Photoionization of Co⁺ and electron-impact excitation of Co^{2 +} using the Dirac R-matrix method Free

Abstract

1 INTRODUCTION

Lowly ionized species of Cobalt are often observed in astrophysical objects such as supernovae (SNe), cool stars (Bergemann, Pickering & Gehren 2010), early-type stars (Smith & Dworetsky 1993) and the solar spectrum (Pickering et al. 1998). These applications necessitate the need for high-quality atomic data which accurately describe the processes of excitation and photoionization. This is further evidenced in SNe by following the nucleosynthesis decay path of ⁵⁶Ni→ ⁵⁶Co→ ⁵⁶Fe, which occurs post explosion. Our principal aim is to facilitate modelling within the astrophysics community with accurate and up-to-date atomic transitions necessary for synthetic spectral analysis, allowing detailed comparisons to be carried out with observation. Stand-alone reports stress both the importance and absence of photon/electron interaction with systems of Iron, Cobalt and Nickel (Ruiz-Lapuente 1995; Hillier 2011; Dessart et al. 2014).

The photoionization cross-sections are to be implemented into the 3D plus time-dependent computer code artis, which models the ejecta of Type Ia SNe as discussed in detail by Sim (2007); Kromer & Sim (2009). Currently, only hydrogenic approximations, or fits to the cross-sections are incorporated into the codes. It is our aim to provide ground and excited state contributions to all possible final states. The code is based on a similar approach by Lucy (2005), where methods for treatment of the transporting radiation, kinetic energies, and ionization energies are considered to compute light curves for particular models and symmetries.

The Opacity Project has been an invaluable source for such data, but is often limited when considering Fe-peak species (Cunto & Mendoza 1992; Cunto et al. 1993). These important Fe-peaks are difficult to investigate due to their open d-shell structure which gives rise to many hundreds of target states for each electronic configuration and typically thousands of closely coupled channels. Hence the target states require substantial configuration interaction expansions for their accurate representation. Fe ii is one such challenging case where over the last decade calculations for this ion have grown in size, complexity and sophistication. Significant differences, however, are still observed in the resulting atomic data as can be seen by the latest two major evaluations for the electron-impact excitation of Fe ii (Ramsbottom et al. 2007; Bautista et al. 2015). Factors of 2 to 3 disparity being the norm at the temperature of maximum abundance 10⁴ K for many of the low-lying forbidden lines.

There have been a number of studies focused on essential atomic data between species of Co i-iii concerning bound transitions. These include oscillator strengths for neutral Cobalt between 2276 and 9357 Å (Cardon et al. 1982), transition probabilities through a multiconfiguration approach for comparison with observed infrared spectra (Nussbaumer & Storey 1988), and also a relativistic Hartree–Fock approach between the lowest 47 levels of Co ii (Quinet 1998). More recently, collision strengths and other radiative data have been calculated for Co ii (Storey, Zeippen & Sochi 2016) and Co iv (Aggarwal et al. 2016). During the preparation of this work, it has come to our attention a detailed study of electron-impact excitation cross-sections for Co iii conducted by Storey & Sochi (2016), which we shall compare with in Sections 2 and 3.

The early ion stages of, and even neutral Cobalt are clearly important as detailed in the literature. Early observations has shown strong Co ii lines in η Carinae (Thackeray 1976), confirmed more recently by Zethson et al. (2001) to be unusually strong, and in the UV regime in ζ Oph (Snow, Weiler & Oegerle 1979), confirmed by Federman et al. (1993). These lines are apparent in the binary star HR5049 (Dworetsky, Trueman & Stickland 1980; Dworetsky 1982) – which previously have been unidentified due to the lack of laboratory data. In this same study, the amount of Cobalt is estimated to be around 3.0 dex overabundant relative to the Sun.

Due to the decay path of ⁵⁶Ni, Cobalt is often observed in various SNe at both early and late epochs. The Type II SNe 1987A, exploded in the large magellanic cloud, providing a study of the expanding ejecta as the ⁵⁶Co decays. A large proportion of Fe ii and Co ii lines are blended due to their similar ionization energies, but the strong 1.547 μm line occurs from the transition a⁵F₅ → b³F₄ (Meikle et al. 1989; Li, McCray & Sunyaev 1993) in Co ii. The Co iii a⁴F_9/2 → a²G_9/2 0.589 μm line in another Type II SNe, 1991bg, is used as a diagnostic to infer the mass of synthesized ⁵⁶Ni. It is also possible to deduce important properties such as the mass of the exploding star (Mazzali et al. 1997).

The Co ii and Co iii ions under discussion in this publication have also received much interest over the last decade. The mid-infrared spectrum of SNe 2003hv and 2005df show strong Co iii line emissions and even emission from Co iv (Gerardy et al. 2007). However, the collisional processes included in the model have been approximated using statistically weighted collision strengths. A study of the near-infrared spectra from SNe 2005df yields strong Co iii emission lines initially but by day 200 the majority of Cobalt has expectedly decayed down to Fe (Diamond, Hoeflich & Gerardy 2015). A number of Co iii lines are still visible at late times for Type Ia SNe as documented in (Ruiz-Lapuente 1995). These lines would be extremely beneficial in particular diagnostic work, but as outlined above, little collisional data exists. In addition the photoionization cross-sections employed in the models are obtained from a central potential approximation (Reilman & Manson 1979) for the ground state only. Co iii is also present in SNe 2014J and the 11.888 μm line is useful for monitoring the time evolution of the photosphere, and again, the mass of synthesized Ni (Telesco et al. 2015). It is evident from these works and the associated applications the importance of conducting sophisticated and complete calculations for the lowly ionized Fe-peak species of Fe, Ni and Co.

In Section 2, we discuss the development of an accurate structure model for Co iii to include in the R-matrix collisional calculations for electron-impact excitation and photoionization. The accuracy of this model will be tested by reassessing energy levels of the target states and the conformity of transition probabilities with previous assessments. In Section 3, we present level-resolved ground and excited state photoionization cross-sections for Co ii and a selection of collision strengths and effective collision strengths for the electron-impact excitation of Co iii. Comparisons will be made where possible with existing data but these are limited. Finally we summarize our findings and conclusions in Section 4.

2 IMPORTANT TRANSITIONS FOR SYNTHETIC SPECTRAL MODELLING

This paper focuses initially on transitions that occur between discrete states of our atomic system, Co iii. In Fig. 1, we graphically present some of the most important lines in the infrared and visible energy bands of the spectrum. Transitions among the ground state term ⁴F and levels of the parent ion Co iii with configuration 3d⁷ are shown on the right-hand side. The neighbouring system Co ii is also shown on the left complete with its fine-structure split J levels to indicate the photoionization process under investigation.

Figure 1.

Important lines in the infrared and visible energy bands between levels of Co iii amongst the 3d⁷ configuration and involving the ground term ⁴F. The neighbouring system of Co ii is to the left with its split J levels to indicate the photoionization process.

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2.1 Target description and bound state transitions

As stated in the introduction, partially filled d-shell systems are difficult due to the hundreds of levels associated with a single configuration. Initially, we include three configurations, 3d⁷ and 3d⁶[4s, 4p] during the optimization process, denoted as model 1, which results in a total of 262 fine structure levels to describe the Co iii ion. The ground state of which is 3d⁷a⁴F_9/2. Next we optimize all orbitals up to 3d on the configurations from the double electron promotions, 3s², 3p² → 3d² and include in the total calculation all configurations from model 1 plus 3p⁶3d⁹ (double promotion from 3s to 3d) and 3s²3p⁴3d⁹ (double promotion from 3p to 3d). This technique can be useful as it alleviates the necessity to include numerous pseudo-states into the calculation. We denote this model 2 which constitutes a total of 292 levels. Finally, by including 3d⁵[4s², 4p²] and 3d⁵4s4p, the number of levels drastically increases to 1259 levels, and we label this as model 3.

We present in Table 1 our ab initio energy levels obtained from grasp0 in eV for the lowest lying 50 levels alongside those observed by Sugar & Corliss (1985). We also present the  per cent difference between Sugar & Corliss (1985) and our model 2, and also provide the lowest 15 levels of Storey & Sochi (2016). Good agreement is found (<10 per cent for the majority of levels) between the present model 2 and model 3 energies and those of Storey & Sochi (2016). As with other Fe-peak ions, the energy levels of the lowest-lying 3d⁷ fine-structure states are notoriously difficult to determine. The highest disparities are found for these levels when compared with Sugar & Corliss (1985), the largest being for the 3d⁷⁴F_9/2 → 3d⁷²H_11/2 (1-12) transition. For levels indexed above 17 the differences are at most 10 to 11 per cent and for many levels by considerably less. The main problem is due to the fact that a single 3d orbital is used to describe the configuration state functions for multiple configurations of type 3d⁷ and 3d⁶4s. Similar differences were reported by Ramsbottom (2009) for the low-lying 3d⁷ fine-structure levels of Fe ii. Despite the high  per cent differences found between these lowest levels in Table 1, overall the average  per cent change across the 171 Sugar & Corliss (1985) Jπ symmetries is a more acceptable 6.2 per cent. The differences between model 2 and model 3 are not significant enough to justify the much larger calculation, and we can also benefit by employing all 292 levels into the close-coupling wavefunction expansion of the Co iii. We therefore adopt our model 2 as the final model for the scattering calculation.

Figure 2.

Present theoretical A-values after the scaling factors in equation (2) have been applied, plotted against the results of Hansen et al. (1984) in s⁻¹ on a log/log scale.

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A much larger calculation for doubly ionized Fe-peak species has been performed recently by Fivet et al. (2016) using the same suite of codes as Hansen et al. (1984), and also by considering the computer package autostructure, (Eissner, Jones & Nussbaumer 1974; Badnell 1986), where the optimization process is carried out with a Thomas–Fermi–Dirac potential using lambda scaling parameters. Numerous doubly ionized ions of Fe, Ni and Co were considered in this publication. Single and double electron promotions out of the 3d, and also single electron promotions out of the 3s to the 5s orbital were included and comprised the basis expansion for both calculations. This is the most sophisticated and complete report for radiative rates in Co iii to date. Within the 3d⁷ complex, we vary approximately 27 per cent on average compared with both methods of Fivet et al. (2016).

2.2 The R-matrix method

To extend this problem to include interactions with photons and electrons, we consider the R-matrix method. A general overview of the theory can be found in Burke (2011). The theory was developed to perform electron scattering by Burke, Hibbert & Robb (1971) and extended for photoionization by Burke & Taylor (1975).

These calculations can be performed using a non-orthogonal B-spline basis set approach (Zatsarinny & Froese Fischer 2000; Zatsarinny & Bartschat 2004), intermediate-coupling frame transformation methods (Griffin, Badnell & Pindzola 1998) and one-body perturbation corrections to the non-relativistic Hamiltonian via the Breit–Pauli operators (Scott & Burke 1980) to name a few. In this paper, we consider the Dirac atomic R-matrix codes (darc) which includes relativistic effects through the Dirac Hamiltonian in equation (1) for this mid to heavy species ion. The wavefunctions obtained from grasp0 are formatted suitable for inclusion into darc. Validity of these methods between the outlined theoretical approaches are always important, and enhance the robustness of the techniques carried out as seen in Tyndall et al. (2016) for Ar⁺ and Fernández-Menchero, Del Zanna & Badnell (2015) for Al^{9 +}. Therefore, it is possible to benchmark sophisticated experimental techniques against theory (Müller et al. 2009; Gharaibeh et al. 2011).

2.3 Photoionization

for initial and final scattering wavefunctions subject to the total dipole contribution. ω = hν is the photon energy or ω → ω⁻¹ when considering the velocity operator, D. α is the fine structure constant, a₀ is the Bohr radius and g_i is the statistical weight of the initial state wavefunction. The summation is then carried out over all dipole-allowed transitions. The total wavefunctions Ψ are obtained from the momenta couplings with those wavefunctions obtained in Section 2.1. We represent these σ^p in units of Mb (≡ 10⁻¹⁸cm²) throughout this work.

The transitions of interest are calculated within the lowest eight fine-structure levels of Co ii pertaining to the two LSπ states as noted in equation (3). We also maintain a closed 3p⁶ core and are only concerned with low-energy transitions above threshold. Therefore, all even and odd allowed dipole symmetries up to J = 6 are calculated subject to the selection rules.

2.4 Electron-impact excitation

We calculate the main contribution to the collision strength defined in equation (4) from the partial waves up to J = 13 of both even and odd parity. These are obtained by considering appropriate total multiplicity and orbital angular momentum partial waves. However, in order to converge transitions at higher energies, we explicitly calculate partial waves up to J = 38 and use top-up procedures outlined in Burgess (1974) to account for further contribution to the total cross-section. The collision strengths can be extended to high-energy limits through an interpolation technique described by Burgess & Tully (1992).

3 RESULTS

As mentioned previously, the photoionization of Co ii and electron-impact excitation of Co iii rely on an accurate description of the Co iii wavefunctions. We are able to retain consistency throughout the calculation, and the fundamental R-matrix conditions apply to both processes. We select a total of 14 continuum basis orbitals per angular momenta to describe the scattered electron. The R-matrix boundary is then set at 10.88 au in order to enclose the radial extent of the 4p orbital. To make comparisons between observation, the target thresholds obtained in Table 1 have been shifted to those of Sugar & Corliss (1985).

3.1 Photoionization

In this section, we detail our results from the photoionization process described by equation (3). There is minimal atomic data for this interaction in the literature as only the total ground state transition exists. The first compilation is from Reilman & Manson (1979), using Hartree–Slater wavefunctions from a central field potential. Secondly, and more recently, Verner et al. (1993) have calculated cross-sections using the Hartree–Dirac–Slater potential and then applying an analytic fitting procedure. Recently, the Los Alamos suite of codes by Fontes et al. (2015) have been used to obtain results by a distorted wave method. For this we look at the photoionization of a 3d electron into the configuration averaged final states. To carefully resolve these spectra, we have employed a total of 200 000 equally spaced energy points over a photoelectron energy range of 20 eV. This ensures the high nl Rydberg resonant states are properly delineated.

The initial bound states that are required, corresponding to the left-hand side of equation (3) are calculated first. In Table 3, we present the energies relative to the ground state 3d⁷ a³F_9/2 of Co iii from our current R-matrix results and compare with those of Pickering et al. (1998). We deviate ≈0.47 eV for the 3d⁸ and ≈0.32 eV for the 3d⁷(⁴F)4s. Despite these discrepancies good agreement is evident for the splitting between all eight fine-structure levels.

In Fig. 3, we present the photoionization cross-section representing the statistically weighted initial state 3d⁸ a³F to all allowed final states. The results in this figure have been convoluted with a 10 meV Gaussian profile at full-width at half-maximum to better compare with experiment. We directly compare here with the results of all three previously mentioned theoretical methods (Reilman & Manson 1979; Verner et al. 1993; Fontes et al. 2015). These methods do not include auto-ionization states and therefore only background cross-sections are presented. It is clear however that the results are in excellent agreement, with only Reilman & Manson (1979) reaching factors of 2 or more difference in the <20 eV region. As over 200 eigenenergies obtained from grasp0 are < 28 eV of photon energy, this above threshold region is dominated by multiple Rydberg resonances series converging on to these states.

Figure 3.

Photoionization cross-sections in Mb against the photon energy in eV. The solid black curve is the total initial ground state, statistically weighted 3d⁸³F to all allowed dipole final states. The crosses are results from Reilman & Manson (1979), the diamonds are from Verner et al. (1993), and finally the circles are those from the distorted wave calculation. (Fontes et al. 2015)

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The second statistically weighted bound state a⁵F of Co ii is from the configuration 3d⁷4s. In Fig. 4, we present the total photoionization cross-section for this metastable level and compare with the earlier work of Reilman & Manson (1979). This cross-section is weighted as a consequence of its five fine-structure split levels 1 < J < 5. Again, the photoionization of the 3d electron is favourable, so we expect the majority of the spectrum to be accounted for from the 3d⁶4s target states, which are accessible at 5.757 62 eV above the ionization threshold. Good agreement is found between the two calculations for the background cross-section, particularly in the higher photon energy range above 25 eV.

Figure 4.

Photoionization cross-sections in Mb against the photon energy in eV. The solid black curve is the photoionization, statistically averaged levels of 3d⁷4s ⁵F to all allowed dipole final states. The crosses are results from Reilman & Manson (1979).

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3.2 Electron-impact excitation

The procedures outlined in Section 2.4 are now applied to obtain accurate collision strengths and the corresponding effective collision strengths describing the electron-impact excitation of Co iii. There has been work carried out recently on both singly ionized Co ii (Storey et al. 2016) and also Co iii (Storey & Sochi 2016). The work of Storey & Sochi (2016) is similar to our current evaluation, but differences are evident in the method considered. Their basis set has been optimized using autostructure, and also includes a |$4\bar{{\rm d}}$| orbital plus additional configuration interaction. However, only a total of 109 fine structure levels have been included in the close coupling target representation. The semi-relativistic Breit–Pauli R-matrix approach was considered during the scattering calculation.

It is necessary to obtain a level of convergence in the spectra of collision strengths by applying a mesh with incremental step sizes in electron energy until convergence is achieved. Initially 5000 equally spaced energy points were considered and it was found by the time we had reached 40 000 energy points, the effective collision strengths had converged for the forbidden transitions among the 292 levels. To extend the energy region we incorporated an additional coarse mesh above the last valence threshold. Due to the long-range nature of the Coulomb potential further contributions to the collision strengths arise from the higher partial waves, particularly for the dipole allowed lines. We compute these additional contributions using the Burgess (1974) sum rule as well as a geometric series for the long-range non-dipole transitions. Hence converged total collision strengths were accurately generated for all 42 486 transitions among the 292 fine-structure levels included in the collision calculation. The corresponding effective collision strengths were obtained by averaging these finely resolved collision strengths over a Maxwellian distribution of electron velocities for electron temperatures ranging from 3800 to 40 000 K.

We present in Fig. 5 the resulting collision strengths and effective collision strengths for a selection of the forbidden near-infrared transitions involving the fine-structure split levels of the ⁴F ground state of Co iii. Panel (a) represents transition from level 1→ 2, panel (b) 1 → 3, and panel (c) 2 → 3. The collision strength is strongest here for the Ω_{1 → 2} transition due to the ΔJ = 0 partial wave.

Figure 5.

The collision strengths, (a) Ω_{1 → 2}, (b) Ω_{1 → 3} and (c) Ω_{2 → 3} presented as a function of electron energy in eV. Also presented are the corresponding effective collision strengths as a function of temperature in K. Solid black lines with stars are the current data set and the dashed line with crosses represent the results from Storey & Sochi (2016).

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Comparison of the effective collision strengths are made for each transition with the work of Storey & Sochi (2016) and good agreement is found for all temperatures above 10 000 K. For temperatures below this value the two calculations deviate somewhat, most probably due to the differing resonance profiles converging on to differing threshold positions. The present calculation has shifted the thresholds to lie in their exact observed positions listed in Table 1. In Fig. 6 we present the effective collision strengths for some higher lying transitions, the 3d⁷⁴F_9/2–3d⁷²G_9/2 (1 → 8), 3d⁷²P_1/2–3d⁷²H_9/2 (11 → 13) and 3d⁷²G_7/2–3d⁷²H_9/2 (9 → 13). Excellent agreement is evident for all three transitions when a comparison is made with the data of Storey & Sochi (2016) across all temperatures where a comparison is possible. The collision strengths and effective collision strengths for all 42 486 forbidden and allowed lines are available from the authors on request but we present in Table 4 the effective collision strengths for all transitions among the lowest 11 3d⁷ fine-structure levels across nine temperatures of astrophysical importance.¹

Figure 6.

The effective collision strengths, ϒ_{1 → 8} (top), ϒ_{11 → 13} (middle) and ϒ_{9 → 13} (bottom), presented as a function of temperature in K (bottom). Solid black lines with stars are the current data set and the dashed line with crosses represent the results from Storey & Sochi (2016).

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3.3 Line ratios

Combining the electron-impact excitation rates with the decay rates (A-values), it is possible to investigate important infrared and visible line ratios. For simplicity and purpose of this study, we neglect the recombination and ionization process from the neighbouring ion stages and focus on the populating mechanisms solely of Co iii. Assuming local thermal equilibrium and detailed balance, we can derive the collisional radiative modelling approach detailed in (Griffin et al. 1997). We can therefore investigate selected transitions that have been suggested as temperature or density diagnostics.

We present in Table 5 a comparison of emissivities (ρ_ij) between our current results, and those from Storey & Sochi (2016) for particular transitions at electron densities N_e = 10⁴ cm⁻³ and N_e = 10⁷ cm⁻³. The emissivities are calculated relative to the H β line [see equation (1) in Storey & Sochi (2016)], where the effective recombination coefficients |$\bar{\alpha }_{4\rightarrow 2}$| can be found in Hummer & Storey (1987) and Storey & Hummer (1995). The wavelengths are those from National Institute of Standards and Technology (NIST) and we use our current set of A-values. In general there is excellent agreement for all emissivities considered with the major outlier ρ_{1 → 2} found to be around 30 per cent at N_e = 10⁴ cm⁻³. This is evident from the strong ϒ_{1 → 2} from Fig. 5(a) at T = 10⁴ K. By replacing this one effective collision strength to ϒ_{1 → 2}(T = 10 000) = 4.18, then the ρ_{1 → 2} differences are within a much more agreeable 8 per cent. A similar argument follows for the large differences in the ρ_{2 → 3} at low electron density.