 288.41 Å, which is mostly due to S xii. It is interesting to note that Behring et al. (1976) indicate that in their high-resolution EUV spectrum the S xii line is blended with a first-order line. This line is therefore our best candidate for the 5–16 transition.

The main decays from the 3s23p54s configuration are the strongest Ni xi lines in the soft X-ray spectrum. One aspect which was puzzling at first is the fact that the line we predict to be strongest at 77.393 Å was barely visible in the quiet Sun spectrum of Behring et al. (1972), and it is also weak in the Manson (1972) spectrum. The 78.744 Å line should be much weaker but in the solar spectra it is much brighter. We thought at first that the identifications were incorrect but then checked the intensities of these lines against the resonance line, and found that the observed intensity of the 77.393 Å is approximately in agreement with our predictions. This means that the 78.744 Å line must be blended, and all the other Ni xi lines in the soft X-ray spectrum of the quiet Sun are extremely weak.

The third strongest line of the soft X-ray spectrum is the 3s23p53d1P1–3s3p64s1S0 (13–440) transition. We identified the same type of transitions (which are quite strong) for Fe x, Fe xi, Fe xii, and Fe xiii (Del Zanna et al. 2012b; Del Zanna 2012a). Our “best” energy estimates predict this line to be around 104.6 Å. There is indeed a weak line (on the blue-wing of another line) in the Manson (1972) and Malinovsky & Heroux (1973) spectra that we measured at 104.67 Å, with the predicted intensity. We therefore suggest this as a tentative identification.

The strongest line from the 3s23p54p is predicted, again using the “best” energies Eb, to be around 112.6 Å. There are two unidentified lines in the Manson (1972) and Malinovsky & Heroux (1973) spectra at 112.55 and 112.75 Å, so it is possible that one of these is the 5–187 transition. Finally, we note that the energy of the 3s23p54d1P1 level (No. 357) is at odds with the energy of the 3s23p54d3P1, so the identification of the former is questionable.

thumbnail Fig. 4

A quiet Sun Hinode spectrum with the Ni xi lines.

3.2. Comparison to EUV observations

We now consider the main density diagnostic in the EUV for this ion, the 207.9 and 211.4 Å lines. These lines are observed by Hinode EIS but they are at the very edge of one of the channels, where the instrument sensitivity is very low. In most Hinode EIS observations, these lines are not visible.

We have found an Hinode EIS full-spectral observation of the quiet Sun on 2010-10-08 at 10:15 UT, with the 2′′ slit and a very long exposure time (120 s), where these lines are visible. This observation was one of those selected for the radiometric calibration of the EIS instrument (Del Zanna 2013). We adopt this calibration here. We selected this observation because it provides typical quiet Sun line radiances. Despite this, the Ni xi lines are weak, as shown in Fig. 4 (units are recorded data numbers).

To compare the EIS radiances with that of the resonance line, we have taken the quiet Sun irradiance measured by the prototype EVE (PEVE) instrument (Chamberlin et al. 2009; Woods et al. 2009) and converted into a radiance, assuming a limb-brightening factor of 1.4, as done in Del Zanna (2013), based on the limb-brightening curves obtained from SOHO CDS described in Andretta & Del Zanna (2014). Despite the uncertainty in the conversion, there is good agreement (within 20%) between predicted and observed radiances, as shown in Fig. 5. This figure shows the “emissivity ratio” curve (1)for each line as a function of the electron density Ne. Iob is the observed intensity of the line, Nj(Ne,Te) is the population of the upper level j relative to the total number density of the ion, calculated at a fixed temperature Te. Aji is the spontaneous radiative transition probability, and C is a scaling constant. This constant is the same for all the lines, and its value is 1.3 × 109, for both plots in Fig. 5. This value was chosen so that the emissivity ratios are near unity, to visually estimate, from the spread in the curves, the relative agreement between observed and predicted intensities for all the lines. In fact, if agreement between experimental and theoretical intensities is present, all lines should be closely spaced. If the plasma is nearly isodensity, all curves should cross at one point, giving the line-of-sight averaged density. The emissivity ratio curves are useful to see at once the density sensitivity of the different emission lines, but are equivalent to the usual single line ratio plots, where the theoretical ratio of two emission lines is plotted as a function of density. The emissivity ratio curves in Fig. 5 are calculated at log Te [K] = 6.1, the temperature of maximum ion abundance in ionization equilibrium, i.e. the temperature where, in normal coronal conditions, the Ni xi lines are formed.

There is good agreement between predicted and observed intensities for all the lines, with the exception of the 198.42 Å very weak line, and the 186.99 Å line which is blended with an Fe xii transition. The emissivities of the 207.9 and 211.4 Å lines indicate that the electron density should be around log Ne [cm-3] = 8.8, although the observation is consistent with any lower density, considering the large uncertainty in the measurement of the weaker 211.4 Å line (say 30%), and the overall uncertainty in the relative Hinode EIS calibration between the two lines (typical uncertainties are 10–20%, see Del Zanna 2013). On the other hand, the CHIANTI v.7.1 model indicates an unreasonably high density of log Ne [cm-3] = 9.3, and a large discrepancy between the quiet Sun radiance of the resonance line and the well-measured 207.9 Å line.

thumbnail Fig. 5

Emissivity ratio curves relative to quiet Sun observations (Hinode EIS and PEVE, see text). Above: with the present atomic data. Below: with the AK07 atomic data as included in CHIANTI version 7.1. The observed intensities Iob are in phot cm-2 s-1 arcsec-2.

4. Conclusions

The present scattering calculations produced the expected results, i.e. enhanced collision strengths for transitions to the 3s23p53d and n = 4 levels (in particular the 3s23p54s), compared to the previous R-matrix and DW results of Aggarwal & Keenan (2007) and Bhatia & Landi (2011). The increases are due to the resonances that arise within the present large-scale target.

This results in significant increases in the predicted intensities of the UV and visible forbidden lines within the 3s23p53d configuration, the strong EUV lines that result form the decays of the 3s23p53d levels to the 3s23p61S0 ground state and the decays from the n = 4 levels, which are visible in the soft X-rays.

The present atomic dataset can now reliably be used for plasma diagnostics. We had a close look at the experimental energies and found the need for some revisions. Further laboratory work is needed to confirm some of the suggested new identifications, and complete the identification of the main levels for this ion.


The present work was funded by STFC (UK) through the University of Cambridge DAMTP astrophysics grant, and the University of Strathclyde UK APAP network grant ST/J000892/1.


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All Tables

Table 1

Target electron configuration basis and orbital scaling parameters λnl.

Table 2

Level energies for Ni xi.

Table 3

List of the strongest Ni xi lines.

All Figures

thumbnail Fig. 1

LS terms of the target configuration basis. The lowest 273 terms which produce levels having energies below the dashed line have been retained for the close-coupling expansion.

In the text
thumbnail Fig. 2

Thermally-averaged collision strengths for a selection of transitions (see text).

In the text
thumbnail Fig. 3

Collision strength for the resonance 148.374 Å transition, averaged over 0.05 Rydbergs. Boxes: the results of our DW calculation. Stars: the results of the DW FAC calculation by Bhatia & Landi (2011).

In the text
thumbnail Fig. 4

A quiet Sun Hinode spectrum with the Ni xi lines.

In the text
thumbnail Fig. 5

Emissivity ratio curves relative to quiet Sun observations (Hinode EIS and PEVE, see text). Above: with the present atomic data. Below: with the AK07 atomic data as included in CHIANTI version 7.1. The observed intensities Iob are in phot cm-2 s-1 arcsec-2.

In the text

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