ELECTRON-IMPACT IONIZATION OF P-LIKE IONS FORMING Si-LIKE IONS

Modeling and interpreting spectra of collisionally ionized astrophysical plasmas requires accurate calculations for the underlying charge state distribution (CSD; Landi & Landini 1999; Kallman & Palmeri 2007; Bryans et al. 2009). Such plasmas are formed in the Sun and other stars, supernova remnants, the interstellar medium, galaxies, and the intracluster medium in clusters of galaxies. These CSD calculations, in turn, depend on reliable data for electron-ion recombination and electron-impact ionization (EII). The past decade or so has seen significant advances in the available theoretical and experimental data for dielectronic recombination, which is the dominant recombination process for most ions in cosmic plasmas (Badnell 2006; Schippers et al. 2010). This is to be contrasted with EII which theoretically has only recently been comprehensively studied (Dere 2007). Experimentally, only in the past few years have unambiguous laboratory results become available (Hahn et al. 2011a, 2011b, 2012a, 2012b, 2013).

EII can occur through a number of different channels. Direct ionization (DI) is a one-step process whereby an electron is ejected from an ion A^{q +} of charge state q +, for any given atom A, thereby forming an ion A^{(q + 1) +}. A number of two-step processes can also contribute to EII. Excitation-autoionization (EA) occurs when the incident electron excites a bound electron to a level which can subsequently autoionize to form A^{(q + 1) +} or higher. The incident electron can also undergo dielectronic capture (DC), forming a doubly excited A^{(q − 1) +} system which can then undergo double autoionization, forming A^{(q + 1) +}. This is known as resonant excitation double autoionization (REDA) if the two electrons are released sequentially (LaGattuta & Hahn 1981; Linkemann et al. 1995) and resonant excitation auto double ionization (READI) if the two electrons are ejected simultaneously (Henry et al. 1982; Pindzola & Griffin 1987; Rinn et al. 1987).

EII calculations are theoretically challenging. They require solving a Hamiltonian which include all the relevant interactions in the scattering event. The potential form to use for the descriptions of target and continuum electrons is often an issue. When multi-step processes can be involved, a large number of intermediate states need to be accounted for along with their radiative and autoionizing branching ratios (BRs). Additionally, the problem necessitates suitable wave function expansions both for the target and continuum states. The most sophisticated method is the close coupling (CC) approach, but this is computationally expensive and unsuitable for generating the vast quantities of data required in astrophysics. For this reason, the distorted wave (DW) method has been most widely employed to generate EII data for astrophysical plasma (Kallman & Palmeri 2007; Dere 2007).

Testing of these theoretical methods has been performed for decades using benchmark experimental measurements. Reviews of such work can be found in Müller (1990) and Müller (2008). However until recently most laboratory measurements have been limited to single pass experiments using ion beams with an unknown population of metastable levels (e.g., Gregory et al. 1987). As a result an exact comparison of theory with experiment has not been possible for ions whose cross sections differ between the metastable levels and the ground level.

The development of ion storage rings combined with merged electron beams has helped to overcome this experimental limitation. Using an ion storage ring one can store the ions long enough so that for many systems essentially all of the metastable levels can radiatively decay to their ground states. The resulting EII data provide unambiguous benchmark data for theory. To date, results from such experiments have been published for Li-like Si^{11 +} and Cl^{14 +} (Kenntner et al. 1995), Be-like S^{12 +} (Hahn et al. 2012b), B-like Mg^{7 +} (Hahn et al. 2010), F-like Fe^{17 +} and Ne-like Fe^{16 +} (Hahn et al. 2013), Na-like Fe^{15 +} (Linkemann et al. 1995), Al-like Fe^{13 +} (Hahn et al. 2013), Si-like Fe^{12 +} (Hahn et al. 2011b), P-like Fe^{11 +} (Hahn et al. 2011a), and S-like Fe^{10 +} and Cl-like Fe^{9 +} (Hahn et al. 2012a).

In Kwon & Savin (2012), we focused on comparing theory and experiment for Fe^{11 +}. Previous experimental work by Hahn et al. (2011a) had found discrepancies with the earlier theoretical work of Dere (2007). The theory was below the experimental results near the 3p DI threshold but above the measurements at higher energies where innershell EA channels opened up. The work of Dere (2007) used the flexible atomic code (FAC) of Gu (2008). In Kwon & Savin (2012) we also performed FAC calculations, but took more channels into account. For that work we included 3ℓ → nℓ' (n = 4–35) EA channels near the threshold and the 2ℓ → nℓ' (n = 3–10) EA channels at higher energies. Particular attention was paid to the detailed BRs. These new FAC results helped to resolve several of the previously seen discrepancies. Moreover, at temperatures where Fe^{11 +} forms in collisional ionization equilibrium (CIE; Bryans et al. 2009) the rate coefficient derived from our calculation lies within 11% of the experimentally derived rate coefficient and is in better agreement with the measurement of Hahn et al. (2011a) than the previous FAC results of Dere (2007) which differed by up to 25% from the measurement.

Here we have extended our EII calculations for Fe^{11 +} to P-like systems from P to Zn^{15 +}. Moreover, for Fe^{11 +} in addition to DI and EA we have also included the REDA process, which we did not include in Kwon & Savin (2012). In Section 2, we describe the calculational approach used to obtain theoretical EII cross sections and rate coefficients. In Section 3 the calculated total EII cross sections and rate coefficients are shown for the selected ions and compared with available experiments and with the previous theoretical works of Dere (2007). Lastly, we summarize our results in Section 4.

We have calculated EII cross sections based on the approach and ionization channels detailed in Kwon & Savin (2012). Here we briefly review those calculations. In addition we describe the method employed for calculating the Fe^{11 +} REDA cross section, which was not part of Kwon & Savin (2012).

Here DI, EA and REDA are treated independently. The total single ionization cross section can then be written as (Badnell & Pindzola 1993)

$$\begin{equation} \sigma _{\rm tot} = \sum _{f} \sigma ^{\rm DI}_f + \sum _{j} \sigma ^{\rm CE}_j B^{\rm a}_j + \sum _{k} \bar{\sigma }^{\rm DC}_k B^{\rm da}_k, \end{equation} \tag{ 1 }$$

where $\sigma ^{\rm DI}_f$ is the DI cross section for any given ion A^{q +} to the level f of A^{(q + 1) +}. The second summation on the right hand side of Equation (1) represents EA where $\sigma ^{\rm CE}_j$ is the collisional excitation (CE) cross section of the initial A^{q +} ions to level j, which can then undergo autoionization by the emission of a single electron to form A^{(q + 1) +} with a BR of $B^{\rm a}_j$. The third summation is due to REDA where $\bar{\sigma }^{\rm DC}_k$ is the energy averaged DC cross section to level k of A^{(q − 1) +}, which can then double autoionize, to form an A^{(q + 1) +} ion with a BR of $B^{\rm da}_k$.

The BR for single autoionization of level j by emission of one electron is given by

$$\begin{equation} B^{\rm a}_j = \frac{\sum _{s} A^{\rm a}_{js} B^{\rm r}_s + \sum _{t} A^{\rm r}_{jt} B^{\rm a}_ t}{\sum _{s} A^{\rm a}_{js} + \sum _{t} A^{\rm r}_{jt}}, \end{equation} \tag{ 2 }$$

where A^a denotes the autoionization rate and A^r denotes the radiative decay rate. The s indices are for the levels of A^{(q + 1) +} and the t indices are for the levels of A^{q +}. Some of the s levels may lie above the ionization limit for A^{(q + 1) +} and some of the t levels may lie above the ionization limit for A^{q +}. The radiative BR $B_s^{\rm r}$ and autoionization BR $B_s^{\rm a}$ account for the fraction of those s and t levels eventually resulting in a net single ionization event. Equation (2) must be solved recursively in order to determine all of the BRs.

The BR for double autoionization of level k by emission of two electrons can be expressed as

$$\begin{equation} B^{\rm da}_k = \frac{\sum _{j^{\prime }} A^{\rm a}_{kj^{\prime }} B^{\rm a}_{j^{\prime }}}{\sum _{j^{\prime }} A^{\rm a}_{kj^{\prime }} + \sum _{t^{\prime }} A^{\rm r}_{kt^{\prime }}}, \end{equation} \tag{ 3 }$$

where $A^{\rm a}_{kj^{\prime }}$ is the autoionization rate of A^{(q − 1) +} from k to any level j' of A^{q +}, $A^{\rm r}_{kt^{\prime }}$ is the radiative decay rate of A^{(q − 1) +} from k to t', and $B^{\rm a}_{j^{\prime }}$ is the BR for autoionization of j' level given by Equation (2).

In order to compare the theoretical REDA results with experimental data we need to define the energy averaged DC cross section $\bar{\sigma }^{\rm DC}_k$. This smooths over the resonances in the DC cross section σ_k and can be related to the experimental energy spread by

$$\begin{equation} \bar{\sigma }^{\rm DC}_k = \frac{\langle \sigma _k v \rangle }{v_0}, \end{equation} \tag{ 4 }$$

where the numerator is defined as

$$\begin{equation} \langle \sigma _k v \rangle = \int \sigma _k (v) v f (v_0, \boldsymbol{v}) d^3 v. \end{equation} \tag{ 5 }$$

Here v₀ is the average longitudinal center-of-mass electron velocity in the experiment and $\boldsymbol{v}$ is the electron velocity vector relative to the ions. The term 〈σ_kv〉 can be expressed analytically as Equation (9) of Kilgus et al. (1992) for the "flattened" Maxwellian distribution typical of ion storage rings, as we discuss more in the results section. In that equation, following the theoretical framework of Shore (1969), the integrated cross section of state k for DC can be expressed as

$$\begin{equation} \hat{\sigma }_k = \frac{2\pi \hbar \mathcal {R}}{E_k}\pi a^2_0 \frac{g_k}{2g_i} A^{\rm a}_{ik}, \end{equation} \tag{ 6 }$$

where ℏ is the Planck constant divided by 2π; $\mathcal {R}$ is the Rydberg energy constant; E_k denotes the resonance energy; a₀ is the Bohr radius; g_k and g_i are the statistical weights of the captured and initial states, respectively; and $A^{\rm a}_{ik}$ is the autoionization rate from the level i to k.

The ionization channels for DI and EA considered here are the same as those of Kwon & Savin (2012) for Fe^{11 +} forming Fe^{12 +} except that we have included some additional EA channels and autoionization decay channels resulting in net double ionization which are energetically viable for some ions with low atomic number Z. For P, S⁺, Cl^{2 +}, Ar^{3 +}, and K^{4 +} the 3ℓ → nℓ' EA channels open up at threshold for the ground state from n = 3 but for the isoelectronic Ca^{5 +}–Zn^{15 +} the 3ℓ EA channels open up starting from n = 4. In addition for P, S⁺, and Cl^{2 +}, the 2ℓ → 3ℓ' EA channel followed by autoionization to 2[s, p]⁸3[s, p]³3d levels can further autoionize to form Al-like ions since some of the 2[s, p]⁸3[s, p]³3d levels lie above the Si-like ionization threshold. Here n[s, p]^m indicates that m electrons are distributed between the s and p orbitals in the n shell. However, these autoionizing levels in the Si-like ions are below the Si-like ionization limit for initially Ar^{3 +}–Zn^{15 +} systems, thereby resulting in no net double ionization. Lastly DI of a 2ℓ electron of the initial P-like system was not included because we calculate that the resulting 2ℓ-hole system for P-like ions considered will autoionize over 93% of the time to form Al-like ions.

In addition to the above, we also calculated REDA for Fe^{11 +}. The REDA channels included are as follows. The first step proceeds via the DC process

$$\begin{equation} e^{-} + 2s^2 {}2p^6 {}3s^2 {}3p^3 {} ^4 S_{3/2} \rightarrow 2[s,p]^7 3s^2 3p^3 n_1 \ell _1 n_2 \ell _2, \end{equation} \tag{ 7 }$$

where for n₁ = 3, n₂ = 4–30 and for n₁ = 4, n₂ = 4. For all cases ℓ₁ < n₁ and ℓ₂ ⩽ 5 were included. The DC cross sections for 2ℓ → n₁ℓ₁ with n₁ > 4 were so small that they can be neglected. The captured states can autoionize by emitting one electron to form Fe^{11 +} via

$$\begin{equation} 2[s,p]^7 3s^2 3p^3 n_1 \ell _1 n_2 \ell _2 \rightarrow \left\lbrace \begin{array}{@{}c@{}}2[s,p]^8 3[s,p]^4 n_3\ell _3 \\ \ms 2[s,p]^8 3[s,p]^3 n_1 \ell _1 n_2 \ell _2 \\\ms 2[s,p]^7 3[s,p]^5 n_3\ell _3\end{array} \right\rbrace + e^-, \end{equation} \tag{ 8 }$$

or can radiatively decay leaving the system as Fe^{10 +} by

$$\begin{equation} 2[s,p]^7 3s^2 3p^3 n_1\ell _1 n_2 \ell _2 \rightarrow \left\lbrace \begin{array}{@{}c@{}}2[s,p]^8 3[s,p]^4 n_1 \ell _1 n_2 \ell _2 \\\ms 2[s,p]^8 3s^2 3p^3 n_3\ell _3\end{array} \right\rbrace + h\nu. \end{equation} \tag{ 9 }$$

The autoionizing states in Equation (8) can then sequentially emit another electron to form Fe^{12 +}

$$\begin{equation} \left\lbrace \begin{array}{@{}c@{}}2[s,p]^8 3[s,p]^4 n_3\ell _3 \\\ms 2[s,p]^8 3[s,p]^3 n_1 \ell _1 n_2 \ell _2 \\\ms 2[s,p]^7 3[s,p]^5 n_3 \ell _3\end{array} \right\rbrace \rightarrow 2[s,p]^8 \left\lbrace \begin{array}{@{}c@{}}3[s,p]^4 \\\ms 3[s,p]^3 n_3 \ell _3\end{array} \right\rbrace + e^-, \end{equation} \tag{ 10 }$$

or radiatively decay via

$$\begin{equation} \left\lbrace \begin{array}{@{}c@{}}2[s,p]^8 3[s,p]^4 n_3\ell _3 \\\ms 2[s,p]^8 3[s,p]^3 n_1 \ell _1 n_2 \ell _2 \\\ms 2[s,p]^7 3[s,p]^5 n_3\ell _3\end{array} \right\rbrace \!\,{\rightarrow}\, 2[s,p]^8 \left\lbrace \begin{array}{@{}c@{}}3[s,p]^5 \\\ms 3[s,p]^4 n_3\ell _3 \\\ms 3[s,p]^3 n_1 {\ell _1}^2\end{array} \right\rbrace + h\nu. \end{equation} \tag{ 11 }$$

For 2ℓ → 3ℓ₁ excitation and capture into n₂ > 30, a simple hydrogenic scaling law (Romanik 1988) was used to extrapolate the resonance energies, autoionization rates, and radiative decay rates of the captured electrons. In the extrapolation the resonance energy for the captured states of n₂ > 30 is given by

$$\begin{equation} E_k (n_2) = E_k (n_s) - (Z-N+1)^2\left(\frac{1}{n_2^2}-\frac{1}{n_s^2}\right)\mathcal {R} \end{equation} \tag{ 12 }$$

for n_s = 30 since the binding energy for high n₂ Rydberg levels can be expressed as

$$\begin{equation} -\frac{(Z-N+1)^2}{(n_2 - \delta)^2} \mathcal {R} \simeq -\frac{(Z-N+1)^2}{n_2^2} \mathcal {R}, \end{equation} \tag{ 13 }$$

where the quantum defect δ is independent of n₂ and n₂ ≫ δ (Cowan 1981). The autoionization and radiative decay rates are given by

$$\begin{equation} A^{\rm a}_{kj^{\prime }}(n_2) = A^{\rm a}_{kj^{\prime }}(n_s) \times \frac{n_s^3}{n_2^3} \end{equation} \tag{ 14 }$$

and by

$$\begin{equation} A^{\rm r}_{kt^{\prime }}(n_2) = A^{\rm r}_{kt^{\prime }}(n_s) \times \frac{n_s^3}{n_2^3}, \end{equation} \tag{ 15 }$$

respectively. The radiative decay rate of the core electron for these higher n₂ levels was set to that of the last n₂ = 30 level for which explicit calculations were carried out.

Based on the good agreement found in Kwon & Savin (2012) with the experimental results, we used the post form of a V^N potential where N is the number of initial target electrons (Pindzola et al. 1995). In the DW approximation, the cross section is obtained by considering the continuum and bound electrons interactions perturbatively. For the first order perturbation theory used in the conventional DW approach, the interaction potential is taken as a Coulomb field generated from an arbitrary effective charge, due to the screening of the nucleus by the continuum and bound electrons, and neglecting the long range interaction between the scattered and ejected continuum electrons (Macek & Botero 1992). Hence, the validity of the assumed potential form in DW relies on benchmarking by experimental results (Pindzola et al. 1995). This is to be contrasted with the more sophisticated CC method where the interaction is treated non-perturbatively, taking into account long-range, three-body Coulomb effects. Clearly a more exact treatment for the potential, for example, using the time-dependent CC approach (Pindzola & Schulz 1996), would be helpful to further verify the proper potential form choice.

More specifically, a single local central potential for the initial and final state radial wave functions was optimized on the 3s²3p³ configuration of initial target ion for DI, EA, and REDA. The alternative optimization on the 3s3p³ + 3s²3p² configurations can lead to different values for the cross section, especially for DI. The sensitivity of the results to the configuration selected increases as the charge state of the initial target ion decreases. However, as we show below, optimization on the 3s²3p³ configuration yielded good agreement with the available experimental results for S⁺, Cl^{2 +}, and Ar^{3 +} as well as for Fe^{11 +}. Hence that is the configuration we selected for optimizing the radial wave functions.

For a Maxwell–Boltzmann electron energy distribution at an electron temperature T_e, the DI and EA rate coefficient α_{DI + EA}(T_e) is generated from the calculated cross section σ(E) using

$$\begin{eqnarray} \alpha _{\rm DI+EA} (T_{\rm e}) &=& \frac{1}{(m_e \pi)^{1/2}} \left(\frac{2}{{\rm {k}_B}T_{\rm e}}\right)^{3/2} \nonumber\\ && \times \int _{0}^{\infty } \sigma _{\rm DI+EA}(E){\rm exp}\left(-\frac{E}{{\rm {k}_B} T_{\rm e}}\right) {d}E, \end{eqnarray} \tag{ 16 }$$

where k_B is the Boltzmann constant and m_e is the electron mass. For REDA the Maxwellian rate coefficient is given by (Shore 1969)

$$\begin{equation} \alpha _{\rm REDA} (T_{\rm e}) = \frac{1}{2g_i}\left(\frac{4 \pi {a^2_0} \mathcal {R}}{{\rm {k}_B} T_{\rm e}} \right)^{3/2} \sum _k g_k A_{ik}^a B^{\rm da}_k \exp \left(-\frac{E_{k}}{{\rm {k}_B} T_{\rm e}} \right). \end{equation} \tag{ 17 }$$

Adding these two gives the total ionization rate coefficient

$$\begin{equation} \alpha _{\rm I}(T_{\rm e}) = \alpha _{\rm DI+EA}(T_{\rm e}) + \alpha _{\rm REDA}(T_{\rm e}). \end{equation} \tag{ 18 }$$

The calculated DI+EA+REDA cross section for ground state Fe^{11 +} is shown in Figure 1. In order to compare our REDA results with ion storage ring data, we must convolve the theoretical data with the flattened Maxwellian distribution of the experiment, which is described by parallel T_|| and transverse T_⊥ temperatures with respect to the electron beam direction (Schippers et al. 2001). For Fe^{11 +} these were k_BT_|| = 13.5 meV and k_BT_⊥ = 180 μeV (Hahn et al. 2011a).

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In our previous EII calculation for ground state Fe^{11 +} (Kwon & Savin 2012), the 2ℓ → 3ℓ' EA channel appeared to turn on about 20 eV higher in energy than the experimental data. We hypothesized that this difference could be due to neither REDA nor READI being accounted for in our previous calculations. As can be seen in Figure 1, including the REDA leads to better agreement with the experiment between ∼680 and 720 eV. However the theory is still below the experiment in the energy range 720–800 eV. This remaining discrepancy is unlikely to be due to REDA. It is true that over the energy range of 640–920 eV DC can proceed via e⁻ + 2ℓ → n₁ℓ₁n₂ℓ₂ (n₁ = 3 or 4 and n₂ > 3). But there are many more sizable resonances concentrated in the energy range 640–750 eV than in the energy range 750–920 eV and the convolved REDA cross section is significant only in the energy range 640–750 eV as shown in Figure 1. We may attribute the remaining difference between theory and experiment over the energy range 720–800 eV to READI which has not yet been incorporated into FAC and therefore has not been considered here. In the energy range of 405–920 eV, READI can occur energetically for DC via e⁻ + 2ℓ → n₁ℓ₁n₂ℓ₂(n₁ = 3 or 4 and n₂ ⩾ 3). The detailed READI cross section calculation is beyond the scope of this paper. READI has been considered for Li-like systems using many-body perturbation theory (Pindzola & Griffin 1987) and a unified R-matrix approach (Müller 2000; Scott et al. 2000). However, it is unclear how to relate those results to the more complex P-like systems considered here.

The REDA contribution to the Maxwellian rate coefficient increases the DI+EA rate coefficient by less than 4% at temperatures where Fe^{11 +} is predicted to form in CIE. This is only a small contribution to the total EII rate coefficient, and so we do not include REDA in the calculated total EII cross sections and rate coefficients for all the P-like ions from P to Zn^{15 +} presented below.

Figures 2–6 show our calculated total EII cross sections for some selected other P-like ions where experimental data is available to compare with. As for comparison to the FAC results of Dere (2007), he did not present data for P, S⁺, and Cl^{2 +}. His FAC calculations for Ar^{3 +} and Ni^{13 +} are shown in Figures 5 and 6, respectively.

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Figure 2 shows our calculated cross section for P. At the peak, it is up to two times larger than the measurements of Freund et al. (1990). At the highest energies it is in good agreement with experiment. The agreement does not improve if we use other potentials for the calculations. Using the V^{N − 1} potential for DI optimized on the 3s3p³ + 3s²3p² configurations reduces both the peak cross section as well as that at high energies. Similarly, the agreement is also poor using a V^N + V^{N − 1} potential optimized on the 3s²3p³ + 3s3p³ + 3s²3p² configurations.

The cause for the discrepancy between experiment and theory is not immediately obvious. The neutral atoms in the experiment were generated using charge-transfer neutralization of fast P⁺. The resulting neutral beam had an unknown metastable population. Clearly, measurements on ground state P would greatly help to resolve the issue by providing unambiguous laboratory benchmark data. Moreover, such benchmark results, could guide the selection of the proper potential form to use for DW calculations.

Another theoretical issue which may play a role involves the approximations used for the calculation of the collision cross section. FAC assumes a weak interaction in order to derive the scattering matrix (Bar-Shalom et al. 1988; Gu 2008; Sampson et al. 2009). Such an approach is usually reliable for highly charged ions but breaks down for neutral atoms. This approximation does not ensure that the scattering matrix is unitary, which implies potentially incomplete conservation of wave function flux before and after scattering. This issue is generally corrected for using a normalization procedure (Sobelman et al. 1981; Clark 1990). Unfortunately the correction is not implemented at present in FAC. Alternatively, it is possible that calculations with the more sophisticated CC R-matrix method, which should yield a scattering matrix closer to unitary, might resolve the discrepancy between experiment and theory.

Our new FAC calculations for S⁺ are in good agreement with the measurements by Yamada et al. (1988) and Djurić et al. (1993) near the ionization threshold and also at peak as shown in Figure 3. But at higher energies our calculations agree better with the results of Djurić et al. (1993) than with those of Yamada et al. (1988).

For Cl^{2 +} our new FAC calculations are in good agreement with the measurement by Mueller et al. (1985) near the ionization threshold and also at peak as shown in Figure 4. At higher energies our calculation is lower than the measurement but still within the total experimental systematic uncertainty.

Figure 5 shows our new FAC calculations for Ar^{3 +}. At peak energy they are in better agreement with the measurements by Gregory et al. (1983) than with those by Müller et al. (1980). At an energy of ∼60 eV, near the 3ℓ ionization threshold, our calculated total EII cross section, which includes EA via 3ℓ into high nℓ', is in better agreement with the experimental data of Gregory et al. (1983) than are the FAC cross section results by Dere (2007).

Our new FAC calculation for Ni^{13 +} near the ionization threshold is in better agreement with the measurement by Cherkani-Hassani et al. (2001) than are the previous FAC results by Dere (2007), as can be seen in Figure 6. However our calculated cross section lies below the measurement at energies of ∼840–2000 eV where 2ℓ → nℓ' EA contributes. Resolving these discrepancies would be greatly aided by laboratory EII measurements on ground state Ni^{13 +}.

Surprisingly, for EII of S⁺–Ar^{3 +} the metastable contamination in the experimental results seems to generate no major discrepancies between our calculations and the laboratory data. We attribute this, in part, to the fact that the metastable states have the same 3s²3p³ valence shell configuration as that of the ground state (See Gregory et al. 1983). According to our calculations, DI is the dominant contribution to the EII cross section for all P-like systems from P through Zn^{15 +}. We find that the peak DI cross sections for the metastable levels are larger by only 8% for S⁺ compared to the ground state. This decreases to 4% by the time one reaches Ar^{3 +}. However, for P the increase is 22%, much larger than for S⁺–Ar^{3 +}. This decreasing difference with increasing Z is due to decrease in the importance of electron–electron interactions as the nuclear charge increases. These low Z results are to be contrasted with the comparison between our results and the measured data for Ni^{13 +}. For this ion any metastable states in the experiment are expected to be mostly in the 3s²3p²3d configuration rather than that of the 3s²3p³ ground term. For Ni^{13 +}, the peak DI cross section increases by 24% as one goes from the ground state to the lowest metastable term of the 3s²3p²3d configuration. This difference is much larger than those of the ground terms for S⁺–Ar^{3 +} and may explain much of the discrepancy between the Ni^{13 +} measurements and our calculations.

The total Maxwellian rate coefficients derived from our calculated EII cross sections are shown in Figure 7 for even Z, P-like ions from P to Zn^{15 +}. Also shown are the rate coefficients from the previous FAC calculations by Dere (2007) for even Z ions from Ar^{3 +} to Zn^{15 +}. The dotted line for S⁺ represents the rate coefficient from the measurement of Yamada et al. (1988) which was included in the CHIANTI atomic database (Dere et al. 1997; Landi et al. 2012).

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Table 1 lists the relative difference between our FAC results and the rate coefficients in the CHIANTI data base, normalize to our calculations. Also listed is the temperature of peak formation for each ion in CIE (Bryans et al. 2009). Our calculated rate coefficient for P shows a +94% difference with the rate coefficient derived from the data of Freund et al. (1990). For S⁺ our calculated rate coefficient shows a +77% difference from the rate coefficient derived from the measurement by Yamada et al. (1988). For Cl^{2 +} our calculated rate coefficient shows a +17% difference from that derived using the data of Mueller et al. (1985). The rate coefficients for the ions Ar^{3 +}–Zn^{15 +} derived from our new FAC calculation show relative differences starting at +50% and decreasing to +7%, compared to the earlier FAC rate coefficients of Dere (2007).

As for which rate coefficients we recommend for modeling, for P we suggest using the experimental results as there are known potential errors in the FAC DW results and the arguments given above suggest that the presence of metastable levels in the experimental results has only a minimal affect for the low Z systems in this isoelectronic sequence. For S⁺, our calculations are in better agreement with the measurements of Djurić et al. (1993) than those of Yamada et al. (1988) and so we recommend the use of our results for this ion. Lastly, for Cl^{2 +}–Zn^{15 +}, we recommend using our data which are more complete than the results of Dere (2007).

For convenience in plasma modeling we have fitted our calculated Maxwellian rate coefficient using the Burgess–Tully (BT) scaling of Dere (2007) and a fifth-order polynomial for the scaled rate coefficient. The electron temperature T_e was scaled as

$$\begin{equation} x = 1-\frac{{\rm ln} 2}{{\rm ln} (t+2)}, \end{equation} \tag{ 19 }$$

where t = k_BT_e/E₀ and E₀ is the ionization potential of the ion. The rate coefficient α_I was scaled as

$$\begin{equation} \rho = \frac{t^{1/2} E_0^{3/2}}{E_1 (1/t)} \alpha _{\rm I} (T_{\rm e}), \end{equation} \tag{ 20 }$$

where E₁ is the first exponential integral. The scaled rate coefficient was fitted by the polynomial form

$$\begin{equation} \rho = \sum _{i=0}^5 a_i x^i. \end{equation} \tag{ 21 }$$

The fit parameters are listed in Table 2 along with the fit temperature range and the maximum fit error over this temperature range.

We have calculated EII for ground state P-like systems from P to Zn^{15 +} forming Si-like ions. The calculations were performed using FAC within a DW approximation. For Fe^{11 +} we have also calculated the REDA cross section and find that it contributes ≲ 4% to the total DI+EA rate coefficient at the CIE temperature of peak formation. For all systems, EA via 3l → nℓ' (n = 4–35) channels near the ionization threshold and via 2ℓ → nℓ' (n = 3–10) channels at high energies are included, along with the proper detailed BRs. Our calculated total DI+EA cross sections are compared with the available experimental results for P, S⁺, Cl^{2 +}, Ar^{3 +}, and Ni^{+13 +} and with the previous FAC calculations for Ar^{3 +}–Zn^{15 +}. We find reasonable agreement with the available experimental data for all systems except for neutral P. For P the calculated peak EII cross section is about two times larger than the available experiment. Further theoretical and experimental work is required to resolve the discrepancy. The differences between our calculated Maxwellian rate coefficients and the previous FAC rate coefficients relative to ours are 50–7% for Ar^{3 +}–Zn^{15 +}, a difference which decreases with increasing atomic number.

We thank M. Hahn for stimulating discussions and P. Defrance for providing the experimental EII data for Ni^{13 +}. This work was supported by the Korean Ministry of Science, ICT and Future Planning (MSIP) and by grants from the NASA Astronomy and Physics Research and Analysis (APRA) Program and the NASA Solar and Heliospheric Physics Supporting Research and Technology Program.