Theoretical Investigation of Electron–Ion Recombination Process of Mg-like Gold

Abstract

The L-shell dielectronic and trielectronic recombinations of highly charged Mg-like gold ions (Au⁶⁷⁺) in the ground state 2s²2p⁶3s^2 1S₀ have been studied systematically. The recombination cross-sections and rate coefficients are carefully calculated for ∆n = 1 (2s/2p → 3l) transitions using a flexible atomic code based on the relativistic configuration interaction method and considering the Breit and QED corrections. Detailed resonance energies and resonance strengths are presented for the stronger resonances of the LMn (n = 3–12) series. It is found that the contributions of the trielectronic recombination to the total cross-section is about 13.75%, which cannot be neglected. For convenience of application, the plasma rate coefficients are also calculated and fitted to a semiempirical formula, and in the calculations, the contributions from the higher excited resonance groups n ≥ 13 are evaluated by an extrapolation method, which is about 2.93% of the total rate coefficient.

1. Introduction

Dielectronic recombination (DR) is a resonant electron–ion recombination process of known importance in astrophysical and fusion plasmas [1,2].A complete DR process can be seen as a resonant two-step process, in which a free electron is captured by an ion under simultaneous excitation of a bound electron, and in the second step, the intermediate excited state decays radiatively [1,2,3,4]. The DR process is a dominant channel occurring in the collisions of electron with highly charged ions (HCIs) and is generally labeled by an inverse Auger process notation. For example, the LMn DR indicates that one L-shell electron is excited during the capture of one free electron, resulting in a resonance-excited state with one electron in the M-shell and another electron in the n-shell (n = M, N, O…). As one important line formation mechanism in plasmas, the cross-sections of DR near resonance energies usually surpass those of other competing processes by orders of magnitude. It strongly influences the total recombination rate and thus the charge state balance, energy-level populations, as well as the radiative spectrum of hot plasmas [1,5]. Hence, accurate DR cross-sections and rate coefficients are needed for modeling astrophysical and fusion plasmas [6,7,8].

Gold is used as a good inertial confinement fusion (ICF) target material, and much attention has been paid to gold plasmas recently [9,10,11,12,13]. The DR processes at different ionization stages of gold have been investigated in recent years. Spies et al. [14] reported the first measurements on the radiative and dielectronic recombination of Li-like gold ions at the heavy-ion storage ring ESR and obtained absolute recombination rates of ∆n = 0 (2s → 2p) transitions with very good energy resolution. Schneider et al. [15] observed the LMM DR resonances of Ne-like gold ions in the electron beam ion trap (EBIT) at the Lawrence Livermore National Laboratory and found that the measured resonance strengths for LMM resonances with 2p⁻¹_3/2 cores are in good agreement with theoretical predictions. Liu et al. [16] investigated the DR of Ne-like gold ions employing a flexible atomic code (FAC) [17] based on the relativistic configuration interaction method and determined its influence on the ionization balance of high-temperature plasma. Yang et al. [18] performed detailed level-by-level calculations for the total DR rate coefficients of Ne-like Au⁶⁹⁺ ions in the ground state using a FAC. Xiong et al. [19] measured the KLL DR resonance strengths of Li-like to O-like gold ions with the Tokyo-EBIT and compared the results with the theoretical predictions, which showed good agreement. Iorga et al. [20] studied the degree of linear polarization in dielectronic recombination Kα satellite lines in Li-like gold ions within a density matrix formalism based on relativistic atomic data computed via relativistic distorted wave approximation.

In this work, we will focus on the L-shell ∆n = 1 (2s/2p → 3l) DR process of Mg-like Au⁶⁷⁺ ions in the initial ground state. Accuracy calculations of the atomic structure of these ions have been carried out and collected spectral data have been collected by many researchers [21,22,23,24]. Konan et al. [21] calculated the energy levels, wavelengths, transition rates and weighted oscillator strengths for strong E1 transitions using the AUTOSTRUCTURE code. Hamasha et al. [22] calculated the energy levels, oscillator strengths and transition rates for the multipole transitions (E1, E2, M1, M2) between the ground and the excited states 3l → nl’ (n = 4, 5, 6, 7). Vilkas et al. [23] reported the energy levels and lifetimes related to ∆n = 0 (n = 3) transitions. Hamasha et al. [24] calculated the atomic structure and spectral data for E1 transitions with ∆n ≠ 0 (n = 3 → 4, 5, 6, 7). To our knowledge, however, theoretical research on the DR processes of Mg-like Au⁶⁷⁺ ions is scare. Using a flexible atomic code [17] based on the relativistic configuration interaction (RCI) method, we studied the DR cross-sections and rate coefficients for inner L-shell 2s/2p-core excitation in isolated resonance approximation. Beyond dominant dielectronic recombination, the higher-order trielectronic recombination is also discussed. In Section 2, the theoretical method is described. In Section 3, the resonant recombination cross-sections and rate coefficients are presented and discussed. Finally, some brief conclusions are given in Section 4.

2. Theoretical Methods

The inner L-shell Δn = 1 resonant recombination process of Mg-like Au⁶⁷⁺ ions in the ground state can be described as follows:

$$\begin{array}{c}A{u}_i^{67+}\left[2s^{2}2p^{6}3s^2\right]+\epsilon e^{-}\to A{u}_d^{66+**}\{\begin{array}{l}\left[{(2s2p)}^{-1}3s^{2}3ln{l}^{\prime }\right](DR)(a)\\ \left[{(2s2p)}^{-1}3s^{-1}3l^{2}n{l}^{\prime }\right](TR)(b)\end{array} \to A{u}_f^{66+*}\left[3s^{2}n{l}^{\prime }+3s3ln{l}^{\prime }+3l^{2}n{l}^{\prime }\right]+h\nu \\ \downarrow \\ A{u}_f^{67+*}\left[3s^2,3snl,n=3-5\right]+{\epsilon }^{\prime }{e}^{-}\end{array}$$

(1)

where Equation (1) (a) describes the dielectronic recombination channels and Equation (1) (b) denotes the trielectronic recombination (TR) channels. The notation (2s2p)⁻¹ indicates a vacancy in the inner L-shell, corresponding to the excitation of either a 2s or 2p electron to the 3l (l = p, d) subshell for the Δn = 1 transition. Here and below, n = 3–12, and all their possible angular momentums l’ are included. Those resonances’ double- and triple-excited states can also autoionize to the ground or to singly excited states of Mg-like Au⁶⁷⁺ ions.

In isolated resonance approximation [25], the DR strength of an individual resonance, involving its excitation from an initial state i via a resonance-excited state d to a final state f, can be given by the following equation [26] (atomic units):

$$S_{idf}={\displaystyle {\int }_0^{\infty }{\sigma }_{idf}}(\epsilon )d\epsilon =\frac{{\pi }^2}{E_{id}}\frac{g_d}{2g_i}{A}_{di}^a{B}_{d,f}^r$$

(2)

where ${\sigma }_{idf}(\epsilon )$ is the DR cross-section as a function of electron energy $\epsilon$; $E_{id}$ denotes the energy separation of the resonance-excited state d from the initial state i, namely, the resonant energy; $g_i$ and $g_d$ are the statistical weights of the states i and d, respectively; $A_{di}^a$ is the autoionization rate from the states d to i; and $B_{d,f}^r$ the radiative branching ratio, written as follows:

$$B_{d,f}^r=\frac{A_{df}^r}{{\displaystyle \sum _{f^{\prime }}{A}_{d{f}^{\prime }}^r}+{\displaystyle \sum _{i^{\prime }}{A}_{d{i}^{\prime }}^a}}$$

(3)

where $A_{df}^r$ represents the radiative rate from the states d to f. In the summations, $i^{\prime }$ and $f^{\prime }$ run over all the possible autoionization and radiative final states from the state d, respectively. $A_{df}^r$ is given by [27]

$$A_{df}^r=\frac{4{\omega }^3{\alpha }^3}{3g_d}{\left|\langle {\psi }_f\Vert T^{(1)}\Vert {\psi }_d\rangle \right|}^2$$

(4)

where $\omega$ is the frequency of the decay photons, $\alpha$ the fine structure constant, $T^{(1)}$ is the electronic dipole operator, and ${\psi }_d={\displaystyle \sum _{\nu }{b}_{d\nu }{\Phi }_{\nu }}$ and ${\psi }_f={\displaystyle \sum _{\mu }{b}_{f\mu }{\Phi }_{\mu }}$ describe the atomic state functions (ASFs) given by mixing the configuration state functions (CSFs) with same symmetries for the states d and f, respectively. Here, the CSFs $\Phi$ are antisymmetric sums of the products of N one-electron Dirac spinors. The autoionization rate can be given by [17]

$$A_{di}^a=2{\displaystyle \sum _{\kappa }{\left|\langle {\psi }_i,\kappa ;J_T{M}_T\left|\left|{\displaystyle \sum _{p<q}\left(V_{\mathrm{Coul} }+V_{\mathrm{Breit} }\right)}\right|\right|{\psi }_d\rangle \right|}^2}$$

(5)

Similar to the previous equation, here, ${\psi }_i$ represents the ASFs of the autoionizing state; $\kappa$ is the relativistic angular quantum number that describes the free electron; $J_T{M}_T$ are the total angular momentum of the coupled system, that is, “the target ion plus the impacting electron”; $V_{\mathrm{Coul} }=1/r_{pq}$ denotes the Coulomb operator of the electron–electron interaction; and $V_{\mathrm{Breit} }$ is the Breit operator [28,29].

For a given resonance state, the strengths $S_{id}$ can be obtained by summing the individual resonance strength $S_{idf}$ over all possible final states f:

$$S_{id}={\displaystyle \sum _f{S}_{idf}}$$

(6)

To compare the theoretical calculations with experimental results, such as those measured in EBIT, the calculated cross-section should be convoluted with a Gaussian distribution describing the experimental electron beam’s energy resolution. Thus, the total DR cross-sections from an initial level i can be obtained by summing the contributions of DR strength $S_{id}$ over all resonance states d [27,30]:

$${\sigma }_{tot}^{DR}\left(\epsilon \right)={\displaystyle \sum _d\frac{2}{\Delta E}}{\left(\frac{\ln 2}{\pi }\right)}^{1/2}\exp \left[-4\ln 2{\left(\frac{\epsilon -E_{id}}{\Delta E}\right)}^2\right]S_{id}$$

(7)

where ΔE is the full width at half maximum (FWHM).

In plasma modeling, the DR rate coefficients are often needed. Suppose that the electron velocity is in the Maxwellian distribution; thus, the DR rate coefficients ${\alpha }_{id}^{DR}$ can be expressed as [30]

$${\alpha }_{id}^{DR}\left(T_e\right)=\frac{4E_{id}}{{\left(2\pi m_e{k}_B{}^3{T}_e^3\right)}^{1/2}}\exp \left(-\frac{E_{id}}{k_B{T}_e}\right)S_{id}$$

(8)

where $T_e$ is the electron temperature and $k_B$ is the Boltzmann constant.

3. Results and Discussion

We performed level-by-level computations to obtain the recombination cross-sections and rate coefficients of Mg-like Au⁶⁷⁺ ions. In the calculations, to explore the effects of electron correlation on resonance energies and strengths, two electron correlation models, labeled as A and B, were adopted. In model A, the important Al-like double-excited configuration complexes 2p⁻¹3s²3lnl’ and 2s⁻¹3s²3lnl’ (l = p, d and n = 3–12, l’ ≤ 5) were considered (in which −1 indicates the hole); that is, only DR channels were included (see Equation (1) (a)), which comprise 11,437 resonant states. Model B included both the resonant double-excited configurations in model A and the resonant triple-excited configuration complexes 2p⁻¹3s⁻¹3l²nl’ and 2s⁻¹3s⁻¹3l²nl’ (l = p, d and n = 3–12, l’ ≤ 5); that is, both the DR and TR channels were included (see Equation (1) (b)), which comprise 147,706 resonant states. The radiative and autoionization final states considered in the present resonance recombination calculations are the same in both model A and model B. Specifically, we included the configurations 3s²nl’, 3s3lnl’ and 3l²nl’ (l = p, d and n = 3–12, l’ ≤ 5) for the radiative final states of Al-like Au⁶⁶⁺ ions, which means that dominant valence–valence, core–valence, and core–core correlations were incorporated. For autoionizing final states, both the ground configuration 2s²2p⁶3s² and the lower single- and double-excited configurations 2s²2p⁶3lnl’ (l = s, p, d and n = 3–5, l’ ≤ n−1) of Mg-like Au⁶⁷⁺ ions were included. Using the FAC code (Version 1.1.5) proposed by Gu [17], which included Berit interactions in the zero-energy limit for the exchanged photons, and quantum electrodynamic (QED) effects [31], we carefully calculated the energy levels and radiative and autoionization rates of Al-like Au⁶⁶⁺ ions. Note that configuration mixing between all the radiative final states and the resonant excited states from DR and TR were included in the calculations. Figure 1 illustrates the energy levels of the Mg-like Au⁶⁷⁺ and Al-like Au⁶⁶⁺ ions calculated in model B. In

Table 1. The calculated energies for the lowest 34 levels and parts of resonance states of Al-like Au⁶⁶⁺ ions relative to the ground state [3s²3p_1/2]_1/2 with the use of models A and B compared to the MRMP results [23] and FAC results [24].

Label	Level	P	2J	This Work		EMRMP [23]	EFAC [24]
				Model A	Model B
0	[3s23p1/2]1/2	1	1	0.00	0.00	0.00	0.00
1	[3s1/2(3p21/2)0]1/2	0	1	187.30	186.88	186.55	186.59
2	[3s23p3/2]3/2	1	3	498.12	498.13	497.56	497.92
3	[(3s1/23p1/2)03p3/2]3/2	0	3	651.39	651.11	650.59	650.47
4	[(3s1/23p1/2)13p3/2]5/2	0	5	671.60	671.27	670.55	670.72
5	[(3s1/23p1/2)13p3/2]3/2	0	3	693.58	693.22	692.65	692.80
6	[(3s1/23p1/2)13p3/2]1/2	0	1	705.64	705.04	703.97	704.87
7	[3s23d3/2]3/2	0	3	753.39	753.22	752.41	753.34
8	[3s23d5/2]5/2	0	5	853.55	853.56	853.25	853.16
9	[(3s1/23p1/2)03d3/2]3/2	1	3	874.47	874.03	873.93	873.24
10	[(3p21/2)03p3/2]3/2	1	3	888.73	888.17	887.10	887.50
11	[(3s1/23p1/2)13d3/2]5/2	1	5	900.18	899.62	899.32	898.92
12	[(3s1/23p1/2)13d3/2]1/2	1	1	929.65	928.98	928.40	928.72
13	[(3s1/23p1/2)13d3/2]3/2	1	3	938.51	937.75	936.90	937.52
14	[(3s1/23p1/2)03d5/2]5/2	1	5	1011.01	1010.55	1010.34	1009.69
15	[(3s1/23p1/2)13d5/2]7/2	1	7	1027.34	1026.79	1026.38	1026.02
16	[(3s1/23p1/2)13d5/2]5/2	1	5	1038.59	1037.87	1037.29	1037.13
17	[(3s1/23p1/2)13d5/2]3/2	1	3	1043.58	1042.73	1042.01	1042.10
18	[(3p21/2)03d3/2]3/2	0	3	1108.54	1107.64	1106.72	1106.85
19	[3s1/2(3p23/2)2]5/2	0	5	1162.12	1161.84	1160.57	1161.11
20	[3s1/2(3p23/2)0]1/2	0	1	1203.57	1203.16	1201.43	1202.58
21	[3s1/2(3p23/2)2]3/2	0	3	1211.30	1210.73	1209.00	1210.42
22	[(3p21/2)03d5/2]5/2	0	5	1236.64	1235.60	1234.60	1234.74
23	[3p1/2(3p23/2)2]5/2	1	5	1366.07	1365.57	1364.44	1364.61
24	[3p1/2(3p23/2)2]3/2	1	3	1372.71	1371.98	1370.40	1371.25
25	[3p1/2(3p23/2)0]1/2	1	1	1391.08	1390.52	1389.35	1389.68
26	[(3s1/23p3/2)23d3/2]3/2	1	3	1398.52	1398.00	1397.05	1397.30
27	[(3s1/23p3/2)23d3/2]1/2	1	1	1400.65	1400.21	1398.81	1399.45
28	[(3s1/23p3/2)23d3/2]5/2	1	5	1404.78	1404.08	1402.80	1403.45
29	[(3s1/23p3/2)23d3/2]7/2	1	7	1406.40	1405.83	1404.84	1405.08
30	[(3s1/23p3/2)13d3/2]3/2	1	3	1438.22	1437.36	1436.11	1436.97
31	[(3s1/23p3/2)13d3/2]5/2	1	5	1451.57	1450.59	1448.92	1450.39
32	[(3s1/23p3/2)13d3/2]1/2	1	1	1458.44	1457.30	1455.55	1457.12
33	[(3s1/23p3/2)23d5/2]9/2	1	9	1486.85	1486.47	1485.85	1485.18
34	[(3s1/23p3/2)23d5/2]5/2	1	5	1511.58	1511.07	1510.17	1510.04
2862	[2p−13/2(3p23/2)0]3/2	1	3	10,550.33	10,546.42
3020	[(2p−13/23d3/2)13d5/2]7/2	1	7	11,098.31	11,100.03
3027	[(2p−13/23d3/2)13d5/2]5/2	1	5	11,105.31	11,109.64
3031	[(2p−13/23d3/2)33d5/2]7/2	1	7	11,106.17	11,117.51
3043	[(2p−13/23d3/2)23d5/2]5/2	1	5	11,122.38	11,136.88
3723	[(2s−11/23p1/2)13d5/2]7/2	1	7	12,801.01	12,794.46
3763	[2p−11/2(3d23/2)2]5/2	1	5	12,858.35	12,884.54
3819	[(2p−11/23d3/2)13d5/2]7/2	1	7	13,002.84	13,015.12

, we compare the energies of Al-like Au⁶⁶⁺ ions calculated in models A and B as well as with the available theoretical results from Refs. [23,24]. It is found that for the lowest 34 levels (Label 1–34) associated with the 3s²3p, 3s3p², 3s3p3d, 3p²3d and 3p³ configurations, the energy levels given by model B show small deviations from the results of model A within about 1.0 eV. The results in model B are closer to the results calculated by Vilkas et al. [23] using the Multi-Reference Møller–Plesset (MRMP) method and also are in good agreement with the FAC results obtained by Hamasha et al. [24] with differences of less than 0.18% and 0.15%, respectively. However, for the resonant excited states (Label 2862–3819), the energy levels given by model B show bigger deviations from the results of model A. Taking the resonant double-excited state [2p⁻¹_1/2(3d²_3/2)₂]_5/2 as an example, the energy is 12,858.35 eV in model A, while it increased to 12,884.54 eV in model B, a difference of 26.19 eV. This indicates that the configuration interactions resulting from taking the TR channels into account have small effects on the lower excited states (radiative final states) but significant effects on the resonance states of the Al-like Au⁶⁶⁺ ions.

In Figure 2, we plot the recombination cross-sections of the LMM processes obtained in model A and B. Here, a Gaussian FWHM of 59 eV is used in the calculations, consistent with the experimental DR measurements for Ne-like to Al-like gold ions reported by Schneider et al. [15]. Compared to the results in model A, it is obvious that the cross-sections in model B are dramatically reduced, especially for the resonances near the resonance energies of 3.24, 4.25, 4.93 and 5.10 keV, which originate from the resonance-excited states [2p⁻¹_3/23d_3/23d_5/2]_J, [2p⁻¹_1/23p_1/23d_3/2]_J, [2s⁻¹_1/23p_1/23d_5/2]_J and [2p⁻¹_1/23d_3/23d_5/2]_J, respectively. This indicates that configuration interactions (CI) between resonances associated with the DR and TR channels strongly affected (reduce) the resonance strengths or recombination cross-sections of Mg-like gold. Based on model B, an extended calculation has also been conducted considering more resonance-excited configuration complexes (2s2p)⁻¹3lnl’ and (2s2p)⁻¹3s⁻¹3l²nl’ (n = 3–12) with l’ up to n−1. We note that this calculation causes very small changes to the total cross-sections. Thus, we employed model B to calculate the resonance recombination strengths, cross-sections and rate coefficients of Mg-like gold.

3.1. Resonance Energy and Strength

In

Table 2. The detailed resonance energies E_id, Auger rate A^a, total width Γ_d, radiative branching ratio B^r_d and resonance strength for dominant DR and TR resonances (S_id > 4.0 × 10⁻²⁰ cm² eV) of Mg-like Au⁶⁷⁺ ions. (a (b) denotes a × 10^b.)

LMn	Intermediate Excited State	Eid (eV)	Aa (s−1)	Γd (s−1)	Brd	Sid (10−19 cm2 eV)
LMM	[(2p−13/23p1/2)23p3/2]7/2	2102.6	9.94 (12)	1.70 (14)	0.88	0.83
	[(2p−13/23p1/2)23p3/2]1/2	2176.1	7.17 (14)	1.86 (15)	0.11	1.74
	[(2p−13/23p1/2)23d5/2]9/2	2446.4	1.81 (13)	1.74 (14)	0.83	1.52
	[(2p−13/23p1/2)23d5/2]1/2	2493.9	7.03 (13)	3.88 (15)	0.96	1.34
	[(2p−13/23p1/2)13d5/2]3/2	2494.2	7.57 (13)	3.88 (15)	0.97	2.90
	[2p−13/2(3p23/2)0]3/2	2635.5	1.47 (13)	1.84 (14)	0.84	0.47
	[2p−13/2(3p23/2)2]3/2	2664.4	4.92 (14)	1.38 (15)	0.14	2.49
	[(2p−13/23p3/2)33d3/2]9/2	2841.7	1.49 (13)	1.77 (14)	0.80	1.04
	[(2p−13/23p3/2)03d3/2]3/2	2923.4	3.93 (14)	1.58 (15)	0.14	1.90
	[(2p−13/23p3/2)33d5/2]9/2	2966.1	1.29 (13)	1.71 (14)	0.84	0.90
	[(2p−13/23p3/2)33d5/2]5/2	2980.8	8.71 (13)	1.28 (15)	0.90	3.93
	[(2p−13/23p3/2)23d5/2]5/2	3001.6	6.79 (13)	2.58 (15)	0.96	3.21
	[(2p−13/23p3/2)33d5/2]3/2	3006.0	4.62 (13)	3.63 (15)	0.97	1.47
	[(2p−13/23p3/2)33d5/2]1/2	3010.1	1.21 (14)	3.61 (15)	0.95	1.90
	[(2p−13/23p3/2)03d5/2]5/2	3031.3	2.67 (14)	1.63 (15)	0.27	3.48
	[(((2p−13/23s−11/2)23p1/2)3/23p3/2)13d5/2]7/2 *	3149.2	7.77 (12)	1.56 (14)	0.86	0.42
	[(2p−13/23d3/2)13d5/2]7/2	3189.1	2.84 (13)	7.49 (14)	0.43	0.76
	[(((2p−13/23s−11/2)13p1/2)1/23p3/2)13d5/2]5/2 *	3190.0	8.22 (13)	2.73 (15)	0.90	3.44
	[(2p−13/23d3/2)13d5/2]5/2	3198.7	4.60 (13)	1.18 (15)	0.69	1.47
	[(2p−13/23d3/2)33d5/2]7/2	3206.6	1.10 (13)	1.93 (14)	0.68	0.46
	[(2p−13/23d3/2)33d5/2]5/2	3218.0	4.80 (13)	9.54 (14)	0.31	0.68
	[(2p−13/23d3/2)23d5/2]7/2	3222.7	2.21 (13)	6.47 (14)	0.32	0.44
	[(2p−13/23d3/2)13d5/2]5/2	3252.3	5.81 (14)	4.36 (15)	0.81	21.61
	[(2p−13/23d3/2)33d5/2]1/2	3256.4	6.99 (13)	4.05 (15)	0.96	1.02
	[2p−13/2(3d25/2)2]7/2	3315.7	1.97 (14)	8.30 (14)	0.73	8.58
	[2p−13/2(3d25/2)4]7/2	3338.5	2.40 (14)	1.86 (15)	0.85	12.08
	[2p−13/2(3d25/2)4]5/2	3358.0	5.89 (13)	4.53 (15)	0.97	2.53
	[2p−13/2(3d25/2)0]3/2	3363.5	9.25 (13)	3.16 (15)	0.95	2.59
	2p−11/2	3491.0	2.49 (14)	5.99 (14)	0.15	0.52
	[(2p−11/23p1/2)03p3/2]3/2	4017.6	3.32 (14)	9.27 (14)	0.10	0.85
	[(2p−11/23p1/2)13d3/2]1/2	4256.9	3.85 (13)	1.62 (15)	0.92	0.41
	[(2p−11/23p1/2)03d3/2]3/2	4260.7	1.29 (14)	8.93 (14)	0.32	0.96
	[(2p−11/23p1/2)03d5/2]5/2	4369.2	1.50 (14)	7.62 (14)	0.14	0.70
	[(2p−11/23p3/2)23d3/2]5/2	4741.1	8.88 (13)	1.99 (15)	0.93	2.58
	[(2s−11/23p1/2)13d5/2]7/2	4883.5	2.12 (13)	9.67 (14)	0.60	0.51
	[(((2p−11/23s−11/2)13p1/2)1/23p3/2)13d3/2]5/2 *	4949.5	3.10 (13)	1.72 (15)	0.75	0.69
	[2p1/2(3d23/2)2]5/2	4973.6	3.18 (13)	5.69 (14)	0.49	0.47
	[(((2p−11/23s−11/2)13p1/2)1/23p3/2)13d5/2]7/2 *	5055.5	5.28 (13)	1.09 (15)	0.40	0.83
	[(2p−11/23d3/2)13d5/2]7/2	5104.2	2.07 (14)	1.91 (15)	0.84	6.73
LMN	[(2p−13/23p1/2)24p3/2]1/2	5496.9	1.60 (14)	5.29 (14)	0.29	0.41
	[(2p−13/23p1/2)14d5/2]3/2	5623.1	3.09 (13)	1.43 (15)	0.96	0.52
LMM	[(2s−11/23d3/2)23d5/2]9/2	5640.6	5.08 (13)	6.92 (14)	0.33	0.74
	[(2s−11/23d3/2)13d5/2]7/2	5653.7	6.00 (13)	5.16 (14)	0.29	0.61
LMN	[(2p−13/23p3/2)34p3/2]3/2	6004.1	1.08 (14)	4.14 (14)	0.39	0.69
	[(2p−13/23p3/2)04p3/2]3/2	6052.7	1.82 (14)	1.46 (15)	0.35	1.03
	[(2p−13/23p3/2)34d5/2]5/2	6121.2	1.82 (13)	3.48 (14)	0.91	0.40
	[(2p−13/23p3/2)04d3/2]3/2	6137.9	7.77 (13)	1.45 (15)	0.46	0.57
	[(2p−13/23d5/2)14p1/2]3/2	6168.4	7.40 (13)	3.41 (15)	0.93	1.10
	[(2p−13/23p3/2)04d5/2]5/2	6188.3	5.93 (13)	1.18 (15)	0.29	0.41
	[(2p−13/23d3/2)24p3/2]3/2	6246.4	4.89 (13)	1.68 (15)	0.85	0.66
	[(2p−13/23d5/2)44p3/2]5/2	6336.1	3.87 (13)	2.82 (14)	0.71	0.65
	[(2p−13/23d5/2)34p3/2]5/2	6349.7	6.99 (13)	5.30 (14)	0.61	1.00
	[(2p−13/23d5/2)44d3/2]7/2	6408.9	1.67 (13)	2.05 (14)	0.85	0.44
	[(2p−13/23d5/2)44d5/2]5/2	6454.7	5.42 (13)	9.45 (14)	0.91	1.14
	[(2p−13/23d5/2)14d3/2]5/2	6458.0	2.23 (14)	3.38 (15)	0.92	4.70
	[(2p−13/23d5/2)44d5/2]7/2	6466.8	7.58 (13)	4.78 (14)	0.81	1.87
	[(((2p−13/23s−11/2)23p1/2)5/23d5/2)54p3/2]7/2 *	6485.0	1.99 (13)	3.06 (14)	0.80	0.49
	[(((2p−13/23s−11/2)13p1/2)1/23d5/2)34p3/2]7/2 *	6497.5	1.74 (13)	1.28 (15)	0.96	0.51
	[(2p−13/23d5/2)14d5/2]7/2	6499.9	5.72 (13)	1.83 (15)	0.95	1.65
	[(2p−13/23d5/2)14d5/2]3/2	6503.2	3.93 (13)	2.86 (15)	0.96	0.57
	[(2p−13/23d5/2)14f5/2]7/2	6573.5	1.02 (14)	3.41 (15)	0.95	2.93
	[(2p−13/23d5/2)14f7/2]9/2	6588.0	8.27 (13)	3.13 (15)	0.96	2.97
	[(2p−13/23d5/2)14f7/2]7/2	6592.8	1.51 (13)	3.53 (15)	0.98	0.44
LMO	[(2p−13/23d5/2)25d5/2]7/2	7901.0	3.64 (13)	3.67 (14)	0.87	0.79
	[(2p−13/23d5/2)15d3/2]5/2	7914.0	7.20 (13)	3.20 (15)	0.96	1.30
	[(2p−13/23d5/2)15d5/2]7/2	7940.1	4.95 (13)	3.99 (15)	0.97	1.20
	[(2p−13/23d5/2)15f5/2]7/2	7972.6	7.78 (13)	3.86 (15)	0.96	1.86
	[(2p−13/23d5/2)15f7/2]9/2	7980.4	5.49 (13)	3.81 (15)	0.97	1.65
LMN	[(2p−11/23d3/2)14d5/2]7/2	8262.6	2.25 (13)	1.44 (15)	0.90	0.49
	[(2p−11/23d5/2)24d3/2]7/2	8299.1	8.76 (13)	1.08 (15)	0.83	1.74
LMP	[(2p−13/23p3/2)06d3/2]3/2	8390.7	4.76 (13)	2.70 (14)	0.79	0.45
	[(2p−13/23d5/2)16d3/2]5/2	8701.6	5.83 (13)	3.76 (15)	0.98	0.98
	[(2p−13/23d5/2)16f5/2]7/2	8733.0	4.82 (13)	3.76 (15)	0.99	1.08
	[(2p−13/23d5/2)16f7/2]9/2	8737.9	3.28 (13)	3.76 (15)	0.99	0.92
LMQ	[(2p−13/23d5/2)17d3/2]5/2	9171.3	3.34 (13)	3.65 (15)	0.99	0.54
	[(2p−13/23d5/2)17f5/2]7/2	9191.3	2.82 (13)	3.67 (15)	0.99	0.60
	[(2p−13/23d5/2)17f7/2]9/2	9194.4	2.04 (13)	3.74 (15)	0.99	0.55
LMR	[(2p−13/23d5/2)18f5/2]7/2	9488.2	1.96 (13)	3.75 (15)	0.99	0.41
LMO	[(2p−11/23d3/2)15d5/2]7/2	9697.5	2.29 (13)	1.36 (15)	0.86	0.40
	[(2p−11/23d5/2)25d3/2]7/2	9751.3	2.53 (13)	4.82 (14)	0.89	0.46

* Represents the TR resonance states.

, the calculated resonance energies, Auger rate A^a, total width ${\Gamma }_d={\displaystyle \sum A^a}+{\displaystyle \sum A^r}$, radiative branching ratio $B_d^r={\displaystyle {\sum }_f{B}_{d,f}^r}$ and the resonance strengths corresponding to each of the individual Al-like resonance states formed by DR and TR processes of Mg-like Au⁶⁷⁺ ions in the initial ground state are tabulated, where only strong resonances with strengths S_id > 4.0 × 10⁻²⁰ cm² eV are listed. It was found that the strength of LMM resonances is dominant, which are mainly produced by Mg-like Au⁶⁷⁺ DR processes through the resonance double-excited configurations (2s2p)⁻¹3l² located in the 2.6–5.6 keV resonance energy range. The strongest resonance comes from the state [(2p⁻¹_3/23d_3/2)₁3d_5/2]_5/2, corresponding to the resonance energy of 3252.3 eV, and the resonance strength of 2.16 × 10⁻¹⁸ cm² eV. Most of the TR resonances show small resonance strengths; however, some resonances, such as [(((2p⁻¹_3/23s⁻¹_1/2)₁3p_1/2)_1/23p_3/2)₁3d_5/2]_5/2 corresponding to a resonance energy of 3190.0 eV in the LMM manifold, have stronger strengths of 3.44 × 10⁻¹⁹ cm² eV that even surpass the strength of many DR resonances.

Figure 3 illustrates the detailed recombination cross-sections for Δn = 1 LMn (n = 3–12) resonance manifolds of Mg-like Au⁶⁷⁺ ions via the various resonant excited configuration complexes (2s2p)⁻¹3lnl’ and (2s2p)⁻¹3s⁻¹3l²nl’ (n = 3–12). As can be seen from the figure, the DR + TR recombination (in model B) spectra of LMn resonance series show similar features in the resonance peaks. For example, the LMM resonances show the largest cross-sections, and the recombination spectra are distributed at 1.5–6.0 keV energy regions, fed by the resonance excitation configuration complexes (2s2p)⁻¹3l² and (2s2p)⁻¹3s⁻¹3l³ (l = p, d). The higher (2s2p)⁻¹3lnl’ and (2s2p)⁻¹3s⁻¹3l²nl’ (n > 3) complexes have weaker DR peaks, appearing at higher energies as n increases and gradually converging to the (2s2p)⁻¹3l and (2s2p)⁻¹3s⁻¹3l² Rydberg limit (which is at about 13.5 keV). Each resonant peak contains a number of individual resonances, and their resonance energy positions are shown as colored vertical bars in the figure. They are also listed together with the resonant strengths in Table 2 (only for some stronger resonances). It is obvious that the LMn resonant spectra belonging to different manifolds overlap each other and become more and more serious as n increases. Such overlapping usually leads to difficulty in clearly identifying the total DR spectra and makes the calculations difficult because of the strong electron–electron correlation effects. To further demonstrate the accuracy of the present results, Figure 4 displays the present calculated LMM DR spectra of Mg-like Au⁶⁷⁺ ions compared with the experimental measurements of the DR spectra for Ne-like to Si-like Au ions obtained by Schneider et al. [15] using the LLNL electron beam ion trap. It can be seen that the DR spectra positions of Mg-like Au⁶⁷⁺ ions in the experiment are predicted well by the theoretical calculations. In our previous work [33], the same electron correlation model, namely, model B, was used to calculate the LMn DR spectra of Mg-like Fe¹⁴⁺ ions. The theoretical results showed very good agreement with the experimental measurements taken by Lukić et al. [34] at the Heidelberg heavy-ion storage ring TSR.

Figure 5 plots the total cross-section for the resonant recombination of Mg-like Au⁶⁷⁺ as a function of electron energy. To clearly show the higher-order TR resonant effects, we identified all DR and TR resonances and obtained the cross-sections by convolving the resonant strength with a Gaussian FWHM of 59 eV. It is found that the DR cross-sections are obviously greater than the TR cross-sections in the whole electron collision energy regions from 2.0 to 12.0 keV. Moreover, the contribution of the TR process to the total cross-section is about 13.75%, which cannot be neglected.

3.2. Rate Coefficients

In Figure 6, the total and partial rate coefficients of Au⁶⁷⁺ ions through the various resonant excited configuration complexes (2s2p)⁻¹3lnl’ and (2s2p)⁻¹3s⁻¹3l²nl’ (n = 3–12) are presented as a function of the electron temperature. It can be seen that the maximum rate coefficients appear at about T_e = 3.4 keV, and the LMM DR processes are the dominant contributors. The (2s2p)⁻¹3lnl’ (DR) + (2s2p)⁻¹3s⁻¹3l²nl’ (TR) (n > 3) complex series (solid curves of different colors) gives the predominant electron–ion resonant recombination contribution. At higher electron temperatures, the electron–ion resonant recombination contributions from the (2s2p)⁻¹3l4l’ and (2s2p)⁻¹3s⁻¹3l²4l’ configuration complex becomes comparable to those of (2s2p)⁻¹3l3l’ and (2s2p)⁻¹3s⁻¹3l²3l’. Our calculations show that when n ≥ 10, the behavior of the total DR rate coefficient follows the n⁻³ scaling law [35]. So, we use the n⁻³ scaling law to extrapolate the rate coefficients for n = 13–1000 and obtain the total rate coefficient. The contribution from the higher-order excited resonance groups n ≥ 13 to the total rate coefficient is about 2.93%. Furthermore, in order to investigate the importance of the TR process on the plasma ionization balance and radiation energy, we also plot the rate coefficients from the TR process (red dashed curve). It is found that the TR process contributes about 12.85% to the total rate coefficients and cannot be neglected.

For the convenient use of the calculated results in plasma modeling, we fitted the total rate coefficients with the following analytical sum expression [36]:

$$\alpha \left(T_e\right)=T_e{}^{-3/2}{\displaystyle \sum _i{c}_i}\times \exp \left(-\frac{E_i}{k{T}_e}\right)$$

(9)

where k is the Boltzmann constant. The fitting parameters of c_i (in cm³ s⁻¹ eV^3/2) and E_i (in eV) are listed in

Table 3. Fitted parameters c_i (in cm³ s⁻¹ eV^3/2) and E_i (in eV) in Equation (9) for the DR rate coefficients of Mg-like Au⁶⁷⁺ ions. (a (b) denotes a × 10^b.)

	ci (cm3 s−1 eV3/2)	Ei (eV)
1	−2.16 (-6)	8383.89
2	1.23 (-6)	11,563.12
3	7.80 (-7)	3224.54
4	2.56 (-6)	5934.46
5	6.62 (-6)	8729.08
6	1.09 (-6)	3260.14
7	1.94 (-7)	2362.51

. Using these parameters in expression (9), it is possible to reproduce the total electron–ion resonant recombination rate coefficients with a good accuracy.

4. Conclusions

In summary, the ∆n = 1 resonant recombination process associated with 2s/2p-core excitation of highly charged Au⁶⁷⁺ ions have been studied theoretically. The resonance energies, resonance strengths, cross-sections and the corresponding electron–ion recombination rate coefficients were carefully calculated for 2s/2p → 3l transitions using an FAC based on the relativistic configuration interaction method and considering Breit and QED corrections. In the calculations, dominant dielectronic recombination and higher-order trielectronic recombination were considered. It was found that the cross-section of the DR process is greater than the TR process and that the contribution of the TR process to the total cross-section is about 13.75%, which cannot be neglected. Furthermore, the plasma rate coefficients are deduced from the calculated recombination spectra, and the contributions from the higher-order excited resonance groups n ≥ 13, which were about 2.93% to total rate coefficient, were evaluated using an extrapolation method. It was also found that the TR processes are important, which contribute 12.85% of the total rate coefficients.