ELECTRON–ION RECOMBINATION OF Fe^{12 +} FORMING Fe^{11 +}: LABORATORY MEASUREMENTS AND THEORETICAL CALCULATIONS

Figure 1 compares the experimental DR MBRRC (filled circles) to the AUTOSTRUCTURE calculations of Badnell (2006) for E < 70 eV. These state-of-the-art theoretical calculations included all low-energy excitations in the model. The theoretical cross section has been multiplied by v_rel and convolved with the energy spread of the experiment. The calculation includes DR contributions from capture into Rydberg states up to n = 1000 and is shown by the dotted line in the figure. However, in the experiment, the ions experience motional electric fields as they pass through the TSR magnets. These fields ionize high-n Rydberg levels (Schippers et al. 2001). Here, field ionization was possible for levels with n ≳ 46. In order to compare the experimental MBRRC to theory, we modified the theoretical results to include the effects of field ionization in the experiment following the procedure of Schippers et al. (2001). The solid line in Figure 1 shows the theoretical MBRRC after accounting for field ionization.

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The MBRRC shows a dense spectrum of resonances dominated by ΔN = 0 core excitations. Some of the expected resonance positions, based on the core excitation energies of Kramida et al. (2013), are illustrated by vertical lines in the figure. For lower energies, below ∼1 eV, it is difficult to assign resonances to individual peaks. The clearest structures in the MBRRC are the series arising from 3s²3p3d ³P_{0, 1, 2} core excitations at about 61 eV. The resonances associated with these excitations are resolved for n up to about 25.

The agreement between theory and experiment is not particularly good below about 40 eV, but they match fairly well near the series limit at 61 eV. One way to quantitatively compare the experimental and DR rate coefficients is by comparing the integrated DR rate coefficients for different energy ranges. We did this using the ratio of the integrated theory to experimental MBRRC:

$$\begin{equation} \kappa = \frac{{\int \left< \sigma _{\mathrm{DR}}v_{\mathrm{rel}} \right>_{\mathrm{theory}} dE}}{{\int \left< \sigma _{\mathrm{DR}}v_{\mathrm{rel}} \right>_{\mathrm{exp}} dE}}, \end{equation} \tag{ 2 }$$

where the theoretical MBRRC used here includes the field ionization model so that it can be directly related to the experiment. The results for selected energy ranges are presented in Table 3. Below about 2.5 eV, the integrated rates are in surprisingly good agreement, despite the fact that theory does a poor job of reproducing the measured DR structure. For the energy range from 2.5 eV to about 47 eV we find κ ∼ 0.6. The details of the resonances are not well reproduced in this energy range, but the agreement improves with increasing energy, especially above about 30 eV. Finally, from 47 to 70 eV there is good agreement in terms of both κ and the detailed structure.

Table 3. Integrated DR Rate Coefficients

Energy Range	∫ < σDRvrel > expdE	∫ < σDRvrel > theorydE	$\kappa = \frac{\int \left< \sigma _{\mathrm{DR}}v_{\mathrm{rel}} \right>_{\mathrm{theory}} dE}{\int \left< \sigma _{\mathrm{DR}}v_{\mathrm{rel}} \right>_{\mathrm{exp}} dE}$
(eV)	(cm3 s−1 eV)
0.002–0.14	4.91 × 10−9	4.63 × 10−9	0.94
0.14–2.5	1.16 × 10−8	9.17 × 10−9	0.79
2.5–12.5	7.79 × 10−9	5.06 × 10−9	0.65
12.5–30.0	8.48 × 10−9	5.07 × 10−9	0.60
30.0–47.0	6.14 × 10−9	3.83 × 10−9	0.62
47.0–70.0	9.96 × 10−9	9.64 × 10−9	0.97
70.0–272.0	6.51 × 10−9	2.53 × 10−9	0.39

Note. The theoretical data used in this comparison accounts for field ionization.

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Figure 2 shows the DR MBRRC for collision energies E > 70 eV. Our results are similar to what was found by Novotný et al. (2012) in the corresponding energy range for Fe^{11 +}. As was seen earlier, we find that the MBRRC for E > 70 eV is much smaller than for lower energies. Most individual resonances are not resolved in this range, except possibly for some resonances at ≈200 eV due to the 3s²3p4d configuration. The threshold for ionization of N = 3 electrons is 361.0 eV, and the DR below this limit is dominated by core excitations of the N = 3 electrons with ΔN ⩾ 1. The energies of the series limits for 3–N' core excitations were estimated using a hydrogenic approximation and are also indicated in the figure. AUTOSTRUCTURE calculations were performed to model the MBRRC due to 3–4 transitions. These are possible up to about 272 eV. The theoretical MBRRC is illustrated by the solid curve in Figure 2. Based on these calculations, we find that κ = 0.39 in the range 70–272 eV. The DR MBRRC drops to a very low value at the ionization threshold of 361 eV.

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Above the ionization threshold, the MBRRC is generally much smaller. Between the Fe^{12 +} ionization limit and the 2–3 series limit at about ∼850 eV there are some clear peaks that are most likely from 2–3 core excitations. A clear drop in the MBRRC is seen near the 2–3 series limit. DR arising from 2–N' core transitions is possible up to the ionization energy from the N = 2 level, which is about 1110 eV (Kaastra & Mewe 1993). Based on a hydrogenic approximation, we estimated the series limits for the various 2–N' core excitations, which are also indicated in Figure 2.

We have derived a Maxwellian DR rate coefficient α_DR(T) from the experimental results. In order to do this, the DR cross section σ_DR must be extracted from the experimentally measured <σ_DRv_rel >. Then α_DR(T) can be found by multiplying σ_DR by the relative velocity and integrating over a Maxwellian (e.g., Schippers et al. 2004). The procedure followed here is similar to that of Schmidt et al. (2008) and Lestinsky et al. (2009), which give additional details.

For low relative energies, E < 0.135 eV, we deconvolved <σ_DRv_rel > by constructing a model σ_DR(E) represented by a sum of δ-function DR resonances $\sigma _{\mathrm{DR}}=\sum _{i}{\bar{\sigma }^{i}_{\mathrm{DR}}\delta (E-E^{i}_{\mathrm{res}})}$ characterized by their centroid positions $E^{i}_{\mathrm{res}}$ and resonance strengths $\bar{\sigma }^{i}_{\mathrm{DR}}$ (Schippers et al. 2004). This cross section was then analytically convolved with the experimental energy distribution characterized by T_⊥ and T_∥ to obtain a model <σ_DRv_rel >. The model used 21 resonances with $E_{\mathrm{res}}^i$ $\bar{\sigma }_{\mathrm{DR}}^i$, T_⊥, and T_∥ as free parameters. The best-fit spectrum is shown in Figure 3. The temperatures quoted in Section 2 were derived from this fit, but note that the derived PRRC is insensitive to the values of T_⊥ and T_∥ since it depends only on σ_DR.

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The low-energy DR data for E ≲ k_BT_⊥ suffers from a known systematic effect related to enhancement of the recombination rate due to motional electric fields in the interaction region (Gwinner et al. 2000; Hörndl et al. 2006). This enhancement does not contribute to recombination in field free regions and therefore should not be included in the plasma rate coefficient. However, it is likely that unresolved DR resonances are present for E < k_BT_⊥. To accommodate either possibility, we estimated the upper and lower bounds, $\alpha _{\mathrm{DR}}^{\mathrm{hi}}$ and $\alpha _{\mathrm{DR}}^{\mathrm{lo}}$, for the plasma rate coefficient. For the upper bound all 21 resonances in the fit were kept. The lower bound was found by omitting all resonances with E_res < k_BT_⊥ ≈ 2 meV. The corresponding MBRRC is shown by the dash-dotted line in Figure 3. This fit is about a factor of 20 smaller than the total measured MBRRC. Typical enhancement factors for measurements with the Cooler electron beam are about 1.5–3 (Wolf & Gwinner 2003) and we expect the Target should have similar values. Thus, the much larger factor found here strongly suggests that much of the very low-energy recombination is from unresolved DR resonances and the lower bound is likely to significantly underestimate the DR rate coefficient. We took the final value for α_DR(T) to be the average value $[\alpha _{\mathrm{DR}}^{\mathrm{hi}}(T) + \alpha _{\mathrm{DR}}^{\mathrm{lo}}(T)]/2$ and used the bounds to estimate the possible systematic error from the field enhancement affect. The resulting uncertainty decreases with temperature, being 9% at 10³ K and dropping to 1% by 10⁴ K and negligible at high temperatures.

For E > 0.135 eV the experimental energy spread was much smaller than the collision energy, so we used the approximation σ_DR = <σ_DRv_rel > /v_rel for the cross section. In this energy range, it was also necessary to correct for field ionization in the experiment. This was done using the theoretical calculations as a guide. We took the ratio of the total theoretical DR MBRRC to the theory with the field ionization model included. This ratio then gives a correction factor as a function of the collision energy that describes the relative increase in <σ_DRv_rel > in the absence of field ionization. The ratio is equal to 1 over most of the energy range, except near the series limits, particularly the 3s²3p3d ³P_{0, 1, 2} series limit at 61 eV (see Figure 1). The experimental MBRRC was multiplied by the correction factor to estimate the MBRRC that we would have measured in the absence of fields. The correction increases the final α_DR(T) compared to not applying the correction. The difference is about 12% for T > 10⁶ K but negligible below about 10⁵ K.

An alternative method for estimating the field ionization contribution, which has been used in previous studies (e.g., Lestinsky et al. 2009), is to replace the energy ranges affected by field ionization in the experiment with theoretical data that was scaled to match experiment in energy ranges not affected by field ionization. For Fe^{12 +} this replacement method seems undesirable due to the clear differences in the resonance structure between the theory and experiment. However, we did use this alternative method to estimate the systematic error from the field ionization correction. The replacement method α_DR(T) is smaller by about 5% compared to the ratio method. Thus, the systematic uncertainty on the plasma rate coefficient due to field ionization is estimated to be about 5% for T ≳ 10⁶ K.

Figure 4 shows the resulting DR plasma rate coefficient. Taking the uncertainties from the 1σ systematic error in the experiment, from field ionization and from the low-energy enhancement, and assuming that the random statistical errors integrate away, we estimate the total 1σ uncertainty of α_DR(T) to be about 8% for T ≳ 10⁴ K. The uncertainty grows to about 12% at 10³ K, due to the possible contribution of unresolved resonances discussed earlier. In Section 2, it was mentioned that this result uses measurements with the Target electron beam for E < 1.8 eV. As a check on the systematic uncertainty, we also derived the plasma rate coefficient using Cooler data for this energy range and found that the two results agree to better than 3%.

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Figure 4 also compares the experimental α_DR(T) to the recommended DR rate coefficient of Arnaud & Raymond (1992). In addition, we compare it to the theoretical rate coefficient of Badnell (2006), modified to include the new AUTOSTRUCTURE calculations for the 3–4 transitions discussed in Section 3.1. The most relevant temperature ranges for comparison are those where Fe^{12 +} is abundant in ionization equilibrium. For photoionized plasmas Fe^{12 +} is greater than 1% abundant over the temperature range 2.0 × 10⁴–1.6 × 10⁵ K (Kallman et al. 2004). The equivalent temperature range for electron-ionized plasmas is 1.1 × 10⁶–1.9 × 10⁶ K (Bryans et al. 2009). These temperature ranges are indicated in Figure 4. As has been seen previously for DR of other M-shell Fe ions, the rate coefficient of Arnaud & Raymond (1992) is orders of magnitude smaller than our experimental result in the temperature range relevant for photoionized plasmas. It is also about 40% smaller than our measurement at temperatures relevant for collisionally ionized plasmas. The modified theoretical rate coefficient of Badnell (2006) is smaller than the experimental value by about 30% for photoionized plasmas and by about 25% for electron-ionized plasmas. This moderate discrepancy is about the same order as found for other systems (Schmidt et al. 2008; Lestinsky et al. 2009).

At high temperatures, one factor that contributes to the discrepancy between theory and experiment is the neglect of ΔN > 1 core excitations in the theoretical calculations. We estimated the contribution of ΔN ⩾ 1 DR to the experimental α_DR(T) by setting σ_DR = 0 for E > 70 eV. We find that ΔN ⩾ 1 DR makes up about 1% of the total rate coefficient at 3 × 10⁵ K, ≈10% at 1 × 10⁶ K, ≈20% at 2 × 10⁶ K, and ≈30% at 1 × 10⁷ K. A similar magnitude effect is seen in the theoretical calculations when higher-energy resonances are included. For example, by including DR arising from 3–4 core excitations to the original theory of Badnell (2006), we find that the rate coefficient is increased by about 15% at 10⁷ K. We expect that including further ΔN > 1 channels would further decrease the discrepancy with experiment at high temperatures.

We have parameterized our experimental DR plasma rate coefficient with a fit that can be used to reproduce our results. The fitting function is

$$\begin{equation} \alpha _{\mathrm{DR}}(T) = T^{-3/2} \sum _{i=1}^{8}{c_{i}\exp (-E_{i}/T)}. \end{equation} \tag{ 3 }$$

The parameters for the fit are given in Table 4. The fit reproduces the experimental result to better than 1% over the temperature range 10²–10⁸ K. Table 5 gives the fitting coefficients for the updated theoretical results, which combine the Badnell (2006) rate coefficient with the new AUTOSTRUCTURE calculations for DR due to 3–4 core excitations.