NLTE Line Formation for Mg i and Mg ii in the Atmospheres of B–A–F–G–K Stars

Energy levels. The model atom includes the Mg i levels belonging to singlet and triplet terms of the 3snl ($n\leqslant 16$, $l\leqslant 8$) and 3p² electronic configurations, the Mg ii levels belonging to doublet terms of the 2p⁶nl ($n\leqslant 10$, $l\leqslant 6$) electronic configurations, and the ground state of Mg iii. Most energy levels were taken from Version 5 of the NIST database³ (Kramida et al. 2018). The Mg i levels with n = 11–16 have small energy differences and their populations must be in LTE relative to one another. These levels were used to make two superlevels. Triplet fine structure was ignored except for the 3s3p³P° splitting, and doublet fine structure was ignored except for the 4p²P° splitting. Energy levels up to 0.02/0.45 eV below the ionization threshold are explicitly included in our Mg i/Mg ii model atom. The term diagrams for Mg i and Mg ii in our model atom are shown in Figures 1 and 2.

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Radiative data. Our model atom includes 673 allowed bound–bound (b−b) transitions. Their transition probabilities were taken from the Opacity Project (OP) database TOPbase⁴ (Luo et al. 1989; Cunto et al. 1993; Hibbert et al. 1993) except for the transitions between levels with n = 7−15 and l = 4−8 in Mg i, for which oscillator strengths were calculated following Green et al. (1957). Photoionization cross-sections for levels of Mg i with $n\leqslant 9$, $l\leqslant 4$ and Mg ii with $n\leqslant 10$, $l\leqslant 4$ were taken from TOPbase, and we adopted the hydrogen-like cross-sections for the higher excitation levels.

Collisional data. For all of the transitions between the lowest 17 energy levels of Mg i (up to 5p ¹P°) and transitions from these levels to the more excited levels up to 3p^{2 1}S—a total of 369 transitions—we adopted accurate electron-impact excitation data from Merle et al. (2015). Sigut & Pradhan (1995) provided accurate electron collision strengths for transitions between the lowest 10 levels of Mg ii (up to 5d ²D). For the remaining transitions, we use the impact parameter method (IPM; Seaton 1962a) in the case of the allowed transitions, and assume that the effective collision strength Ω_ij = 1 in the case of the forbidden transitions.

The electron-impact ionization was considered using the formula of Seaton (1962b) and the threshold photoionization cross-sections. As discussed in the literature, the SE of Mg i in cool atmospheres is affected by inelastic collisions with neutral hydrogen atoms (e.g., Mashonkina 2013). We used detailed quantum mechanical calculations of Barklem et al. (2012) for H i impact excitations and charge transfer processes.

To solve the radiative transfer and SE equations, we used the code DETAIL (Butler & Giddings 1985) based on the accelerated Λ-iteration method (Rybicki & Hummer 1991). The DETAIL opacity package was updated by Przybilla et al. (2011) by including bound–free opacities of neutral and ionized species. The obtained departure coefficients, b_i =n_NLTE/n_LTE, were then used by the code synthV_NLTE (Ryabchikova et al. 2016) to calculate the synthetic NLTE line profiles. Here, n_NLTE and n_LTE are the SE and thermal (Saha–Boltzmann) number densities, respectively. Comparison with the observed spectrum and spectral line fitting were performed with the binmag code⁵ (Kochukhov 2010, 2018).

Calculations were performed using plane-parallel (1D) and chemically homogeneous model atmospheres. For consistency with our NLTE studies of C i–C ii (Alexeeva & Mashonkina 2015; Alexeeva et al. 2016), Ti i–Ti ii (Sitnova et al. 2016), and Ca i–Ca ii (Sitnova et al. 2018), here we use exactly the same model atmosphere for each star, as in the earlier papers. For late-type stars (3900 K < ${T}_{\mathrm{eff}}$ < 7000 K), these are the model atmospheres, which were interpolated in the MARCS model grid (Gustafsson et al. 2008) for a given ${T}_{\mathrm{eff}}$, log g, and [Fe/H] using a FORTRAN-based routine written by Thomas Masseron.⁶ For A–B-type stars, the model atmospheres were calculated with the code LLmodels (Shulyak et al. 2004). An exception is Sirius, for which the model atmosphere was computed by R. Kurucz.⁷

Table 1 lists lines of Mg i and Mg ii used in our abundance analyses together with the adopted line data. For lines of Mg i in the visible spectral range, oscillator strengths were taken from Pehlivan Rhodin et al. (2017) and, for the IR lines, were calculated following Green et al. (1957). The NIST data (Kramida et al. 2018) were mostly employed for lines of Mg ii. It is worth noting that the Mg i line arising from the ground state, 4571 Å, was not used for late-type stars of close-to-solar metallicity due to possible effects by stellar chromospheres (see, for example, Altrock & Cannon 1975; Sasso et al. 2017).

Table 1.  Lines of Mg i and Mg ii Used in Abundance Analysis and Solar Magnesium Abundances, log (Columns 10–12)

λ	Transition	${E}_{\mathrm{exc}}$	log gf	References	log Γ4/Ne	log Γ4/Np	References	log Γ6/NH	References	LTE	NLTE
(Å)		(eV)			(rad s−1 cm3)			(rad s−1 cm3)			M15	IPM
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)
Mg i
3829.36	$3{\rm{p}}\,{}^{3}{{\rm{P}}}_{0}^{^\circ }$–3d 3D1	2.71	−0.214	1	−4.67	−5.06	5	−7.29	7
3832.29	$3{\rm{p}}\,{}^{3}{{\rm{P}}}_{1}^{^\circ }$–3d 3D1	2.71	−0.339	1	−4.67	−5.06	5	−7.29	7
3832.30	$3{\rm{p}}\,{}^{3}{{\rm{P}}}_{1}^{^\circ }$–3d 3D2	2.71	0.138	1	−4.67	−5.06	5	−7.29	7
3838.29	$3{\rm{p}}\,{}^{3}{{\rm{P}}}_{2}^{^\circ }$–3d 3D1	2.71	−1.515	1	−4.67	−5.06	5	−7.29	7
3838.29	$3{\rm{p}}\,{}^{3}{{\rm{P}}}_{2}^{^\circ }$–3d 3D3	2.71	0.409	1	−4.67	−5.06	5	−7.29	7
3838.29	$3{\rm{p}}\,{}^{3}{{\rm{P}}}_{2}^{^\circ }$–3d 3D2	2.71	−0.339	1	−4.67	−5.06	5	−7.29	7
4167.27	$3{\rm{p}}\,{}^{1}{{\rm{P}}}_{1}^{^\circ }$–7d 1D2	4.35	−0.746	1	−3.51	−5.00	5	−6.89	8	7.47	7.48	7.50
4571.09	3s2 1S0–$3{\rm{p}}\,{}^{3}{{\rm{P}}}_{1}^{^\circ }$	0.00	−5.397	1	−6.15		8	−7.77	9	7.45	7.47	7.49
4702.99	$3{\rm{p}}\,{}^{1}{{\rm{P}}}_{1}^{^\circ }$–5d 1D2	4.35	−0.456	1	−4.17	−4.75	5	−6.69	7	7.35	7.35	7.39
4730.03	$3{\rm{p}}\,{}^{1}{{\rm{P}}}_{1}^{^\circ }$–6s 1S0	4.35	−2.379	1	−4.23	−4.91	5	−7.12	8	7.77	7.77	7.72
5167.32	3p 3P0°–4s 3S1	2.71	−0.865	1	−5.43	−6.06	5	−7.27	7	7.51	7.57	7.54
5172.68	$3{\rm{p}}\,{}^{3}{{\rm{P}}}_{1}^{^\circ }$–4s 3S1	2.71	−0.387	1	−5.43	−6.06	5	−7.27	7	7.48	7.53	7.51
5183.60	$3{\rm{p}}\,{}^{3}{{\rm{P}}}_{2}^{^\circ }$–4s 3S1	2.72	−0.166	1	−5.43	−6.06	5	−7.27	7	7.49	7.52	7.53
5528.41	$3{\rm{p}}\,{}^{1}{{\rm{P}}}_{1}^{^\circ }$–4d 1D2	4.35	−0.513	1	−4.63	−5.19	5	−6.98	7	7.44	7.45	7.45
5711.08	$3{\rm{p}}\,{}^{1}{{\rm{P}}}_{1}^{^\circ }$–5s 1S0	4.35	−1.742	1	−4.71	−5.38	5	−7.29	8	7.58	7.59	7.64
8806.76	$3{\rm{p}}\,{}^{1}{{\rm{P}}}_{1}^{^\circ }$–3d1D2	4.35	−0.128	1	−5.46	−5.90	5	−7.41	7	7.53	7.53	7.58
Mean										7.52	7.54	7.54
σ										0.11	0.11	0.09
73736.40	5g 1,3G–6h 1,3H°	7.09	1.780	4	−3.06	−3.08	5	−6.56	9	e	7.92	7.66
122240.00	6g 1,3G–7h 1,3H°	7.27	1.670	4	−2.39	−2.41	5	−6.54	9	e	7.74	7.63
123217.00	6h 1,3H°–7i 1,3I	7.27	1.947	4	−2.55	−2.55	5	−6.58	9	e	7.74	7.63
Mg ii
3848.21	3d 2D5/2–$5{\rm{p}}\,{}^{2}{{\rm{P}}}_{3/2}^{^\circ }$	8.86	−1.580	2	−4.80	−5.48	6	−7.42	8
3848.34	3d 2D3/2–$5{\rm{p}}\,{}^{2}{{\rm{P}}}_{3/2}^{^\circ }$	8.86	−2.534	2	−4.80	−5.48	6	−7.42	8
3850.38	3d 2D3/2–$5{\rm{p}}\,{}^{2}{{\rm{P}}}_{1/2}^{^\circ }$	8.86	−1.842	2	−4.80	−5.48	6	−7.42	8
4384.64	$4{\rm{p}}\,{}^{2}{{\rm{P}}}_{1/2}^{^\circ }$–5d 2D3/2	9.99	−0.790	2	−4.20	−4.86	6	−7.34	8
4390.51	$4{\rm{p}}\,{}^{2}{{\rm{P}}}_{3/2}^{^\circ }$–5d 2D3/2	9.99	−1.478	2	−4.20	−4.86	6	−7.34	8
4390.57	$4{\rm{p}}\,{}^{2}{{\rm{P}}}_{3/2}^{^\circ }$–5d 2D5/2	9.99	−0.530	2	−4.20	−4.86	6	−7.34	8
4427.99	$4{\rm{p}}\,{}^{2}{{\rm{P}}}_{1/2}^{^\circ }$–6s 2S1/2	9.99	−1.208	2	−4.46	−5.38	6	−7.32	8
4433.98	$4{\rm{p}}\,{}^{2}{{\rm{P}}}_{3/2}^{^\circ }$–6s 2S1/2	9.99	−0.910	2	−4.46	−5.38	6	−7.32	8
4481.13	3d 2D5/2–$4{\rm{f}}\,{}^{2}{{\rm{F}}}_{7/2}^{^\circ }$	8.86	0.749	2	−4.70	−5.58	6	−7.56	8
4481.15	3d 2D5/2–$4{\rm{f}}\,{}^{2}{{\rm{F}}}_{5/2}^{^\circ }$	8.86	−0.560	2	−4.70	−5.58	6	−7.56	8
4481.33	3d 2D3/2–$4{\rm{f}}\,{}^{2}{{\rm{F}}}_{5/2}^{^\circ }$	8.86	0.590	2	−4.70	−5.58	6	−7.56	8
4739.59	4d 2D5/2–$8{\rm{f}}\,{}^{2}{{\rm{F}}}_{7/2}^{^\circ }$	11.57	−0.660	2	−2.53		8	−7.00	8
4739.71	4d 2D3/2–$8{\rm{f}}\,{}^{2}{{\rm{F}}}_{5/2}^{^\circ }$	11.57	−0.816	2	−2.53		8	−7.00	8
6545.94	$4{\rm{f}}\,{}^{2}{{\rm{F}}}_{5/2}^{^\circ }$–6g 2G7/2	11.63	0.040	3	−2.98		8	−7.25	8
6545.99	$4{\rm{f}}\,{}^{2}{{\rm{F}}}_{7/2}^{^\circ }$–6g 2G7/2	11.63	0.150	3	−2.98		8	−7.25	8
7877.05	$4{\rm{p}}\,{}^{2}{{\rm{P}}}_{1/2}^{^\circ }$–4d 2D3/2	9.99	0.391	2	−4.61	−4.61	6	−7.52	8	7.63	7.59	7.59
7896.04	$4{\rm{p}}\,{}^{2}{{\rm{P}}}_{3/2}^{^\circ }$–4d 2D3/2	9.99	−0.308	2	−4.61	−4.61	6	−7.52	8
7896.37	$4{\rm{p}}\,{}^{2}{{\rm{P}}}_{3/2}^{^\circ }$–4d 2D5/2	9.99	0.643	2	−4.61	−4.61	6	−7.52	8	7.66	7.59	7.59
9218.25	4s 2S1/2–$4{\rm{p}}\,{}^{2}{{\rm{P}}}_{3/2}^{^\circ }$	8.65	0.268	2	−5.06	−5.06	6	−7.61	8	7.90	7.68	7.68
9244.26	4s 2S1/2–$4{\rm{p}}\,{}^{2}{{\rm{P}}}_{1/2}^{^\circ }$	8.65	−0.034	2	−5.06	−5.06	6	−7.61	8
10914.24	3d 2D5/2–$4{\rm{p}}\,{}^{2}{{\rm{P}}}_{3/2}^{^\circ }$	8.86	0.038	2	−5.11	−5.11	6	−7.61	8	7.67	7.52	7.52
10951.78	3d 2D3/2–$4{\rm{p}}\,{}^{2}{{\rm{P}}}_{1/2}^{^\circ }$	8.86	−0.219	2	−5.11	−5.11	6	−7.61	8	7.68	7.56	7.56
Mean										7.72	7.59	7.59
σ										0.11	0.05	0.05
Mean of Mg i and Mg ii										7.60	7.56	7.56
σ										0.14	0.10	0.09

References. (1) Pehlivan Rhodin et al. (2017), (2) Kramida et al. (2018), (3) Kurucz & Peytremann (1975), (4) Green et al. (1957), (5) Dimitrijevic & Sahal-Brechot (1996), (6) Dimitrijević & Sahal-Bréchot (1995), (7) Barklem et al. (2000), (8) Kurucz (http://kurucz.harvard.edu/atoms.html), (9) Osorio et al. (2015). Γ₄/N_e,p and Γ₆/N_H are given for T = 10,000 K. Emission lines are marked by the symbol e. σ means standard deviation.

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For most lines of Mg i and Mg ii, Stark collisional data are taken from Dimitrijevic & Sahal-Brechot (1996) and Dimitrijević & Sahal-Bréchot (1995), respectively. They are given separately for collisions with electrons and protons in logarithmic form for T = 10,000 K. The SynthV_NLTE code was changed to account for both colliders. In the atmospheres of cool stars, both ${{\rm{\Gamma }}}_{4e}$ and ${{\rm{\Gamma }}}_{4p}$ may be coadded and multiplied by the number of electrons only, while in hotter stars, collisions should be treated separately and coadded after multiplying by the number of colliders.

In cool atmospheres, profiles of the Mg i lines are sensitive to the van der Waals broadening. For most neutral magnesium lines, collisional broadening due to collisions with neutral H is described via cross-sections and velocity parameters from the ABO theory (Anstee & O'Mara 1995; Barklem & O'Mara 1997). The van der Waals broadening constants, Γ₆/N_H, were taken from Barklem et al. (2000) and are presented in Table 1 in logarithmic form for T = 10,000 K. For IR lines of Mg i, van der Waals broadening parameters were taken from Osorio et al. (2015). For the remaining lines of Mg i and for all Mg ii lines, the corresponding broadening data were extracted from Kurucz' site.^{8 ,9}

Figure 3 displays fractions of Mg i, Mg ii, and Mg iii in the model atmospheres of different effective temperatures from the LTE and NLTE calculations. In the atmospheres with ${T}_{\mathrm{eff}}$ ≲12,000 K, Mg ii is the dominant ionization stage in the line-formation region, with small admixtures of Mg i (typically a few parts in a thousand) and Mg iii, while Mg ii becomes the minority species in the hotter atmospheres. NLTE leads to a depleted population of Mg i in the cool atmosphere (${T}_{\mathrm{eff}}$ log g/[Fe/H] =5780/3.7/−2.3), while only small deviations from LTE for Mg i are obtained in the 9550/3.95/−0.5 model. In the 17,500/3.8/0 model, NLTE leads to an enhanced number density of Mg ii.

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Figure 4 shows the departure coefficients for the selected levels in the models 5780/3.8/−2.3, 9550/3.95/−0.5, and 12,800/3.8/0.0, which represent the atmospheres of HD 140283, Vega, and π Cet, respectively. As expected, the departure coefficients are equal to unity deep in the atmosphere, where the gas density is large and collisional processes dominate, enforcing the LTE.

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In the 5780/3.8/−2.3 and 9550/3.95/−0.5 models, the Mg i 3p ³P° level (λ_thr = 2510 Å) and the ground state (λ_th = 1622 Å) are subject to UV overionization, and the population depletion is extended to the remaining Mg i levels due to their coupling to the low-excitation levels. Since singly ionized magnesium dominates the element abundance over all atmospheric depths, the Mg ii ground state keeps its LTE populations.

The Mg ii excited levels are overpopulated relative to their LTE populations in the 5780/3.8/−2.3 model, outward log τ = 0, due to the radiative pumping of ultraviolet transitions from the ground state, but the levels 4p²P°, 4d²D, and 4f²F° have depleted populations in the hotter 9550/3.95/0.5 model, outward log τ = −1, due to photon losses in the near-IR and visible lines of Mg ii arising in the transitions 4s²S–4p²P° (9218 Å), 4p²P°–4d²D (7877 Å), and 3d²D–4f²2F° (4481 Å).

In the 12,800/3.8/0.0 model, outward log τ = −1.0, NLTE leads to depleted populations of the Mg ii levels because of UV overionization. The formation depths of the Mg ii lines shift to the outer layers, where the deviations from LTE are strong.

The NLTE effects for a given spectral line can be understood from the analysis of the departure coefficients of the lower (b_l) and upper (b_u) levels at the line-formation depths. An NLTE strengthening of lines occurs, if ${b}_{l}\gt 1$ and (or) the line source function is smaller than the Planck function, that is, ${b}_{l}\,\gt$ b_u. The Mg i and Mg ii lines used in this study are listed in Table 1. First, we consider lines of Mg i at 8806 Å and 4571 Å in the 5780/3.8/−2.3 model. The Mg i 8806 Å line is strong, and its core forms around $\mathrm{log}\,{\tau }_{5000}=-0.7$, where the departure coefficient of the upper level 3d ¹D₂ drops rapidly (Figure 4) due to photon escape from the line itself, resulting in the line source function dropping below the Planck function and enhanced absorption in the line core. In contrast, in the line wings, absorption is weaker compared with the LTE case due to overall overionization in deep atmospheric layers. As a result, the NLTE effect on the total energy absorbed in this line tends to be minor. The Mg i 4571 Å line is weak, and its core forms at atmospheric depths ($\mathrm{log}\,{\tau }_{5000}\simeq -0.05$) dominated by overionization of Mg i, resulting in the weakened line and positive abundance correction of Δ_NLTE = log _NLTE–log _LTE = +0.17 dex.

In the 12,800/3.8/0.0 model, we consider Mg ii 4481, 7877, and 4390 Å. The Mg ii 4390 Å line is weak and forms in the layers around $\mathrm{log}\,{\tau }_{5000}=-0.7$, where ${b}_{l}\,\lt$ 1 and ${b}_{l}\,\lt$ b_u, resulting in the line weakening and a positive NLTE abundance correction of 0.04 dex. In contrast, the Mg ii 4481 and 7877 Å lines are strong, and their cores form around $\mathrm{log}\,{\tau }_{5000}=-2.5$ and $\mathrm{log}\,{\tau }_{5000}=-1.7$, where ${b}_{l}\,\gt$ b_u and the line source function drops below the Planck function, resulting in strengthened lines and negative NLTE abundance corrections of Δ_NLTE = −0.22 dex and −0.21 dex, respectively.

Our NLTE calculations in the grid of model atmospheres with 4000 K ≤ ${T}_{\mathrm{eff}}$ ≤ 7000 K, 4.0 ≤ log g ≤ 2.0, and 0 ≤ [Fe/H] ≤ −3 predict several Mg i lines, which arise between the high-excitation levels, to appear in emission. This concerns the Mg i IR lines at 7.736 μm (5g–6h), 11.789 μm (6f–7g), 12.224 μm (6g–7h), and 12.321 μm (6h–7i). Figure 5 shows how a variation in ${T}_{\mathrm{eff}}$ affects the theoretical profiles of the Mg i IR lines and in which atmospheric parameters the line absorption is changed by the emission. For different lines, an emission feature appears at different temperatures: ${T}_{\mathrm{eff}}$ ≥6000 K for 11.789 μm and ${T}_{\mathrm{eff}}$ ≥ 5000 K for 7.736 μm. The 12.321 and 12.224 μm lines are quite strong, and they reveal emission profiles over the 4000–7000 K temperature range.

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Carlsson et al. (1992) were the first to compute the 12 μm emission peaks by employing standard plane-parallel NLTE modeling for Mg i with a theoretical model atmosphere without a chromosphere. They showed that competitive processes of the overionization of the low-excitation levels of Mg i and close collisional coupling of the Rydberg levels to the Mg ii ground state, which contains most of the element, result in depleted level populations of Mg i compared with the LTE case and a systematic departure divergence pattern with the higher levels closer to LTE (b_u > b_l for any pair l < u). In the infrared, even tiny population divergences have large effects on the line source function due to the increasing importance of stimulated emission. The 12 μm line source function increases outwards in the atmosphere, resulting in an emission profile.

The NLTE calculations of Uitenbroek & Noyes (1996), Ryde & Richter (2004), Sundqvist et al. (2008), and this study show that similar physical processes produce emission not only in 12 μm, but also in the other Mg i IR lines and not only in the solar atmosphere, but also in the extended range of atmospheric parameters.

It is worth noting that, in the hotter atmospheres, the emission phenomena are observed in the near-IR lines of C i, at 8335, 9405, 9061–9111, and 9603–9658 Å, and Ca ii, at 8912–27 and 9890 Å, and they were well reproduced using the standard plane-parallel NLTE modeling of C i–C ii (Alexeeva et al. 2016) and Ca i–Ca ii (Sitnova et al. 2018).

Our sample includes 17 bright stars with well-known atmospheric parameters (Table 2).

Table 2.  Atmospheric Parameters of the Selected Stars and Sources of the Data

Star	Name	Sp. T.	${T}_{\mathrm{eff}}$	log g	[Fe/H]	${\xi }_{t}$	References
			(K)	CGS	(dex)	(km s−1)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
7000 K < ${T}_{\mathrm{eff}}$ ≤ 17,500 K
HD 160762	ι Her	B3 IV SPB	17,500	3.8	0.02	1.0	1
HD 17081	π Cet	B7 IV E	12,800	3.8	0.0	1.0	2
HD 22136	⋯	B8 V C	12700	4.2	−0.28	1.1	3
HD 209459	21 Peg	B9.5 V C	10400	3.5	0.0	0.5	2
HD 48915	Sirius	A1V+DA	9850	4.3	0.4	1.8	4
HD 72660	⋯	A0 V C	9700	4.1	0.4	1.8	5
HD 172167	Vega	A0 Va C	9550	3.95	−0.5	2.0	6
HD 73666	40 Cnc	A1V C	9380	3.78	0.16	1.9	7
HD 32115	⋯	A9 V C	7250	4.2	0.00	2.3	8
3900 K ≤ ${T}_{\mathrm{eff}}$ < 7000 K
HD 61421	Procyon	F5 IV–V	6580	4.00	−0.03	2.0	9
HD 84937	⋯	F8 V	6350	4.09	−2.08	1.7	10
HD 140283	⋯	F9 V	5780	3.7	−2.38	1.6	11
HD 103095	⋯	K1 V	5130	4.66	−1.26	0.9	11
HD 62509	Pollux	K0 IIIb	4860	2.90	0.13	1.5	12
HD 122563	⋯	G8 III	4600	1.6	−2.56	2.0	10
HD 124897	Arcturus	K1.5 III	4290	1.6	−0.52	1.7	12
HD 29139	Aldebaran	K5+III	3930	1.11	−0.37	1.7	12

References. (1) Nieva & Przybilla (2012), (2) Fossati et al. (2009), (3) Bailey & Landstreet (2013), (4) Hill & Landstreet (1993), (5) Sitnova et al. (2016), (6) Przybilla et al. (2000), (7) Fossati et al. (2007), (8) Fossati et al. (2011), (9) Boyajian et al. (2013), (10) Mashonkina et al. (2011), (11) Sitnova et al. (2015), (12) Heiter et al. (2015). Spectral types are extracted from the SIMBAD database.

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All stars can be divided into three groups. The first group includes 10 hot stars of F, A, and B spectral types with effective temperatures from 6580 to 17,500 K.

• ι  Her (HD 160762) is the hottest star in our sample. It is a single bright star, with V sin i ∼ 6 km s⁻¹. Its atmospheric parameters were determined spectroscopically from all available H and He lines and multiple ionization equilibria, and they were confirmed via the spectral energy distribution and the Hipparcos distance (Nieva & Przybilla 2012).
• The star π Cet (HD 17081) is an SB1 star with V sin i ∼ 20 km s⁻¹. This star was used by Smith & Dworetsky (1993) as a normal comparison star in the abundance study of chemically peculiar stars.
• The stars 21 Peg (HD 209459) and HD 22136 are "normal" single stars with V sin i ∼ 4 km s⁻¹ and 15 km s⁻¹, respectively (Sadakane 1981; Fossati et al. 2009; Bailey & Landstreet 2013). For 21 Peg and π Cet, their effective temperatures and surface gravities were spectroscopically determined from the Balmer lines (Fossati et al. 2009). The fundamental parameters of HD 22136 were derived by Bailey & Landstreet (2013) based on the Geneva and uvbyβ photometry
• The Sirius binary system (HD 48915 = HIP 32349) is composed of a main sequence A1V star and a hot DA white dwarf. It is an astrometric visual binary system at a distance of only 2.64 pc. Their physical parameters are available with the highest precision and were taken from Hill & Landstreet (1993). Sirius A is classified as a hot metallic-line (Am) star.
• Similar to Sirius, HD 72660 was classified as a hot Am star in the catalog of Renson & Manfroid (2009), but later it was reclassified as a transition object between HgMn and hot Am stars (Golriz & Landstreet 2016). Its parameters 9700/4.10/0.45/1.8 were derived in Sitnova et al. (2016) by fitting the 4400–5200 Å and 6400–6700 Å spectral regions with the SME (Spectroscopy Made Easy) program package (Valenti & Piskunov 1996; Piskunov & Valenti 2017).
• The effective temperature and surface gravity of Vega (HD 172167) were determined by Przybilla et al. (2000) from the Balmer line wings and Mg i and Mg ii lines. Vega is a rapidly rotating star seen pole-on. Rapid rotation causes a change of the surface shape from a spherical to an ellipsoidal one; therefore, the temperature and surface gravity vary from the pole to equator (Hill et al. 2010). We ignore the non-spherical effects in our study and analyze Vega's flux spectrum using the average temperature and gravity. Vega was also classified as a mild λ Bootis-type star (Venn & Lambert 1990).
• HD 73666 is a Blue Straggler and a member of the Praesepe cluster. It was previously considered as an Ap (Si) star, but appears to have the abundances of a normal A-type star (Fossati et al. 2007). This star is a primary component of SB1, as is the case for many other Blue Stragglers (Leonard 1996). The flux coming from the secondary star is negligible, as previously checked by Burkhart & Coupry (1998), so the star was analyzed ignoring the presence of the secondary. The physical parameters were taken from Fossati et al. (2007). Both Fe i excitation (for the effective temperature) and Fe i/Fe ii ionization (for log g) equilibria were used in the parameter determination.
• HD 32115 is another single-line spectroscopic binary. From radial velocity variations, Fekel et al. (2006) derived an orbital period of 8.11128 days, and concluded that the companion is either a late K-type or an early M-type dwarf. It allows us to consider that the spectral lines of the primary are not affected by the companion at the spectral range of our study, so one can ignore the contribution of the secondary to the total flux. The atmospheric parameters of HD 32115 were determined by Fossati et al. (2011) based on photometry, hydrogen line profiles, and wings of Mg i lines.
• Procyon is a known spectroscopic binary system, consisting of an F5 IV–V star (Procyon A) and a faint white dwarf (Procyon B). The proximity of Procyon to Earth makes it possible to measure directly its angular diameter. We adopted ${T}_{\mathrm{eff}}$ = 6582 K and log g = 4.00 found by Boyajian et al. (2013) using Hipparcos parallaxes and measured bolometric fluxes.

The second group includes four metal-poor stars in the −2.56 ≤ [Fe/H] ≤ −1.26 metallicity range. Their characteristics were taken from Alexeeva & Mashonkina (2015).

The third group is three K-giants from the sample of Gaia FGK benchmark stars: Pollux (HD 62509), Arcturus (HD 124897), and Aldebaran (HD 29139). Their V magnitudes range from −0.05 to 1.1. The stars are all nearby, and their parameters should be fairly accurate. We adopted the ${T}_{\mathrm{eff}}$ and log g from Heiter et al. (2015), where the determinations were obtained in a systematic way from a compilation of angular diameter measurements and bolometric fluxes, and from a homogeneous mass determination based on stellar evolution models. The NLTE [Fe/H] values for these stars were taken from Jofré et al. (2014).

For 12 stars in the sample, in the visible spectral range (3690–10480 Å) we used the normalized spectra obtained with the Echelle SpectroPolarimetric Device for the Observation of Stars (ESPaDOnS) attached at the 3.6 m telescope of the Canada–France–Hawaii (CFHT) observatory. For 11 of them, spectra were extracted from the ESPaDOnS archive,¹⁰ while the spectrum of HD 72660 was kindly provided by V. Khalack. Spectral observations (3300–9900 Å) of five stars were carried out with the UltraViolet and Visible Echelle Spectrograph (UVES) at the 8 m Very Large Telescope (VLT) of the European Southern Observatory (ESO). The spectra of Procyon, HD 84937, HD 122563, and HD 140283 were taken from the ESO Ultraviolet and Visual Echelle Spectrograph Paranal Observatory Project archive (Bagnulo et al. 2003). The spectrum of HD 103095 was obtained with the Hamilton Echelle Spectrograph mounted on the Shane 3 m telescope of the Lick Observatory (see for details Sitnova et al. 2015). All spectra were obtained with a high spectral resolving power, R, and high signal-to-noise ratio (S/N; more than 100). The characteristics of the observed spectra for individual stars are given in Table 3. The infrared spectra of Procyon, Pollux, Arcturus, and Aldebaran were kindly provided by Ryde et al. (2004) and Sundqvist et al. (2008). The observations were made with the Texas Echelon-cross-echelle Spectrograph attached at the 3 m NASA Infrared Telescope Facility (IRTF; Lacy et al. 2002). S/Ns in the spectra vary but are generally high, reaching S/N ∼ 450 per pixel for Pollux and $\approx 300$ for Arcturus and Aldebaran, in regions around the 12.22 μm line.

In this section, we derive the magnesium abundances of 10 selected stars with effective temperatures from 6580 to 17,500 K using lines of Mg i and Mg ii in the visible and near-IR spectral range. The Mg ii 9218 and 9244 Å lines are sitting at the ends of consecutive echelle orders, where continuum fitting is problematic. Moreover, these lines are lying in the wings of the Paschen P9 line at 9229.7 Å. This makes the formation of Mg ii 9218 and 9244 Å very sensitive to the correct modeling of the hydrogen Paschen line formation. As was shown by Sitnova et al. (2018), the different treatment of hydrogen opacity in the Paschen-series region implemented in the codes for spectral synthesis leads to significant differences in the Paschen line absorption, thus introducing difficulties for the careful analysis of spectral lines falling close to the Paschen line cores. We analyze the Mg ii 9218 and 9244 Å lines (Table 4) but do not include them in the calculations of the mean abundances.

For Sirius and HD 72660, we use also the UV spectra kindly provided by J. Landstreet. The detailed description of the spectra and their reduction is given in Landstreet (2011) and in Golriz & Landstreet (2016). We examined the Mg ii 1737.62 and 2798.00 Å and Mg i 1827.93 and 2852.13 Å lines and found them to be nearly unblended and suitable for abundance analysis. Atomic parameters were taken from Kramida et al. (2018; oscillator strengths) and Dimitrijević & Sahal-Bréchot (1995, 1996; Stark damping data).

The abundance results are presented in Tables 4 and 5. For each star, NLTE provides a smaller line-to-line scatter that is well illustrated by Figure 7. For nine stars where lines of both ionization stages are available, the NLTE abundances derived from the Mg i and Mg ii lines agree within the error bars, while in LTE, the abundance differences vary between +0.23 dex and −0.21 dex. For most stars, the obtained Mg NLTE abundance is close to the solar value. The exceptions are Vega, HD 72660, and HD 73666, whose classification indicates a peculiar origin. Below we comment briefly on some individual stars.

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Table 5.  Lines of Mg i and Mg ii in the UV Spectra of HD 72660 and Sirius and the Obtained NLTE Abundances

λ	Transition	${E}_{\mathrm{exc}}$	log gf	log Γ4/Ne	log Γ4/Np	HD 72660	Sirius
(Å)		(eV)		(rad s−1 cm3)		log Mg
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
Mg i
1827.93	3s2 1S0–$5{\rm{p}}\,{}^{1}{{\rm{P}}}_{1}^{^\circ }$	0.00	−1.667	−4.47	−4.97	7.89	7.53
2852.13	3s2 1S0–$3{\rm{p}}\,{}^{1}{{\rm{P}}}_{1}^{^\circ }$	0.00	0.240	−5.78	−6.20	8.05	7.55
Mg ii
1737.61	$3{\rm{p}}\,{}^{2}{{\rm{P}}}_{3/2}^{^\circ }$ –4d 2D3/2	4.43	−1.814	−4.65	−5.43	⋯	⋯
1737.63	$3{\rm{p}}\,{}^{2}{{\rm{P}}}_{3/2}^{^\circ }$ –4d 2D5/2	4.43	−1.859	−4.65	−5.43	7.69	7.42
2797.93	$3{\rm{p}}\,{}^{2}{{\rm{P}}}_{3/2}^{^\circ }$ –3d 2D3/2	4.43	−0.426	−5.32	−6.44	⋯	⋯
2798.00	$3{\rm{p}}\,{}^{2}{{\rm{P}}}_{3/2}^{^\circ }$ –3d 2D5/2	4.43	0.528	−5.32	−6.44	7.84	7.50

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ι Her. This is the hottest star of our sample; therefore the magnesium abundance, log _Mg = 7.55 ± 0.07, was derived from the Mg ii lines only. The NLTE corrections are negative for Mg ii 4481, 7877, and 7896 Å and positive for Mg ii 4384, 4390, 4427, 4433, and 4739 Å. Positive corrections do not exceed 0.06 dex for the first four lines.

π Cet. This star is 4700 K cooler than ι Her, and three weak lines of Mg i can be measured. The NLTE effects for the Mg ii lines are similar to those for ι Her, although they are slightly smaller.

HD 22136. This star was found to be an SPB-type variable (Molenda-Zakowicz & Polubek 2005), and the asymmetric line profiles seen in its spectrum seem to support this classification. The asymmetry causes uncertainties in the line-fitting procedure, depending on the line intensity. Nevertheless, the magnesium NLTE abundances based on 4 Mg i and 11 Mg ii lines are consistent within 0.09 dex. The Mg NLTE abundance does not differ significantly from the solar value; however, it is worth noting that it is 0.2 dex higher than the overall stellar metallicity (see Table 2).

21 Peg. In this star, the Mg i/Mg ii ionization equilibrium is achieved in both NLTE and LTE; however, with a much smaller line-to-line scatter in NLTE (σ_{Mg I} = 0.05 dex, σ_{Mg II} = 0.04 dex) than in LTE (σ_{Mg I} = 0.15 dex, σ_{Mg II} =0.16 dex). This is illustrated well by Figure 7. Hereafter, the statistical abundance error is the dispersion in the single-line measurements about the mean: σ = Σ (x − x_i)²/(N − 1). The NLTE abundance corrections are slightly positive (${{\rm{\Delta }}}_{\mathrm{NLTE}}\,\leqslant$ 0.08 dex) for Mg i 4702 and 5528 Å; close to zero for Mg i 4167 Å and Mg ii 3848, 3850, 4384, 4390, 4427, 4433, 4739 Å; and slightly negative (${{\rm{\Delta }}}_{\mathrm{NLTE}}\,\leqslant$ 0.18 dex in absolute value) for Mg i b, 3829, 5167 Å and Mg ii 4481, 7896.04 Å; however, pronounced NLTE effects are found for the Mg i 3832, 3838 Å and Mg ii 7877, 7896.37 Å lines, with negative NLTE corrections up to Δ_NLTE = −0.48 dex.

Sirius. The NLTE abundances from lines of the two ionization stages, Mg i and Mg ii, are consistent, and they are close to the solar value, although the star is classified as a hot Am star with the metallicity of +0.4 dex (Table 2). We find that abundances derived from the lines in the UV region (Table 5) agree with those derived from the visible and near-IR region (Table 4), which means that the model atom developed in this paper provides consistent magnesium abundance determinations over a wide spectral range from UV to IR (Figure 8). The NLTE abundance corrections for the investigated UV lines of Mg i and Mg ii are smaller than 0.05 dex.

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HD 72660. This star has the same metallicity as Sirius but, in contrast to Sirius, the Mg NLTE abundance, averaged over 11 Mg i and 10 Mg ii lines, exceeds the solar value by 0.2 dex. This perhaps reflects a classification of HD 72660 as a transition object between HgMn and hot Am stars. The NLTE abundance derived from the Mg i UV lines, log _Mg = 7.97 ± 0.11 (Table 5), is 0.22 dex higher than that based on the Mg i visible and near-IR lines (Table 4), although the discrepancy does not exceed 2σ. For Mg ii, the NLTE abundances based on the UV (log _Mg = 7.76 ± 0.10) and visible (log _Mg = 7.74 ± 0.07) lines are consistent.

HD 73666. For Mg i, the NLTE abundance corrections are either slightly positive (${{\rm{\Delta }}}_{\mathrm{NLTE}}\,\leqslant$ 0.02 dex for 4167, 4702, 5528, and 5711 Å) or negative, with ${{\rm{\Delta }}}_{\mathrm{NLTE}}$ up to −0.55 dex. For most investigated lines of Mg ii, the NLTE effects are minor. The exceptions are 4481, 7877, and 7896.04 Å, with Δ_NLTE = −0.12, −0.27, and −0.26 dex, respectively.

HD 32115. NLTE leads to weakened lines of Mg i and, in contrast, strengthened lines of Mg ii, such that the obtained NLTE abundances from the two ionization stages are consistent within 0.02 dex (Figure 8). The LTE abundance based on the Mg ii lines is 0.18 dex lower compared with that for Mg i.

Vega. The NLTE effects for the Mg lines are very similar to those found for HD 73666. The magnesium NLTE abundance is subsolar and corresponds to Vega's metallicity, such that [Mg/Fe] = −0.02.

Procyon. The best fits to the Mg i 5183 Å and Mg ii 4481 Å lines in Procyon are shown in Figure 9. Our NLTE calculations reproduce well the Mg i 12.22 and 12.32 μm lines observed in emission (Figure 10 for 12.32 μm). The best fits are achieved with log _Mg = 7.50, in line with the abundances derived from the absorption lines of Mg i and Mg ii. The NLTE abundances from Mg i (eight lines) and Mg ii (three lines) are fairly consistent (Figure 8), while the LTE abundance difference amounts to Mg i – Mg ii = −0.21 dex.

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Methodical notes and recommendations. We can recommend the Mg ii 3848, 3850, 4384, 4390, 4427, and 4433 Å lines for Mg abundance determinations even at the LTE assumption in a wide range of effective temperatures, from 7000 to 17,500 K, due to the small NLTE effects for these lines. The Mg i 4167, 4571, 4702, 5528, 5167, 5172, and 5183 Å lines can be safely used in the LTE analysis for stars with 7000 K < ${T}_{\mathrm{eff}}$ ≤ 8000 K. For the hotter stars, with ${T}_{\mathrm{eff}}$ from 8000 to 9500 K, the NLTE effects are minor only for Mg i 4167, 4702, and 4528 Å. We caution against using the Mg ii 9218, 9244 Å lines for Mg abundance determination due to their position in the wings of the Paschen P9 line and also their pronounced NLTE effects, with negative NLTE corrections up to one order of magnitude.

For seven stars with 3900 K ≤ ${T}_{\mathrm{eff}}$ < 6350 K, the abundance results are presented in Table 4. For each star, NLTE provides a smaller line-to-line scatter (Figure 7). For the three stars, abundances were determined from lines of the two ionization stages, Mg i and Mg ii. The best fit to Mg ii 4481 Å in HD 84937 is shown in Figure 9. The only line of Mg ii, at 9244 Å, is observed in Pollux, and it gives the NLTE abundance, which differs from that for Mg i by only 0.01 dex. In LTE, we obtain Mg i – Mg ii = −0.16 dex.

However, Mg i/Mg ii ionization equilibrium is not achieved in HD 140283 and HD 84937, independent of either NLTE or LTE. In these two stars, we can measure the only line of Mg ii, at 4481 Å, and it gives a higher abundance than that from the Mg i lines, by 0.24 and 0.19 dex, respectively. Although Mg ii 4481 Å is contaminated by blends of Ti i 4481.259 Å, Gd ii 4481.054 Å, and Ni i 4481.119 Å, which contribute no more than 15% of the total equivalent width, this can hardly be a reason for the obtained abundance discrepancy. The NLTE effects for the Mg i and Mg ii lines in these stars are small and also cannot solve the problem. Unfortunately, we were unable to find literature investigations of the Mg ii 4481 Å line in the atmospheres of cool stars. In order to understand an origin of such a discrepancy, it would be useful to perform full 3D NLTE calculations with our comprehensive model atom.

As found by Uitenbroek & Noyes (1996) and Sundqvist et al. (2008), the IR spectra of our three K-giants reveal the emission in the Mg i 12.22 and 12.32 μm lines (Figure 10 shows Arcturus and Pollux), which is much stronger than that in the corresponding solar and Procyon lines. Our NLTE calculations predict emission lines; however, they are too weak. We suggest that the formation of Mg i 12.22 and 12.32 μm in Aldebaran, Arcturus, and Pollux is affected by the chromospheric temperature rise, which is not accounted for by our classical radiative-equilibrium model atmospheres.