Photospheric silicon abundances of upper main-sequence stars derived from Si II 6347/6371 doublet lines

Silicon abundances were determined by applying the spectrum-fitting technic to the Si II doublet lines at 6347 and 6371A for a sample of 120 main-sequence stars in the T_eff range of ~7000-14000K (comprising not only normal stars but also non-magnetic chemically peculiar stars) with an aim of investigating their behaviors (e.g., correlation with stellar parameters and abundances of other elements such as Fe or C) and the background physical mechanisms involved therein, where attention was paid to taking into account of the non-LTE effect and to assigning a reasonable value of microturbulence. The following trends were revealed from the analysis: (i) The resulting [Si/H] values, mostly ranging from ~-0.5 to ~+0.3, show a positive correlation with [Fe/H]. (ii) A kind of anti-correlation exists between Si and C as seen from the tendency of [C/Si] steeply decreasing with [Si/H]. (iii) Si abundances do not show any clear dependence upon T_eff or vsini, while Am and HgMn stars appear to show comparatively higher [Si/H] than normal stars. Although it is not straightforward to explain these observational facts, different physical processes (gas-dust separation and atomic diffusion) are likely to be intricately involved in producing these characteristic behaviors of Si composition in the surface of late A through late B dwarfs.


Introduction
It is known that a significant fraction of late A through late B-type main sequence stars show anomalous spectra indicative of surface abundance anomalies. Those chemically peculiar (CP) stars are divided into several groups according to their features as summarized in the review paper by Preston (1974). So far, the abundance characteristics of many elements in CP stars have been investigated in comparison with normal stars to discuss the origin of their anomalies.
However, regarding silicon, an important abundant element used as the fiducial reference in geo-or cosmo-chemistry, its abundance behavior in upper mainsequence stars is not yet well understood. While conspicuous overabundance of Si is known to be observed in magnetic CP stars (CP2; the group 2 in CP stars classified by Preston 1974), how it behaves in non-magnetic CP stars (CP1 -Am stars; CP3 -HgMn stars) or in normal stars is not clear. As a matter of fact, we still do not know whether any Si anomaly ever exists in these stars. According to the atomic diffusion theory (see, e.g., Michaud et al. 2015), which is considered to be a promising mechanism to explain the origin of abundance characteristics in CP stars, Si is expected to be somewhat underabundant (Richer et al. 2000;Talon et al. 2006). Meanwhile, such a trend is not necessarily seen in spectroscopically determined Si abundances of A-and late B-type dwarfs, which are rather diversified around the normal (solar) abundance (somewhat overabundant or underabundant depending on cases; e.g., Niemczura et al. 2015;Ghazaryan, Alecian 2016;Mashonkina et al. 2020b;Saffe et al. 2021), though Si abundance determination significantly depends upon which line is to be used.
Another noteworthy aspect characterizing the importance of Si abundance is that this element is a typical refractory species (being easily fractionated into dust) in contrast to the volatile elements such as C, N, and O. Interestingly, Holweger and Stürenburg (1993) reported that even normal early A-type stars (like λ Boo-type stars) show anti-correlation between the abundances of Si and C ([C/Si] systematically decreases with [Si/H]), which means that some kind of gas-dust separation process (its degree being different from star to star) would have operated in the star formation phase and altered the primordial composition of gas. Is such an effect observed also stars of other types (i.e., late A through late B stars including CP1 and CP3 stars)? This is an interesting problem to be clarified.
Conveniently, Takeda et al. (2018; hereinafter referred to as T18) recently published the C, N, and O abundances for a large sample of 100 main-sequence stars (comprising normal as well as non-magnetic CP stars) covering 7000 < ∼ T eff < ∼ 11000 K. It would be worthwhile, therefore, to determine the Si abundances for these stars. This would enable to clarify the behaviors of both [Si/H] and [C/Si], by which the nature of abundance peculiarity of Si (if any exists) in late A through late B-type stars and the involved physical process may be investigated. This is the aim of the present study.
In the past Si abundance determinations in upper main-sequence stars so far, it appears that neutral Si i lines were mainly used in late-mid A-type stars (including classical Am stars) while once-ionized Si ii lines were primarily employed in early A and late B stars (because Si i lines quickly fade out with an increase in T eff ). Since the mixed use of lines of different ionization stages is not advantageous because of inevitable line-by-line abundance discrepancies (see, e.g., Mashonkina 2020a), we decided to invoke in this study only the Si ii doublet lines at 6347 and 6371Å, which are of high quality (i.e., almost free from blending) and sufficiently strong over the whole relevant T eff range. Nevertheless, some disadvantages are involved in using these strong Si ii lines; that is, the resulting Si abundances suffer an appreciable non-LTE affect and are sensitive to the microturbulence parameter. Accordingly, special attention had to be paid to these two points.

Observational data
Regarding the program stars in this study, all the 101 targets (including the reference star Procyon) in T18 (cf. Section 2 therein) were adopted without change. In addition, in order to back up the range of 11000 < ∼ T eff < ∼ 14000 K (which was not covered in T18), 19 late B-type stars (among which ∼ 40% are CP3 stars) were newly included. As such, our targets are 120 late B-type through early F-type stars on or near to the main sequence (luminosity classes of III-V) which have slow to moderately-high rotational velocities (0 km s −1 < ∼ v e sin i < ∼ 100 km s −1 ). Among these, about ∼ 1/3 are non-magnetic CP stars: 25 Am stars, 13 HgMn (or Mn) stars, and 2 λ Boo stars. Besides, our sample includes 16 Hyades A-type stars. The list of these 120 stars is given in Table 1, while the data source and the basic information of the observational materials are summarized in Table 2 . The 120 program stars are plotted on the theoretical HR diagram (log(L/L ) vs. log T eff ), where T eff was derived from colors (cf. Section 3) and L was evaluated from visual magnitude (corrected for interstellar extinction by following Arenou et al. 1992), Hipparcos parallax (van Leeuwen 2007), and bolometric correction (Flower 1996). Theoretical solar-metallicity tracks for 7 different masses (1.5, 1.7, 2, 2.5, 3, 4, and 5 M ), which were computed by Lejeune and Schaerer (2001), are also depicted by solid lines for comparison.   (1) HD number.

Stellar parameters
As in T18, the effective temperature (T eff ) and the surface gravity (log g) for each star were determined from colors of Strömgren's uvbyβ photometric system by using Napiwotzki et al.'s (1993) calibration.
Especially important parameter we should care about is the microturbulence (ξ). We basically adopted (as done in T18) the analytical T eff -dependent relation derived by Takeda et al. (2008) (where A ≡ [log(10000/8000)]/ √ ln 2, ξ is in km s −1 , and T eff is in K) for stars with T eff < 11000 K, while ξ = 1 km s −1 was assumed at T eff > 11000 K (where this equation yields ξ < 1 km s −1 ). Such formula-based values are called as the "standard" microturbulence (designated as ξ std ) in order to clarify the difference from another choice of microturbulence described later (cf. Section 6.2).
The only exception is the standard star Procyon (HD 61421), 1 for which we used Takeda et al.'s (2005b) spectroscopically determined values (T eff = 6612 K, 1 The reason why Procyon was chosen as the reference standard (as done in our previous studies) is to carry out abundance determination by "differential analysis" where the resulting relative abundances are unaffected by uncertainties in the gf values of spectral lines. That is, Procyon (F5 IV-V) is more suitable than the Sun (whose T eff is too low in comparison with those of A and late B stars to be used for such a purpose), while its chemical abundances are log g = 4.00, and ξ std = 1.97 km s −1 ) to maintain consistency with Takeda et al. (2008).
The adopted values of T eff , log g, [Fe/H], 2 and ξ std are summarized in Table 1. All the program stars are plotted on the log L vs. log T eff diagram (theoretical HR diagram) in Fig. 1, where theoretical evolutionary tracks corresponding to different stellar masses are also depicted. This figure indicates that the masses of our sample stars are in the range between ∼ 1.5M and ∼ 5M . More detailed data regarding the targets and their stellar parameters are given in the electronic table (tableE.dat) available online at https://www.astro.sk/caosp/Eedition/FullTexts/vol52no1/pp5-31.dat/.
The model atmosphere corresponding to each star was constructed by interpolating Kurucz's (1993a) ATLAS9 model grid (for ξ = 2 km s −1 ) in terms of T eff , log g, and [Fe/H].

Non-LTE calculation for Si
The statistical-equilibrium calculations for silicon atom were carried out by using the non-LTE code described in Takeda (1991). The atomic model of Si adopted in this study was constructed based on Kurucz and Bell's (1995) compilation of atomic data (gf values, levels, etc.), which consists of 34 Si i terms (up to 4d 1 F o at 58893.4 cm −1 ) with 222 Si i radiative transitions, 31 Si ii terms (up to 3p 3 4 S o at 123033.5 cm −1 ) with 109 Si ii radiative transitions, and 23 Si iii terms (up to 4p 3 P at 248073 cm −1 ; included only for conservation of total Si atoms).
Regarding evaluations of photoionization rates, the cross-section data taken from TOPbase (Cunto, Mendoza 1992) were used for the lower 10 Si i terms and 10 Si ii terms (while hydrogenic approximation was assumed for higher terms).
As to the collisional rates, the theoretical results of Aggarwal and Keenan (2014) were invoked for the bound-bound electron impact rates between the lower 10 Si ii terms. Otherwise, the recipe described in Sect. 3.1.3 of Takeda (1991) was followed (inelastic collisions due to neutral hydrogen atoms were formally included as described therein, though insignificant in the atmosphere of earlytype stars under question).
practically the same as those of the Sun (cf. the references quoted in Section IV(c) of Takeda et al. 2008). 2 These Fe abundances were already established in our previous papers (cf. the references given in Table 2) based on the spectrum-fitting method in the wavelength region (∼ 20-30Å wide) centered around ∼ 6155Å (where the Fe ii 6147/6149 doublet lines are the important indicators of Fe abundance).
The depth-dependent non-LTE departure coefficients to be used for each star were then evaluated by interpolating this grid in terms of T eff and log g.

Abundance determination
The non-LTE Si abundances were determined (as done in T18 for CNO abundances) based on Takeda's (1995) numerical algorithm by accomplishing the best fit between the synthetic and observed spectra in the 6340-6380Å region while varying the abundances of Si and some other elements showing appreciable lines (especially Fe, plus other elements such as Mg, Mn, Zn depending on cases), v M (macrobroadening velocity corresponding to instrumental/rotational broadening and macroturbulence) and ∆λ (radial velocity or wavelength shift) but the microturbulence being fixed at ξ std . Since the relevant wavelength region of the raw spectra is more or less contaminated by weak telluric lines, they were removed in advance by dividing by the spectrum of a rapid rotator as demonstrated in Fig. 2. The atomic data of spectral lines comprising in this region were exclusively taken from Kurucz and Bell's (1995) compilation (those of relevant Si ii doublet lines are summarized in Table 3), though some pre-adjustments 3 were necessary in order to achieve an satisfactory fit. The accomplished fit in the neighborhood of both lines for each star is displayed in Fig. 3. Note. These data are were taken from Kurucz and Bell's (1995) compilation, while those parenthesized are the default values calculated by Kurucz's (1993a) WIDTH9 program. Followed by first four self-explanatory columns, damping parameters are given in the last three columns: Gammar is the radiation damping width (s −1 ), log γ rad . Gammas is the Stark damping width (s −1 ) per electron density (cm −3 ) at 10 4 K, log(γe/Ne). Gammaw is the van der Waals damping width (s −1 ) per hydrogen density (cm −3 ) at 10 4 K, log(γw/NH).
Then, the equivalent widths (W 6347 and W 6371 ) of the Si ii 6347 and 6371 lines were inversely evaluated from the best-fit solution of A N std (Si) with the same model and atmospheric parameters as used in the spectrum-fitting analysis. From such evaluated W , the non-LTE abundance (A N ), LTE abundance (A L ) and non-LTE correction (∆ ≡ A N −A L ) were derived for each line. Besides, W can be further used to estimate the abundance uncertainties due to typical ambiguities of atmospheric parameters by perturbing the standard values interchangeably. Such derived W 6347 /W 6371 (equivalent widths), ∆ 6347 /∆ 6371 (non-LTE corrections), A N std (non-LTE Si abundance), δ T ± (abundance changes for T eff perturbations by ±3%), δ g± (abundance changes for log g perturbations by ±0.1 dex), and δ ξ± (abundance changes for ξ perturbations by ±30%) are plotted against T eff in Fig. 4.
These standard abundances, expressed in the form of [Si/H] std (≡ A N std (star) − A N std (Procyon)) are given in Table 1. More complete results including W and ∆ are presented in "tableE.dat" of the online material.  Figure 3. Synthetic spectrum fitting analysis for Si abundance determination from Si ii 6347/6371 lines, In the left, middle, and right panels are shown the results for 40 stars in 6500 K < T eff < 8000 K, 39 stars in 8000 K < T eff < 9500 K, and 41 stars in 9500 K < T eff < 14000 K, respectively. The best-fit theoretical spectra (in the selected ranges of 6344-6350Å and 6367.5-6375Å comprising the relevant Si ii lines) are depicted by blue solid lines, while the observed data are plotted by pink symbols (the masked data excluded in judging the goodness of fit are highlighted in green). In each panel, the spectra are arranged in the descending order of v e sin i, and an offset of 0.2 is applied to each spectrum (indicated by the HD number) relative to the adjacent one. The case of Procyon (standard star) is separately displayed at the bottom of the left panel.   , where the error bar denotes ±δ T gv (root-sumsquare of δ T , δ g , and δ ξ , where δ T is the mean of |δ T + | and |δ T − |; etc.), (d) δ T + and δ T − (abundance variations for Si ii 6347 in response to T eff changes of +3% and −3%), (e) δ g+ and δ g− (abundance variations for Si ii 6347 in response to log g changes by +0.1 dex and −0.1 dex), and (f) δ ξ+ and δ ξ− (abundance variations for Si ii 6347 in response to perturbing the ξ std value by +30% and −30%). The abundance of Procyon (A N std = 7.367), which is adopted as the reference, is indicated by the horizontal dashed line in panel (c).

Characteristics of the non-LTE effect
As seen from the results derived in Section 5, the Si ii 6347/6371 lines suffer an appreciable non-LTE effect. According to Fig. 4b, their non-LTE abundance corrections (∆) are negative (which means that the non-LTE effect strengthens the lines) and typically a few tenths dex (|∆ 6347 | ∼ 0.2-0.5 dex, |∆ 6371 | ∼ 0.1-0.4 dex; naturally the former is larger because of the stronger line forming in comparatively shallower layer). The maximum of |∆| is around T eff ∼ 10000 K.  (τ ) (the non-LTE-to-LTE line-center opacity ratio; almost equal to b 1 ) and S L (τ )/B(τ ) (the ratio of the line source function to the Planck function; nearly equal to b 2 /b 1 ) for the transition relevant to the Si ii 6347/6371 lines (b 1 and b 2 are the non-LTE departure coefficients for the lower and upper terms), which were computed on the models of representative T eff and log g values. As seen from this figure, while l NLTE 0 /l LTE 0 > 1 (overpopulation) holds in the line-forming region at T eff < ∼ 10000 K (A-type tars), this inequality suddenly turns to be reversed (underpopulation) at higher T eff (late B-type stars) because of the beginning of Si ii overionization (once-ionized Si is not the dominant ionization stage any more in such a higher T eff regime). Although the non-LTE effect still acts to intensify lines (∆ remains negative) at 10000 K < ∼ T eff < ∼ 14000 K due to the dilution of S L (< B) (see the lower panels in Fig. 5), |∆| progressively decreases with an increase in T eff (see also the Appendix A where the behavior of ∆ in B-type stars is further discussed).
How the theoretical W and ∆ computed for these two Si ii lines depend upon the atmospheric parameters (T eff , log g, and ξ) is illustrated in Fig. 6, which reasonably explains the trends observed in Figs. 4a and 4b (the maximum of W is seen around T eff ∼ 8000 K because the peak of ξ is attained there).

Consistency check of microturbulence
As is evident from the lower three panels (d-f) of Fig. 4, an uncertainty ξ has the most significant impact on the Si abundance among the three atmospheric parameters especially at T eff < ∼ 10000 K, since the Si ii 6347/6371 lines are strong and saturated (on the flat part of the curve of growth). Therefore, particular attention should be paid to whether or not an appropriate choice of ξ has be done. As a matter of fact, Takeda et al. (2009) reported that considerably underestimated Na abundance would result from the strongly saturated Na i 5889/5895 D lines if the microturbulence given by Equation (1) is used, which may be attributed to the depth-dependence of ξ (cf. Section 5 therein) Does such an inadequacy similarly exist also for the case of Si ii 6347/6371 lines?
In order to examine this problem, another solution of microturbulence was determined from these doublet lines themselves by taking the advantage that their log gf strengths are different by 0.3 dex. That is, spectrum fitting analysis was retried (taking A N std and ξ std as the starting solutions) while allowing both A N (Si) and ξ to vary. These refit solutions (which are referred to as A N refit and ξ refit ) were successfully converged for 97 stars (about ∼ 80%), though failed for the remaining 23 stars.
The resulting A N refit and ξ refit (given in Table 1) are compared with A N std and ξ std in Fig. 7, where the following characteristics are observed. - Fig. 7a indicates that consistency between ξ refit (dots) and ξ std (solid line) is not necessarily bad, though considerable discrepancy (quite a few ξ refit values  Figure 7. (a) Microturbulence directly determined by spectrum refitting (ξ refit ) plotted against T eff by dots, while the T eff -dependent standard microturbulence (ξ std ) given by Equation (1) is shown by the red solid line. (b) Non-LTE Si abundance (A N refit ) resulting from refitting (corresponding to ξ refit ) plotted against T eff . (c) Difference between "std" and "refit" abundances (A N std − A N refit ) plotted against T eff . (d) Difference between [Si/H] std (derived in this study based on Si ii 6347/6371 lines by using the standard ξ std ) and [Si/H] Si I (derived by Takeda et al. 2009) based on the spectrum fitting applied to the region comprising Si i lines) plotted against T eff . tending to be appreciably lower than ξ std ) is seen around T eff ∼ 8000 K. 4 -As a result, A N std tends to be lower than A N refit at T eff ∼ 8000 K. The differences are typically a few tenths dex (four stars show especially large discrepancies of ∼ 0.5-0.6 dex (cf. Fig. 7c).
-It is worth noting that the [Si/H] N std values also exhibit similar discrepancies when compared with Takeda et al.'s (2009) [Si/H] results derived from the spectrum fitting in the 6140-6170Å region comprising Si i lines (Fig. 7d).
-Accordingly, we may state that the abundances derived from Si ii 6347/6371 lines by using Equation (1)-based ξ std are apt to be underestimated around T eff ∼ 8000 K corresponding to late-to-mid A-type stars.
-However, this problem is not so serious as the case of Na i 5889/5895 lines addressed by Takeda et al. (2009). Actually, the appearance of A N refit vs. T eff plot (Fig. 7b) is not significantly different from the case of A N std (Fig. 4c). In the figures illustrating the behaviors of Si abundances to be discussed in the next section, both ("std" and "refit") results will be shown, so that they may be compared with each other.  Fig. 8, where two kinds of results based on A N std and A N refit are presented in parallel in each panel. Besides, the same correlation plots as Fig. 8 but only for the selected 16 Hyades stars are depicted in Fig. 9. The following characteristics can be read from these figures.
• The resulting Si abundances (relative to Procyon) for most stars are in the range of −0.5 < ∼ [Si/H] < ∼ +0.3 (tending to be rather Si-deficient than Sirich).
• As for the relation to stellar parameters, any clear dependence upon  (Fig. 8c), which is also observed for the selected sample of Hyades stars (Fig. 9c)   How can we interprete these results? As mentioned in Section 1, two physical processes may be considered for the possible cause of chemical abundance anomalies: (i) atomic diffusion and (ii) gas-dust separation. Regarding the former diffusion process, although considerable uncertainties still exist, available theoretical calculations predict a deficiency of C, a slight underabundance of Si, and an overabundance of Fe (cf. Richer et al. 2000;Talon et al. 2006). As to the latter gas-dust separation process, Si as well as Fe (both are refractory elements and should behave similarly) are expected to be anti-correlated with C (volatile species).
Although [C/Si] Fig. 8h is too steep to be identified with Holweger and Stürenburg's (1993) Fig. 2 (the slope is ∼ −45 • ). Actually, the considerably wide range of [C/Si] (∼ 2 dex) is mainly due to the diversified deficiency of C (−1.5 < ∼ [C/H] < ∼ 0 especially seen in Am stars of lower T eff ; cf. Fig. 8d) while the contribution of [Si/H] is comparatively minor. Therefore, the main cause of C deficiency is attributed to (not the dust-gas separation but) to atomic diffusion as discussed in T18.
Regarding  (Fig. 8c, Fig. 9c; just like Holweger and Stürenburg's Fig. 3) may indicate that the gas-dust separation process (acting both Si and Fe in a same direction) is involved at least partly, because atomic diffusion would differently affect Si and Fe according to the currently available calculations. However, it is unlikely that the diffusion process does not play any role in affecting the surface abundances of Si and Fe, since Takeda and Sadakane (1997) reported the v e sin i-dependence of over-/under-abundance of Fe/O in Hyades A-type stars (cf. Fig. 7 therein) which is reasonably interpreted as the result of atomic diffusion being more effective for slower rotators.
Consequently, given the available information alone, it is hardly possible to find any satisfactory interpretation regarding the observed behavior of Si abundances (particularly in relation to the abundances of C and Fe). It may be possible that both processes (atomic diffusion and gas-dust separation) concurrently operate in an intricate manner (the former being more significant?). Unfortunately, the current diffusion calculations appear to still suffer considerable uncertainties (e.g., in the choice of parameters concerning turbulent mixing or mass loss). 6 Further progress in this field is desirably awaited, so that it may shed light to this complicated situation.

A. Non-LTE effects on Si ii 6347/6371 lines in B-type stars
Recently, Mashonkina (2020a) carried out an extensive study on the non-LTE line formation for silicon (Si i, Si ii, and Si iii) in main-sequence stars of A-and B-type covering the T eff range between 7000 and 20000 K. Mashonkina's calculation includes the Si ii 6347/6371 lines and the non-LTE corrections for these lines derived by her for late A through late B stars (T eff ∼ 7000-13000 K; negative ∆ with extents of several tenths dex) are more or less consistent with the results of this investigation.
However, her calculation failed to explain the formation of these Si ii doublet lines in the early B-type star ι Her (T eff = 17500 K), because of the positive non-LTE corrections resulting in unacceptably large non-LTE Si abundances (∆6347 = +0.60, ∆6371 = +0.67, A N 6347 = 8.38, A N 6371 = 8.27; cf. Table 4 in her paper). Although such an early B-type star is outside of the scope of this study, it is interesting to examine whether similar inconsistency emerges in our calculations at the higher T eff regime (> 15000 K). For this purpose, additional non-LTE calculations were performed for the log g = 4 models with extended T eff up to 20000 K. The resulting runs of l NLTE 0 /l LTE 0 and SL/B with depth for the Si i 6347/6371 lines (from T eff = 8000 K through 20000 K) are depicted in Figs. 10a and 10b; and Figs. 10c and 10d display how W6371 and ∆6371 (for the weaker line of the doublet) vary with T eff .
Our calculations suggest that the extent of the (negative) non-LTE abundance correction (|∆|) progressively decreases with an increase of T eff in the regime of Btype stars (T eff > ∼ 10000 K), until it eventually reaches ∆ ∼ 0 at the critical T eff of ∼ 19000 K; thereafter ∆ turns into positive (cf. Fig. 10d). In other words, the line is strengthened by the non-LTE effect (W N > W L ) at T eff < ∼ 19000 K while weakened (W N < W L ) at T eff > ∼ 19000 K, as can be confirmed in Fig. 10c.
As mentioned in Section 6.1, the behavior of ∆ is mainly controlled by the line source function; that is, as long as the inequality SL < B (SL dilution) holds in the line-forming region, ∆ remains negative. However, according to Fig. 10b, as T eff is ever increased, SL becomes comparable with or even outweighs B , which explains why ∆ approaches zero or even turns into positive at higher T eff (∼ 20000 K). Fig. 10d suggests that the non-LTE correction for ι Her (T eff = 17500 K) expected from our calculation is ∆6371 ∼ −0.1 dex, Then, since the LTE abundance is A L 6371 = 7.60 (cf. Table 4 in Mashonkina 2020a), the non-LTE Si abundance for ι Her would make A N 6371 ∼ 7.5, which is almost consistent with the solar abundance. Mashonkina's (2020a) ∆6371 vs. T eff relation (taken from Table 9 of her paper) is also overplotted for comparison in Fig. 10d. We can see from this figure that the upturn of Mashonkina's ∆6371 is considerably steeper and ∆6371 ∼ 0 is attained already at T eff ∼ 13000-14000 K, which is in marked contrast to our calculation (critical T eff for ∆6371 ∼ 0 is at ∼ 19000 K). The reason for this discrepancy is not clear. An inspection of the bottom panel of Mashonkina's (2020a) Fig. 1 (in comparison with our Fig. 10a) suggests that Si ii levels are largely underpopulated (presumably due to more enhanced Si ii overionization) in her calculation. We suspect that her procedure of evaluating UV photoionizing radiation field may have been rather different, for which we used the opacities included in Kurucz's (1993a) ATLAS9 program along with Kurucz's (1993b) line opacity distribution function as described in Section 3.1.2 of Takeda (1991). In panel (d), the T eff -dependence of ∆ 6371 calculated by Mashonkina (2020a) is also shown for comparison.