Plasma environment effects on K lines of astrophysical interest I. Atomic structure, radiative rates and Auger widths of oxygen ions

TL;DR

This study investigates how plasma environments influence the atomic structure and radiative properties of oxygen ions relevant to astrophysical phenomena, using advanced relativistic calculations to quantify changes in ionization and emission characteristics.

Contribution

It introduces a novel application of the Debye-Huckel potential within the relativistic MCDF framework to assess plasma effects on oxygen ion K-vacancy states.

Findings

Modified ionization potentials and K thresholds in plasma environments.

Altered Auger widths and radiative transition wavelengths and rates.

Quantitative data across a range of plasma temperatures and densities.

Abstract

Aims. In the context of black-hole accretion disks, the main goal of the present study is to estimate the plasma environment effects on the atomic structure and radiative parameters associated with the K-vacancy states in ions of the oxygen isonuclear sequence. Methods. We use a time-averaged Debye-Huckel potential for both the electron-nucleus and the electron-electron interactions implemented in the fully relativistic multiconfiguration Dirac-Fock (MCDF) method. Results. Modified ionization potentials, K thresholds, Auger widths and radiative transition wavelengths and rates are reported for O I - O VII in plasma environments with electron temperature and density ranges 10E5 - 10E7 K and 10E18 - 10E22 cm-3 .

Tables

Table 1. Table 1: Reference configurations used to build up the MCDF active space for O i – O vii .

Ion	Reference configurations
O i	$2p^4$, $\left[2s\right]2p^5$, ${\left[2s\right]}^{2}2p^6$, $\left[1s\right]2p^5$, $\left[1s\right]\left[2s\right]2p^6$
O ii	$2p^3$, $\left[2s\right]2p^4$, ${\left[2s\right]}^{2}2p^5$, $\left[1s\right]2p^4$, $\left[1s\right]\left[2s\right]2p^5$, $\left[1s\right]{\left[2s\right]}^{2}2p^6$
O iii	$2p^2$, $\left[2s\right]2p^3$, ${\left[2s\right]}^{2}2p^4$, $\left[1s\right]2p^3$, $\left[1s\right]\left[2s\right]2p^4$, $\left[1s\right]{\left[2s\right]}^{2}2p^5$
O iv	$2p$, $\left[2s\right]2p^2$, ${\left[2s\right]}^{2}2p^3$, $\left[1s\right]2p^2$, $\left[1s\right]\left[2s\right]2p^3$, $\left[1s\right]{\left[2s\right]}^{2}2p^4$
O v	$2s^2$, $2s2p$, $2p^2$, $\left[1s\right]2s^{2}2p$, $\left[1s\right]2s2p^2$, $\left[1s\right]2p^3$
O vi	$2s$, $2p$, $\left[1s\right]2s2p$, $\left[1s\right]2p^2$
O vii	$1s^2$, $1s2s$, $1s2p$

Table 2. Table 2: Plasma screening parameter μ 𝜇 \mu (a.u.) for different electron temperatures T e subscript 𝑇 𝑒 T_{e} and densities n e subscript 𝑛 𝑒 n_{e} .

	${10}^{18}$	${10}^{19}$	${10}^{20}$	${10}^{21}$	${10}^{22}$
${10}^5$	$\mu =0.0024$	$\mu =0.0077$	$\mu =0.024$	$\mu =0.077$	$\mu =0.24$
${10}^6$	$\mu =0.00077$	$\mu =0.0024$	$\mu =0.0077$	$\mu =0.024$	$\mu =0.077$
${10}^7$	$\mu =0.00024$	$\mu =0.00077$	$\mu =0.0024$	$\mu =0.0077$	$\mu =0.024$

Table 3. Table 3: Plasma screening effects on the IP E 0 ​ ( μ ) subscript 𝐸 0 𝜇 E_{0}(\mu) in oxygen ions determined with the MCDF method.

Ion	NISTa	MCDF
	$E_0$	$E_0\left(0\right)$	$E_0\left(0.1\right)$	$E_0\left({0.1}^b\right)$	$E_0\left(0.25\right)$
O i	13.61804(7)	13.07	10.33	$-$6.85	6.42
O ii	35.12111(6)	35.00	29.68	14.65	22.30
O iii	54.93554(12)	54.80	46.90	34.26	36.12
O iv	77.41350(25)	77.31	66.82	56.60	52.52
O v	113.8989(5)	112.81	99.75	92.04	82.14
O vi	138.1189(21)	138.04	122.41	117.21	101.37
O vii	739.32679(6)	739.86	720.99	718.32	693.36

Table 4. Table 4: Plasma screening effects on the K threshold E K ​ ( μ ) subscript 𝐸 𝐾 𝜇 E_{K}(\mu) in oxygen ions computed with MCDF.

Ion	$E_K\left(0\right)$	$E_K\left(0.1\right)$	$E_K\left(0.25\right)$
O i	543.58	540.62	535.47
O ii	570.89	565.31	556.49
O iii	593.27	585.08	572.74
O iv	626.49	615.64	599.52
O v	664.10	650.57	630.58
O vi	699.64	683.44	659.64
O vii	739.86	720.99	693.36

Table 5. Table 5: Plasma environment effects on the wavelengths and transition probabilities of K α 𝛼 \alpha lines in oxygen ions computed with MCDF.

Ion	Transition	$\lambda$ (Å)				$A(j,i)$ (s-1)
		$\mu =0.0$	$\mu =0.1$	$\mu =0.25$		$\mu =0.0$	$\mu =0.1$	$\mu =0.25$
O i	$[$1s$]$2p5 1P1 – 2p4 1D2	23.3180	23.3252	23.3684		2.918E+12	2.885E+12	2.732E+12
	$[$1s$]$2p5 3P1 – 2p4 3P2	23.3752	23.3832	23.4301		6.648E+11	6.529E+11	5.993E+11
	$[$1s$]$2p5 3P0 – 2p4 3P1	23.3753	23.3833	23.4302		1.597E+12	1.568E+12	1.440E+12
	$[$1s$]$2p5 3P1 – 2p4 3P1	23.3759	23.3839	23.4308		3.992E+11	3.921E+11	3.601E+11
	$[$1s$]$2p5 3P1 – 2p4 3P0	23.3763	23.3843	23.4311		5.326E+11	5.231E+11	4.806E+11
	$[$1s$]$2p5 3P2 – 2p4 3P2	23.3763	23.3843	23.4312		1.197E+12	1.176E+12	1.080E+12
	$[$1s$]$2p5 3P2 – 2p4 3P1	23.3770	23.3851	23.4319		3.994E+11	3.924E+11	3.605E+11
	$[$1s$]$2p5 1P1 – 2p4 1S0	23.3940	23.4001	23.4382		5.923E+11	5.884E+11	5.704E+11
O ii	$[$1s$]$2p4 2P1/2 – 2p3 2D3/2	23.1144	23.1241	23.1788		2.534E+12	2.513E+12	2.407E+12
	$[$1s$]$2p4 2P3/2 – 2p3 2D3/2	23.1159	23.1256	23.1802		2.541E+11	2.521E+11	2.426E+11
	$[$1s$]$2p4 2P3/2 – 2p3 2D5/2	23.1159	23.1256	23.1803		2.342E+12	2.321E+12	2.224E+12
	$[$1s$]$2p4 2S1/2 – 2p3 2P1/2	23.1204	23.1303	23.1858		6.302E+11	6.231E+11	5.890E+11
	$[$1s$]$2p4 2S1/2 – 2p3 2P3/2	23.1204	23.1303	23.1858		1.372E+12	1.358E+12	1.289E+12
	$[$1s$]$2p4 2D5/2 – 2p3 2D3/2	23.1636	23.1734	23.2286		9.126E+10	9.033E+10	8.579E+10
	$[$1s$]$2p4 2D5/2 – 2p3 2D5/2	23.1636	23.1734	23.2287		1.389E+12	1.375E+12	1.307E+12
	$[$1s$]$2p4 2D3/2 – 2p3 2D3/2	23.1638	23.1736	23.2288		1.376E+12	1.362E+12	1.293E+12
	$[$1s$]$2p4 2D3/2 – 2p3 2D5/2	23.1638	23.1736	23.2289		1.259E+11	1.247E+11	1.191E+11
	$[$1s$]$2p4 2P1/2 – 2p3 2P1/2	23.1759	23.1852	23.2376		1.054E+12	1.047E+12	1.013E+12
	$[$1s$]$2p4 2P1/2 – 2p3 2P3/2	23.1759	23.1852	23.2376		5.130E+11	5.093E+11	4.910E+11
	$[$1s$]$2p4 2P3/2 – 2p3 2P1/2	23.1774	23.1867	23.2390		2.676E+11	2.657E+11	2.563E+11
	$[$1s$]$2p4 2P3/2 – 2p3 2P3/2	23.1774	23.1867	23.2391		1.237E+12	1.229E+12	1.188E+12
	$[$1s$]$2p4 4P1/2 – 2p3 4S3/2	23.2124	23.2227	23.2807		9.524E+11	9.400E+11	8.801E+11
	$[$1s$]$2p4 4P3/2 – 2p3 4S3/2	23.2131	23.2234	23.2814		9.523E+11	9.399E+11	8.801E+11
	$[$1s$]$2p4 4P5/2 – 2p3 4S3/2	23.2143	23.2246	23.2825		9.521E+11	9.397E+11	8.800E+11
	$[$1s$]$2p4 2D5/2 – 2p3 2P3/2	23.2254	23.2348	23.2877		5.013E+11	4.973E+11	4.781E+11
	$[$1s$]$2p4 2D3/2 – 2p3 2P1/2	23.2256	23.2350	23.2879		3.980E+11	3.949E+11	3.800E+11
	$[$1s$]$2p4 2D3/2 – 2p3 2P3/2	23.2256	23.2350	23.2879		8.319E+10	8.247E+10	7.888E+10
O iii	$[$1s$]$2p3 1P1 – 2p2 1D2	22.8154	22.8267	22.8887		1.982E+12	1.967E+12	1.890E+12
	$[$1s$]$2p3 3S1 – 2p2 3P0	22.8748	22.8862	22.9485		4.582E+11	4.554E+11	4.428E+11
	$[$1s$]$2p3 3S1 – 2p2 3P1	22.8754	22.8868	22.9491		1.458E+12	1.449E+12	1.405E+12
	$[$1s$]$2p3 3S1 – 2p2 3P2	22.8766	22.8880	22.9503		2.736E+12	2.717E+12	2.621E+12
	$[$1s$]$2p3 3P1 – 2p2 3P0	22.8897	22.9011	22.9637		4.266E+11	4.225E+11	4.011E+11
	$[$1s$]$2p3 3P2 – 2p2 3P1	22.8902	22.9016	22.9642		2.660E+11	2.636E+11	2.515E+11
	$[$1s$]$2p3 3P0 – 2p2 3P1	22.8903	22.9017	22.9642		1.143E+12	1.132E+12	1.079E+12
	$[$1s$]$2p3 3P1 – 2p2 3P1	22.8903	22.9017	22.9643		3.956E+11	3.918E+11	3.702E+11
	$[$1s$]$2p3 1D2 – 2p2 1D2	22.8908	22.9021	22.9636		3.479E+12	3.455E+12	3.340E+12
	$[$1s$]$2p3 3P2 – 2p2 3P2	22.8914	22.9028	22.9653		8.768E+11	8.684E+11	8.277E+11
	$[$1s$]$2p3 3P1 – 2p2 3P2	22.8915	22.9029	22.9654		3.373E+11	3.342E+11	3.225E+11
	$[$1s$]$2p3 1P1 – 2p2 1S0	22.9227	22.9338	22.9943		1.535E+12	1.525E+12	1.480E+12
	$[$1s$]$2p3 3D1 – 2p2 3P0	22.9663	22.9776	23.0395		6.370E+11	6.319E+11	6.066E+11
	$[$1s$]$2p3 3D2 – 2p2 3P1	22.9669	22.9782	23.0400		8.592E+11	8.522E+11	8.181E+11
	$[$1s$]$2p3 3D1 – 2p2 3P1	22.9669	22.9782	23.0401		4.554E+11	4.518E+11	4.341E+11
	$[$1s$]$2p3 3D3 – 2p2 3P2	22.9680	22.9792	23.0410		1.119E+12	1.110E+12	1.067E+12
	$[$1s$]$2p3 3D2 – 2p2 3P2	22.9681	22.9794	23.0412		2.607E+11	2.587E+11	2.488E+11
	$[$1s$]$2p3 3D1 – 2p2 3P2	22.9682	22.9794	23.0413		2.776E+10	2.755E+10	2.651E+10
O iv	$[$1s$]$2p2 2S1/2 – 2p 2P1/2	22.5066	22.5192	22.5869		3.799E+11	3.772E+11	3.636E+11
	$[$1s$]$2p2 2S1/2 – 2p 2P3/2	22.5087	22.5214	22.5890		9.693E+11	9.607E+11	9.194E+11
	$[$1s$]$2p2 2P3/2 – 2p 2P1/2	22.5567	22.5695	22.6379		6.296E+11	6.259E+11	6.072E+11
	$[$1s$]$2p2 2P1/2 – 2p 2P1/2	22.5587	22.5716	22.6399		2.703E+12	2.686E+12	2.602E+12
	$[$1s$]$2p2 2P3/2 – 2p 2P3/2	22.5588	22.5717	22.6400		3.323E+12	3.303E+12	3.203E+12
	$[$1s$]$2p2 2P1/2 – 2p 2P3/2	22.5609	22.5737	22.6420		1.246E+12	1.239E+12	1.204E+12
	$[$1s$]$2p2 2D3/2 – 2p 2P1/2	22.6310	22.6437	22.7113		1.102E+12	1.094E+12	1.058E+12
	$[$1s$]$2p2 2D5/2 – 2p 2P3/2	22.6330	22.6457	22.7132		1.286E+12	1.277E+12	1.235E+12
	$[$1s$]$2p2 2D3/2 – 2p 2P3/2	22.6331	22.6458	22.7134		1.862E+11	1.850E+11	1.791E+11
O v	$[$1s$]$2p 1P1 – 2s2 1S0	22.2088	22.2229	22.2960		2.884E+12	2.868E+12	2.786E+12
O vi	$[$1s$]$2s2p 2P3/2 – 2s 2S1/2	21.7832	21.7998	21.8840		6.424E+11	6.368E+11	6.109E+11
	$[$1s$]$2s2p 2P1/2 – 2s 2S1/2	21.7836	21.8002	21.8844		6.792E+11	6.735E+11	6.467E+11
	$[$1s$]$2s2p 2P3/2 – 2s 2S1/2	21.9706	21.9860	22.0646		2.694E+12	2.680E+12	2.609E+12
	$[$1s$]$2s2p 2P1/2 – 2s 2S1/2	21.9730	21.9884	22.0669		2.657E+12	2.643E+12	2.573E+12
O vii	1s2p 1P1 – 1s2 1S0	21.5642	21.5821	21.6707		3.702E+12	3.680E+12	3.574E+12

Table 6. Table 6: Plasma environment effects on the energy and Auger widths of K-vacancy states in oxygen ions computed with MCDF.

Ion	Level	$E$ (eV)				$A_a\left(j\right)$ (s-1)
		$\mu =0.0$	$\mu =0.1$	$\mu =0.25$		$\mu =0.0$	$\mu =0.1$	$\mu =0.25$
O i	$[$1s$]$2p5 3P2	530.39	530.20	529.14		2.606E+14	2.528E+14	2.104E+14
	$[$1s$]$2p5 3P1	530.41	530.23	529.17		2.613E+14	2.536E+14	2.124E+14
	$[$1s$]$2p5 3P0	530.43	530.24	529.18		2.587E+14	2.513E+14	2.110E+14
	$[$1s$]$2p5 1P1	534.13	533.94	532.87		2.086E+14	2.037E+14	1.832E+14
O ii	$[$1s$]$2p4 4P5/2	534.08	533.85	532.52		2.489E+14	2.405E+14	2.202E+14
	$[$1s$]$2p4 4P3/2	534.11	533.87	532.54		2.474E+14	2.392E+14	2.191E+14
	$[$1s$]$2p4 4P1/2	534.12	533.89	532.55		2.475E+14	2.394E+14	2.194E+14
	$[$1s$]$2p4 2D3/2	539.22	538.96	537.51		3.004E+14	2.928E+14	2.680E+14
	$[$1s$]$2p4 2D5/2	539.22	538.96	537.51		3.011E+14	2.935E+14	2.686E+14
	$[$1s$]$2p4 2P3/2	540.33	540.07	538.63		2.050E+14	1.975E+14	1.804E+14
	$[$1s$]$2p4 2P1/2	540.37	540.11	538.66		2.041E+14	1.969E+14	1.798E+14
	$[$1s$]$2p4 2S1/2	541.65	541.38	539.85		3.014E+14	2.924E+14	2.578E+14
O iii	$[$1s$]$2p3 3D1	539.86	539.59	538.14		2.969E+14	2.903E+14	2.719E+14
	$[$1s$]$2p3 3D2	539.86	539.59	538.14		2.957E+14	2.891E+14	2.709E+14
	$[$1s$]$2p3 3D3	539.86	539.59	538.15		2.974E+14	2.908E+14	2.724E+14
	$[$1s$]$2p3 3P1	541.66	541.39	539.92		2.746E+14	2.682E+14	2.500E+14
	$[$1s$]$2p3 3P0	541.66	541.39	539.92		2.733E+14	2.669E+14	2.488E+14
	$[$1s$]$2p3 3P2	541.66	541.39	539.92		2.746E+14	2.681E+14	2.499E+14
	$[$1s$]$2p3 3S1	542.02	541.74	540.27		1.050E+14	1.031E+14	9.827E+13
	$[$1s$]$2p3 1D2	544.59	544.30	542.76		2.562E+14	2.505E+14	2.350E+14
	$[$1s$]$2p3 1P1	546.38	546.09	544.52		2.389E+14	2.333E+14	2.177E+14
O iv	$[$1s$]$2p2 2D3/2	547.85	547.54	545.91		2.617E+14	2.552E+14	2.392E+14
	$[$1s$]$2p2 2D5/2	547.86	547.55	545.92		2.622E+14	2.556E+14	2.395E+14
	$[$1s$]$2p2 2P1/2	549.61	549.29	547.63		1.054E+14	1.033E+14	9.766E+13
	$[$1s$]$2p2 2P3/2	549.66	549.34	547.68		1.054E+14	1.032E+14	9.762E+13
	$[$1s$]$2p2 2S1/2	550.88	550.57	548.92		2.239E+14	2.179E+14	2.029E+14
O v	$[$1s$]$2p 1P1	558.26	557.92	556.08		1.196E+14	1.168E+14	1.095E+14
O vi	$[$1s$]$2s2p 2P1/2	564.25	563.87	561.86		3.902E+13	3.677E+13	3.360E+13
	$[$1s$]$2s2p 2P3/2	564.32	563.93	561.92		3.651E+13	3.434E+13	3.128E+13
	$[$1s$]$2s2p 2P1/2	569.16	568.73	566.55		9.593E+13	9.544E+13	9.314E+13
	$[$1s$]$2s2p 2P3/2	569.17	568.74	566.56		9.701E+13	9.649E+13	9.414E+13

Equations

$$\Psi (\gamma ,P,J,M)=\sum _i{c}_i\text{leavevmode}\Phi ({\alpha }_i,P,J,M),$$

$${\phi }_{n\kappa m}(r,\theta ,\varphi )=\frac{1}{r}\left(\begin{array}{c}{P}_{n\kappa }(r)\text{leavevmode}{\chi }_{\kappa m}(\theta ,\varphi )\\ i\text{leavevmode}{Q}_{n\kappa }(r)\text{leavevmode}{\chi }_{-\kappa m}(\theta ,\varphi )\end{array}\right).$$

$$H_{DC}=\sum _{i}c\text{@vec}{\alpha }_i\cdot \text{@vec}{p}_i+{\beta }_i{c}^2-\frac{Z}{r_i}+\sum _{i>j}\frac{1}{r_{ij}}.$$

$$H_{DC}^{DH}=\sum _{i}c\text{@vec}{\alpha }_i\cdot \text{@vec}{p}_i+{\beta }_i{c}^2-\frac{Z}{r_i}{e}^{-\mu r_i}+\sum _{i>j}\frac{1}{r_{ij}}{e}^{-\mu r_{ij}},$$

$$\mu =\frac{1}{{\lambda }_D}=\sqrt{\frac{4\pi n_e}{k{T}_e}}.$$

$$\Gamma =\frac{e^2}{4\pi {ϵ}_{0}dk{T}_e}$$

$$d={\left(\frac{3}{4\pi n_e}\right)}^{1/3},$$

$$\frac{1}{r_{ij}}{e}^{-\mu r_{ij}}=-\mu \sum _{l=0}^{\infty }\text{leavevmode}(2l+1)\text{leavevmode}{j}_l(i\mu r_{<})\text{leavevmode}{h}_l^1(i\mu r_{>})\text{leavevmode}{P}_l(cos\text{leavevmode}{\omega }_{ij}),$$

Full text

*Methods. *We use a time-averaged Debye–Hückel potential for both the electron–nucleus and the electron–electron interactions implemented in the fully relativistic multiconfiguration Dirac–Fock (MCDF) method.

Results. Modified ionization potentials, K thresholds, Auger widths and radiative transition wavelengths and rates are reported for O i – O vii in plasma environments with electron temperature and density ranges $10^{5}{-}10^{7}$ K and $10^{18}{-}10^{22}$ cm-3.

1 Introduction

High-density plasma effects (free–free heating at electron densities $n_{e}>10^{19}$ cm*-3*) may explain the apparent supersolar Fe abundances inferred from the X-ray spectra of accreting black holes (García et al., 2018). However, the currently available atomic data to model astronomical spectra do not take into account high-density effects, and are therefore limited to densities below $10^{18}$ cm*-3* (García et al., 2016). In this respect Schnittman et al. (2013) recently carried out MHD simulations of a 10 M*⊙* black hole accreting at a 10% rate, and predicted plasma conditions in the accretion disk characterized by electron temperatures and densities spanning the ranges $10^{5}{-}10^{7}$ K and $10^{18}{-}10^{22}$ cm*-3*. Fields et al. (2007) have determined supersolar abundances for carbon, nitrogen, oxygen and iron from a high-resolution Chandra spectrum of the warm absorbing gas outflowing from the AGN Mrk 279 compact region, where especially the oxygen abundance is estimated at ${\sim}8$ times solar. This latter environment type is also expected to have densities greater than $10^{19}$ cm*-3*.

García et al. (2005) reported a complete set of atomic data relevant to the modeling of oxygen K lines formed in astrophysical photoionized plasmas. As underlined by Smith & Brickhouse (2014) in the interpretation of X-ray spectra taken by current space observatories, none of these atomic parameters takes into account high-density plasma embedding, where the atomic structure and processes (e.g. emissivities, opacities and ionization balance) could be significantly distorted by the extreme temperature and density.

In the present study we provide a complete set of structure and radiative data for the modeling of oxygen K lines that include plasma environment effects. In Section 2, we describe the atomic physics computational approach, and in Section 3 we validate the time-averaged potential used to model the plasma environment by means of three test cases. In Section 4 we discuss in detail our results, and finally our conclusions are drawn in Section 5.

2 Theoretical approach

In the MCDF method (Grant et al., 1980; McKenzie et al., 1980; Grant, 1988) the atomic state function (ASF) $\Psi$ is represented by a linear combination of configuration state functions (CSF) $\Phi$ of the same parity ($P$), total angular momentum ($J$) and angular-momentum projection ($M$)

[TABLE]

where the CSFs are antisymmetrized products of orthonormal monoelectronic spin-orbitals of the form

[TABLE]

In Eq. (2) $P_{n\kappa}(r)$ and $Q_{n\kappa}(r)$ are the large and small radial orbitals, respectively, and the angular functions $\chi_{\kappa m}(\theta,\phi)$ are spinor spherical harmonics. These spin-orbitals are optimized self-consistently based on the Dirac–Coulomb Hamiltonian

[TABLE]

For an atom embedded in a weakly coupled plasma, the Hamiltonian of Eq. (3) is replaced with the Debye-Hückel (DH) screened Dirac–Coulomb Hamiltonian (Saha & Fritzsche, 2006)

[TABLE]

where $r_{ij}=|\@vec{r}_{i}-\@vec{r}_{j}|$ and the plasma screening parameter $\mu$ is the inverse of the Debye shielding length $\lambda_{D}$, which can be expressed in atomic units (a.u.) as a function of the plasma electron density $n_{e}$ and temperature $T_{e}$ as

[TABLE]

Typical plasma conditions in black-hole accretion disks are $T_{e}\sim 10^{5}{-}10^{7}$ K and $n_{e}\sim 10^{18}{-}10^{22}$ cm*-3* (Schnittman et al., 2013). For weakly coupled plasmas they correspond to screening parameters $0.0\leq\mu\leq 0.24$ a.u. and, for a completely ionized hydrogen plasma (with plasma ionization $Z^{*}=1$), to plasma coupling parameters

[TABLE]

with

[TABLE]

in the range $0.0003\leq\Gamma\leq 0.6$.

The last term of Eq. (4) has the angular dependence (Saha & Fritzsche, 2006)

[TABLE]

where $r_{>}=max(r_{i},r_{j})$, $r_{<}=min(r_{i},r_{j})$, $j_{l}$ is a Bessel function, $h^{1}_{l}$ denotes a Hankel function of the first kind and $P_{l}$ is a Legendre polynomial that depends on the angle $\omega_{ij}$ between the two position vectors $\@vec{r}_{i}$ and $\@vec{r}_{j}$. This screening reduces the electron–electron repulsion and, hence, increases the binding of the electron by the nucleus.

We use the active space (AS) method to obtain the MCDF expansions of Eq. (1) for O i – O vii, whereby electrons from reference configurations are excited to a given active set of orbitals. For these oxygen ions, the AS was built up by considering all the single and double excitations of the reference configurations listed for each species in Table 1 to configurations including $n=2$ and $n=3$ orbitals.

In the isolated atom approximation, the relativistic orbitals $P_{n\kappa}(r)$ and $Q_{n\kappa}(r)$, along with the expansion coefficients $c_{i}$ in Eqs. (1)–(2), were optimized using GRASP2K (Parpia et al., 1996) with the extended average level (EAL) option, where the $(2J+1)$-weighted trace of the Dirac–Coulomb Hamiltonian (see Eq. 3) is minimized to determine energy levels, wavelengths and radiative and Auger rates. To take into account core-relaxation effects on the K-vacancy states (García et al., 2005), we introduce non-orthogonal orbitals optimized separately in two distinct level groups: a first group of exclusively valence levels where the K shell is full; and a second group of levels with at least a single K-vacancy. For the ionization potential (IP) and K threshold, the orbitals were respectively optimized on the ground level and on the lowest K-vacancy level of each ion using the optimal level (OL) option of GRASP2K.

Plasma effects are included perturbatively in a second step where we use the RATIP code of Fritzsche (2012) to improve the expansion coefficients, i.e. the ASF, energy levels and radiative rates by solving the secular equation with the Debye-Hückel screened Dirac–Coulomb Hamiltonian (Eq. 4), the two-body Breit interaction and the quantum electrodynamic corrections (self-energy and vacuum polarization). Plasma screening parameters in the range $0.00\leq\mu\leq 0.25$ a.u. were adopted, the upper-limit choice, justified in Section 3.3, corresponding to the extreme plasma conditions found in accretion disks.

3 Validation of the Debye–Hückel model potential

Stark shifts of the dipole-allowed spectral lines emitted by an ion in a dense plasma are measured. In a semi-classical picture, neighboring electrons and ions give rise to effective microscopic electric fields that result in level energy shifts due to induced dipole moments in the emitting ion. Alternatively, Stark shifts can also be predicted quantum mechanically with a Debye–Hückel potential (Rozsnyai, 1975). To test the validity of our Debye–Hückel model potential, we have performed a series of calculations with the GRASP2K/RATIP code to compare with available laboratory Stark shifts. Such comparisons are of common pratice, see e.g. Rozsnyai (1975); Neiger & Griem (1976) and more recently Belkhiri et al. (2015). The three test cases investigated are detailed in the following Sections 3.1–3.3.

3.1 Stark shifts in valence transitions of O ii

Djenize et al. (1998) measured Stark shifts for the valence-shell transitions ${\rm 2p^{2}3s\leavevmode\nobreak\ ^{4}P_{3/2}}$ – ${\rm 2p^{2}3p\leavevmode\nobreak\ ^{4}D^{o}_{5/2}}$ (${\lambda}4641.81$) and ${\rm 2p^{2}3s\leavevmode\nobreak\ ^{4}P_{1/2}}$ – ${\rm 2p^{2}3p\leavevmode\nobreak\ ^{4}D^{o}_{3/2}}$ (${\lambda}4638.85$) in O ii, respectively $0.03\pm 0.02$ Å and $0.05\pm 0.02$ Å, at a temperature $T_{e}=54\,000$ K and density $n_{e}=2.8\times 10^{17}$ cm*-3*. To reproduce these shifts theoretically, we considered intravalence and core–valence correlations up to $n=5$ to represent the respective ASFs in the two transitions. The MCDF shifts obtained with $\mu=0.0017$ a.u. are both 0.05 Å in good agreement with experiment. Moreover, if the DH screening of the electron–electron Coulomb potential is switched off (i.e. $\mu=0$ in the last term of Eq. 4), these shifts become much larger: 0.26 Å and 0.25 Å, respectively. This results confirm that the DH electron–electron plasma screening cannot be neglected. It should be also emphasied that Djenize et al. (1998) compared their measurements with the semi-classical calculations of Griem (1974) and Dimitrijević (1982) (see multiplet No. 1 in their Figure 4). The three sets of Stark shifts (Djenize et al., 1998; Griem, 1974; Dimitrijević, 1982) disagree to each other, with the theoretical values having opposite signs and being about $\sim$ 0.15 Å (Griem, 1974) and $\sim-$0.05 Å (Dimitrijević, 1982). In that respect, our time-independent quantum model confirms the measurements of Djenize et al. (1998).

3.2 Stark shifts in valence transitions of Na i

The Stark shifts of the valence-shell Na i D doublet at 5889.95/5895.90 Å have been measured in a plasma with $T_{e}=38\,000$ K and $n_{e}=3.5\times 10^{17}$ cm*-3* (Sreckovic et al., 1996): $0.38\pm 0.09$ Å and $0.41\pm 0.09$ Å, respectively. Our MCDF calculations with $\mu=0.0023$ a.u. including both intravalence and core–valence correlations ($n\leq 4$) give 0.43 Å and 0.51 Å, respectively, in satisfactory agreement with experiment. Again, if the electron–electron screening is neglected, the MCDF shifts are way off at 4.42 Å and 4.50 Å.

3.3 Ti K$\alpha$ line pressure shift

Our last test case concerns the K$\alpha$ line ${\rm 1s^{2}\leavevmode\nobreak\ {}^{1}S_{0}-1s2p\leavevmode\nobreak\ ^{1}P^{o}_{1}}$ of He-like Ti xxi at 4749.73 eV. Khattak et al. (2012) measured a line shift of $3.4\pm 1.0$ eV, and inferred an electron temperature greater than ${\sim}3$ keV and an electron density exceeding $10^{24}$ cm*-3* from a hydrodynamic simulation of their laser-produced plasma. This experimental shift was then reproduced theoretically using an ion-sphere model by Belkhiri et al. (2015) leading to an estimate of 3.4 eV at $T_{e}=3$ keV and $n_{e}=4.2\times 10^{24}$ cm*-3*. In the same plasma conditions adopted by Belkhiri et al. (2015), namely $\mu=0.27$ a.u., our MCDF-DH model yields a line shift of 3.3  eV in good agreement with both the measurement and the ion-sphere calculation. We are therefore confident to consider plasma parameters up to $\mu=0.27$ a.u.

4 Results and discussion

In the following Sections 4.1–4.3 we examine the DH plasma screening effects on the oxygen K-line characteristics by considering screening parameters in the range $0\leq\mu\leq 0.25$ a.u., which, as shown in Table 2, can be associated to electron temperatures and densities in the ranges $10^{5}\leq T_{e}\leq 10^{7}$ K and $10^{18}\leq n_{e}\leq 10^{22}$ cm*-3*.

4.1 Ionization potentials and K thresholds

The IPs ($E_{0}$) and K thresholds ($E_{K}$) for O ions determined with MCDF for three values of the plasma screening parameter—$\mu=0$, 0.1 and 0.25 a.u—are listed in Tables 3–4. For isolated species ($\mu=0$), the NIST IPs (Kramida et al., 2016) are reproduced to within 1% except for O i, for which only a 4% accuracy was attained due to the well-known slow convergence of the CI expansion (Eq. 1) for neutrals. As expected, a substantial reduction of the absolute values of $E_{0}$ and $E_{K}$ increasing with $\mu$ is obtained. For the case of $\mu=0.1$ it is shown in Table 3 that, if the electron–electron DH screening is neglected ($\mu=0$ in the last term of Eq. 4), the continuum lowering, particularly for the lowly ionized species, is much larger yielding an unphysical negative IP for O i. We therefore stress again the DH electron–electron screening must-have.

In Fig. 1 we plot the trends of the IP lowering $\Delta E_{0}(\mu)=E_{0}(\mu)-E_{0}(\mu=0)$ with $Z_{\rm eff}=Z-N+1$, which are found to be practically linear except for $Z_{\rm eff}=7$ (O vii) due to the absence of the DH electron–electron screening. We also include for each species the DH limit $\Delta E^{\rm DH}_{0}\equiv-Z_{\rm eff}\,\mu$ as $\Gamma\rightarrow 0$ determined by Stewart & Pyatt (1966) and Crowley (2014). For O vii, $\Delta E^{\rm DH}_{0}=-47.6$ eV close to the MCDF IP lowering for $\mu=0.25$.

K-threshold lowering ($\Delta E_{K}(\mu)=E_{K}(\mu)-E_{K}(\mu=0)$) trends with $Z_{\rm eff}$ are very similar to those of $\Delta E_{0}$ as shown in Fig. 2, a predominantly linear decrease up to $\Delta E_{K}\approx\Delta E_{0}\approx-50$ eV. This finding is significant inasmuch as the DH screened photoionization cross sections will only involve approximately constant downward energy shifts of the thresholds leading series truncations rather than variant line wavelengths and resonance energy positions, points that are further discussed in Sections 4.2–4.3.

4.2 Radiative data

Oxygen K-line wavelengths and transition probabilities ($A$-values) computed with MCDF with $\mu=0$, 0.1 and 0.25 a.u. are reported in Table 5. For isolated systems ($\mu=0$) our radiative data are in good general accord with the HFR and MCBP results of García et al. (2005); more precisely, present K-line wavelengths are shorter by 0.1% for the highly charged ions to little less than 1% for the lower ionization stages. In addition, they are in excellent agreement with the few spectroscopic reports available; for example, for the strong K$\alpha$ line in O vii, our predicted wavelength agrees within 0.2% with the measurement by Engström & Litzén (1995), and within 0.7% and 0.2% with those reported by Schmidt et al. (2004) for O v and O vi, respectively. Regarding radiative rates, our MCDF results agree with García et al. (2005) on average to within 10% except for a few weak transitions.

Plasma effects on the radiative parameters are found to be small for $\mu=0.1$ a.u. but more conspicuous for $\mu=0.25$ a.u (see Fig. 3). In fact for $\mu=0.25$ a.u. K-line wavelengths appear to be shifted by ${\sim}50{-}100$ mÅ with respect to the isolated atom ($\mu=0$) with a trend increasing with the ion effective charge $Z_{\rm eff}$ as shown in Fig. 3 (upper panel). Although small, such wavelength shifts can be resolved by present and next-generation satellite borne X-ray spectrometers. $A$-values are generally also slightly modified; for instance, they are reduced on average by 5% with $\mu=0.25$ a.u. (see lower panel of Fig. 3), which would make negligible differences in astrophysical modeling.

4.3 K-vacancy level energies and Auger widths

MCDF level energies and Auger widths for oxygen K-vacancy levels are presented for $\mu=0$, 0.1 and 0.25 a.u. in Table 6. Our MCDF energies for $\mu=0$ a.u. are slightly lower ($\sim$0.5% on average) with respect to those computed with the HFR and MCBP by García et al. (2005), and our Auger widths are on average shorter by $\sim$25% and $\sim$20%, respectively. Level-energy decrements are in general found to be small ($|\Delta E|\lesssim 3.0$ eV) as illustrated in Fig. 4 (top panel) for $\mu=0.1$ a.u and $\mu=0.25$ a.u. with trends increasing with $Z_{\rm eff}$. Regarding the Auger widths as shown in Fig. 4 (bottom panel), they are reduced on average by up to $\sim 10\%$ for $\mu=0.25$ a.u., which might have an impact on spectral K-line modeling. Neutral oxygen ($Z_{\rm eff}=1$) has a different situation where MCDF predicts a 20 % decrease for $\mu=0.25$ a.u. This may illustrate the difficulty of our atomic structure model to compute accurate rates (better than 20%) at the neutral end of an isonuclear sequence.

5 Summary and conclusions

We have studied plasma embedding effects on the atomic structure of oxygen ions, namely the K-shell radiative parameters and Auger widths, as a function of the screening parameter. Such plasma effects were modeled perturbatively in the MCDF framework with a time-independent DH potential. Our main findings and conclusions can be summarized as follows.

The validity of our DH model has been benchmarked with Stark line-shift measurements (Sreckovic et al., 1996; Djenize et al., 1998; Khattak et al., 2012) for screening parameters as large as $\mu=0.27$ a.u. The latter value has been associated with the extreme density conditions found in accretion disks around compact objects (see Table 2 and García et al., 2016). To obtain the desirable degree of agreement with experiment, the DH electron–electron screening must be taken into account. 2. 2.

We have studied plasma screening effects for $\mu\leq 0.25$ a.u. finding considerable lowering (up to $\sim 50$ eV) of both the IPs and K thresholds. Such shifts could arguably enhance the ionization fractions and K-vacancy state populations or, at least, lead to erroneous spectral line identifications. 3. 3.

Only a modest impact on the radiative and Auger data has been detected for $\mu\leq 0.1$ a.u., but it becomes more acute under the extreme plasma conditions of $\mu=0.25$ a.u.: K-line wavelengths are systematically redshifted by up to ${\sim}0.1$ Å with potential consequences on out/inflow velocity determinations; and Auger rates might decrease by up to $\sim 20\%$ in O i. 4. 4.

These new atomic data will be incorporated in the atomic the database of the XSTAR modeling code (Bautista & Kallman, 2001) for future spectral analysis of accretion disks around compact objects. 5. 5.

We believe the perturbative approach adopted here for $\mu\leq 0.25$ is well supported by the relatively small effects on and smooth trends found in the atomic parameters, as well as by the good agreement with the experimentally determined Stark shifts for a few ionic species. More extreme conditions may require a non-perturbative inclusion of the DH potential; this work is underway and would be the subject of a subsequent report.

Acknowledgements.

JD is a Research Fellow of the Belgian Fund for Research in Industry and Agriculture FRIA. PP & PQ are, respectively, Research Associate and Research Director of the Belgian Fund for Scientific Research F.R.S.–FNRS. Financial supports from these organizations, as well as from the NASA Astrophysics Research and Analysis Program (grant 80NSSC17K0345) are gratefully acknowledged. We are indented to Professor Nigel R. Badnell (Strathclyde University, UK) for lengthy and useful discussions on the validity the Debye–Hückel potential and its implementations in atomic structure calculations. JAG acknowledges support from the Alexander von Humboldt Foundation.