Chemical Abundances of Main-sequence, Turnoff, Subgiant, and Red Giant Stars from APOGEE Spectra. II. Atomic Diffusion in M67 Stars

We adopted the effective temperatures derived from ASPCAP DR14. We used the purely spectroscopic raw T_eff values from ASPCAP (given in the FPARAM array in DR14). For comparison, we also determined photometric temperatures by adopting the calibration of González-Hernández & Bonifacio (2009) and using five different colors, B – V, V – J, V – H, V – K_s, and J – K_s, with an adopted cluster reddening of $E(B-V)$ = 0.041 mag (Sarajedini et al. 2009) and a metallicity of [Fe/H] = 0.00. González-Hernández & Bonifacio (2009) provide photometric calibrations for red giant and dwarf stars; we adopted the coefficients for giants for those stars with log g < 4.00 dex and for dwarfs for those stars with higher gravities. Good agreement between the photometric and the adopted raw ASPCAP T_eff scales is obtained, where $\langle \delta ({T}_{\mathrm{eff}}(\mathrm{ASPCAP}-\mathrm{GHB})\rangle =-25\pm 106$ K. The effective temperatures obtained from the ASPCAP pipeline have an internal precision of ±50 K (Holtzman et al. 2015, García Pérez et al. 2016).

We determined surface gravities from the fundamental Equation (1), where the adopted T_eff values are from the raw ASPCAP DR14 values, with stellar masses and bolometric magnitudes obtained from interpolation in the MIST isochrones (Choi et al. 2016; [Fe/H] = 0.00; age = 4.00 Gyr; $E(B-V)$ = 0.041; distance modulus (μ) = 9.60). The adopted solar values are log g_⊙ = 4.438 dex, ${T}_{\mathrm{eff},\odot }=5772$ K, and ${M}_{\mathrm{bol},\odot }=4.75$, following the IAU recommendations in Prša et al. (2016):

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}g & = & {\mathrm{log}}_{10}{g}_{\odot }+{\mathrm{log}}_{10}\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right)\\ & & +\,4{\mathrm{log}}_{10}\left(\displaystyle \frac{{T}_{\star }}{{T}_{\odot }}\right)+0.4({M}_{\mathrm{bol},\star }-{M}_{\mathrm{bol},\odot }).\end{array}\end{eqnarray} \tag{ 1 }$$

We adopted the surface gravities derived from Equation (1) in the abundance analysis in this study. The uncertainties in the determined surface gravities are similar to the ones reported in Souto et al. (2018), where σ = ±0.10 dex. The comparison between the derived log g values in this work with those from ASPCAP confirms the log g offset, where we obtain $\langle$δ(log g(Physical-ASPCAP)$\rangle$ = −0.18 ± 0.16 dex for red giants, −0.16 ± 0.11 dex for subgiants, −0.19 ± 0.07 dex for turnoff, and 0.17 ± 0.13 dex for main-sequence stars.

Figure 3 (right panel) shows the T_eff–log g values adopted in this study. The effective temperatures for the studied stars are well spread out in the H-R diagram, with effective temperatures ranging between 4200 and 6250 K. The surface gravity values for the studied stars span a range in log g = 1.78–4.71.

In this work, we derive individual abundances for 15 elements: C, N, O, Na, Mg, Al, Si, K, Ca, Ti, V, Cr, Mn, Fe, and Ni. Individual abundances were determined with the qASPCAP code. The qASPCAP code basically corresponds to the ASPCAP pipeline, but for custom work, providing flexibility to change the analysis parameters. The methodology in the analysis is the same as adopted in ASPCAP, and the optimization is based on the FERRE code.

The procedure for determining individual abundances and microturbulent velocities with qASPCAP is similar to the one in ASPCAP. The ASPCAP pipeline (described in detail in García Pérez et al. 2016) uses a grid of synthetic spectra (Zamora et al. 2015) computed with the turbospectrum code (Alvarez & Plez 1998, Plez 2012) using KURUCZ model atmospheres (Castelli & Kurucz 2004, Mészáros et al. 2012) and the APOGEE DR14 line list, which is an updated version of the one published in Shetrone et al. (2015). The stellar parameters and chemical abundances are obtained by χ² minimization with the FERRE code (Allende Prieto et al. 2006) controlled by an IDL wrapper (the qASPCAP in this work).

In a first phase, seven parameters are determined through a 7D optimization (T_eff, log g, [M/H], [C/Fe], [N/Fe], [α/Fe], and ξ) using the entire wavelength range of the APOGEE spectra. During the second phase, individual abundances are obtained by repeating the fitting in predetermined windows that are sensitive to elemental abundances using the set of atmospheric parameters determined in the previous phase. It is possible to determine individual abundances for more than 26 elements from the APOGEE spectra; see Holtzman et al. (2018), Hasselquist et al. (2016, for Nd), and Cunha et al. (2017, for Ce). In this work, we adopt the same molecular and atomic lines as Souto et al. (2018) to derive individual abundances (see also Smith et al. 2013 and Souto et al. 2016). Even though Souto et al. (2018) reported Na and Cr abundances for main-sequence and turnoff stars, we opt in this work to not present these abundances (for these stellar classes) as the comparisons between the observed and synthesis were not satisfactory due to the weakness of the Na i and Cr i lines.

All M67 targets studied here have similar v sin (i), in the range 0 ≤ v sin (i) ≤ 7 km s⁻¹. In fact, the threshold to detect the star's v sin (i) from APOGEE spectra is 7–8 km s⁻¹. The effect of macroturbulence on the line profiles is similar to that of stellar rotation, and, as an approximation, qASPCAP treats rotation and macroturbulence as a single Gaussian profile.

The stellar parameters adopted in this work are shown in Table 2, with individual abundances presented in Table 3. The uncertainties in the derived abundances adopted in this work are the same as the ones reported in Table 4 of Souto et al. (2018). We note that, using ASPCAP calibrated abundances, the average $\langle \delta A(\mathrm{El})\rangle$ between the results derived in this work minus ASPCAP is smaller by 0.10 dex for all elements.

We split our sample into four different classes based on the stars' evolutionary stage. We selected as main-sequence stars those with log g ≥ 4.20, turnoff stars those with surface gravity 3.90 < log g < 4.20, subgiants those having 3.60 ≤ log g ≤ 3.90, and red giant stars those with log g < 3.60. (We note that the cut in surface gravity is similar to the one in color and magnitude, as can be seen in the right panel of Figure 1.)

Probing the level of homogeneity in open clusters is important to understanding their formation and for evaluating the possibility of performing chemical tagging in stellar populations. Chemical homogeneity in open clusters (as well as in globular clusters) is a critical assumption to understand changes in the abundances across evolutionary stages. Bovy (2016) and Price-Jones & Bovy (2018), using APOGEE spectra, found tight constraints on the chemical homogeneity of M67 using a sample of red giant stars. Bovy (2016) analyzed 24 red giant stars in M67, finding one-dimensional sequences with a spread in the elemental initial cluster abundances lower than 0.03 dex (2σ of uncertainty) for all elements studied in this work. It is worth noting that the Bovy (2016) results were derived in a way that is insensitive to the effects of atomic diffusion, mixing, and other physical processes that may modify the stellar surface abundances.

One straightforward way to evaluate if samples of stars have similar abundances is to apply a Kolmogorov–Smirnov test (K-S test). The K-S test is usually invoked to find out if two samples are drawn from the same distribution. We perform a study of chemical homogeneity of M67 stars using the derived abundances through a K-S test, and we apply it to the same classes, for example, red giants × red giants. To be able to compare the derived abundances for the same classes using the K-S test, we randomly split each group into two samples and then we apply the K-S test. To ensure we do not choose a random split that favors homogeneity, for each group, we have run the test in 1000 random splits. This result shows that the abundances of each stellar class are indistinguishable, with the derived median p value >0.50 for all elements in the four stellar classes. This is a complementary result to Bovy (2016), finding chemical homogeneity of M67 stars in the same evolutionary stage based on the stellar abundances derived in this work.

We also applied the K-S test using the derived abundances for the 15 studied elements comparing stars in the different groups: G dwarf main sequence (MS) × red giant, G dwarf (MS) × subgiant, G dwarf (MS) × turnoff, red giant × subgiant, red giant × turnoff, and subgiant × turnoff stars.

In Figure 4 we present the results of the K-S two-sided test comparing the individual abundances for each stellar class. The vertical axis represents the [X/H] derived here, and the horizontal axis represents the subgroups being compared. Each cell shows the p value of the K-S test and is colored as shown in the side color bar. We designed the color scale to give a blue color if the samples are clearly distinct, a yellow color if the p value is near 0.05, and a red color if we cannot reject the null hypothesis, that is, the samples are not distinguishable. Note that we have applied a false discovery rate (Benjamini & Hochberg 1995) correction in order to account for the fact that we are performing many hypothesis tests simultaneously, and spurious rejections of the null hypothesis are therefore expected. Regardless of the threshold that we use, we obtain outstanding segregation for red giant and turnoff stars based on their abundances. The K abundance is the one with higher p values (>0.03) for all scenarios. On the other hand, the two classes most difficult to separate based on their abundances are the main-sequence and the turnoff stars. The abundances of Mg, Ca, V, and Fe are the best ones to distinguish between these classes. The Mg abundances show significant differences among all stellar classes (with p values <0.10 for all comparisons).

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4.1.1. As a Function of Stellar Parameters

In Figure 5 we display the derived individual abundances as a function of surface gravity for the 15 elements studied. We use the same symbol notation as in Figure 1, but with open symbols instead of filled. We also show the line-by-line manual abundance results from Souto et al. (2018), our control sample. Atomic diffusion models computed for this work (see Section 6) are overplotted for each element (C and N including mixing processes). We note that the diffusion models for Na and Mg abundances were slightly shifted in order to better fit the observed abundances.

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From visual inspection—and in agreement with the results from the K-S test—we can organize the element variations as a function of surface gravity (as well as T_eff and M_⋆) into three groups of elements: (1) C and N, with abundances displaying a different behavior for the evolved subgiant and red giant stars (as a consequence of dredge-up mechanisms); (2) O, Na, and Cr as their abundances are not reliable for the main-sequence and turnoff stars because their spectral lines become too weak; (3) the elements showing a dip, either sinuous or small, in the elemental abundance close to log g = 4.00 dex (Fe, Mg, Al, Si, K, Ca, Ti, V, Mn, and Ni). The derived abundances of Mg, Al, and Si present the most significant changes between the stellar classes (excluding N), where the red giant abundances are 0.10 to 0.20 dex higher than those from the subgiants.

In Figure 6, we present the abundance results as a function of T_eff in M67 stars, with diffusion models also shown. Overall, the behavior seen in Figure 6 indicates an abundance increase (in the range 0.00–0.40 dex) as T_eff decreases from 6000 to 4000 K. The elements showing a smooth increase or decrease in abundance as functions of T_eff are Fe, Ca, and Mn. The elements most sensitive to T_eff, Na, Mg, Al, and Si, show a monotonic increase in their individual abundances. Similar to the trends with log g, C shows a particular behavior, and the abundance variation of N shows a maximum value around T_eff ∼ 4700 K and then decreases for higher and lower values of T_eff. The elements presenting the least sensitivity to T_eff are K, Cr, and Ni. Ti and V show the most significant abundance scatter in the analysis as a function of both log g and T_eff.

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Souto et al. (2018) showed that atomic diffusion processes can explain the abundance variations of M67 stars across the different evolutionary stages. However, other physical processes are also relevant in the context of abundance variations, where the most significant sources of deviations, not precisely in order, are non-LTE effects, 1D or 3D treatment of the model atmosphere, stellar rotation (v sin i), mixing process (e.g., first dredge-up), and atomic diffusion processes. In the following sections, we discuss the impact of these possible deviations in our results.

Deviations from the LTE have been studied mostly at optical wavelengths, where strong deviations are found to occur in metal-poor, evolved red giant stars (Asplund 2005; Asplund et al. 2009). In the near infrared (NIR), in particular in the H band, the works of Cunha et al. (2015) and Zhang et al. (2016, 2017) have investigated non-LTE effects in Na i, Mg i, and Si i lines in the APOGEE spectra, finding deviations from non-LTE in these elements to be usually smaller than 0.05 dex (see also the discussion in Souto et al. 2018). Using the results from Bergemann & Gehren (2008) and Bergemann et al. (2012a, 2013, 2015) compiled from a Maria Bergemann website (nlte.mpia.de), we created a grid of non-LTE deviations for five elements: Fe, Mg, Si, Ti, and Mn. The deviations were estimated for each stellar class, assuming a solar metallicity and T_eff = 4700 K, log g = 2.40, and ξ = 1.60 km s⁻¹ for red giants; T_eff = 5400 K, log g = 3.70, and ξ = 1.25 km s⁻¹ for subgiants; T_eff = 6100 K, log g = 3.90, and ξ = 1.15 km s⁻¹ for turnoff stars; and T_eff = 5850 K, log g = 4.40, and ξ = 1.00 km s⁻¹ for main-sequence stars. We adopted 1D plane-parallel models computed with MAFAGS-OS for all stellar classes. In Table 4 we summarize the average non-LTE correction for each stellar class and element.

In Figure 7 we show the non-LTE corrected abundances for the five elements studied (Fe, Mg, Si, Ti, and Mn). The top panel displays the abundance differences from [X/H]_non-LTE–[X/H]_LTE, and in the bottom panel we show a plot similar to Figure 5, but now using the [X/H]_non-LTE.

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The iron abundances do not show significant non-LTE deviations, as seen in Table 4, where δ(non-LTE–LTE) are smaller than 0.01 dex for all stellar classes. For Mg and Si, the deviation is very similar for main-sequence stars, both positive, being almost null for Mg. For subgiant and red giant stars, we obtain small, negative non-LTE corrections. The deviations for Ti and Mn are more significant in this study. For Ti, the deviations are positive for the stellar classes studied here, with the major deviation observed in turnoff stars (δ(non-LTE–LTE) = 0.11 dex). When applying non-LTE corrections, we do not see a strong change in the abundance versus log g diagram, when compared to the LTE one presented in Figure 5. The abundances of Ti are shifted in all classes, resulting in a higher scatter as a function of log g. The Mn corrections show the most significant differences, δ(non-LTE–LTE) ∼ 0.13 dex for main sequence, turnoff, and subgiants, and δ(non-LTE–LTE) ∼ 0.30 dex for red giants. The inclusion of non-LTE corrections in the analysis does not erase the observed abundance trends in the different stellar classes.

Stellar atmospheres are 3D and time-dependent; however, by convenience, we usually treat model atmospheres as having 1D plane-parallel or spherical geometry in hydrostatic equilibrium. This approximation simplifies the analysis, but can lead to systematic errors in the derived quantities (atmospheric parameters or chemical abundances).

The use of a 1D treatment of the stellar atmosphere requires the inclusion of "ad hoc" parameters to account for velocities that broaden the profiles at microscopic (microturbulence) and macroscopic (macroturbulence) levels. A precise determination of the microturbulence parameter minimizes the deviations from the results obtained with 3D models.

As in non-LTE studies, 3D effects are also transition dependent, and analyses for NIR H-band transitions have been limited. The studies of Asplund (2005), Asplund et al. (2009), and Caffau et al. (2011) have summarized various effects and corrections for elemental abundances using optical spectra as a reference. In this section, we will summarize these effects for solar-metallicity stars to verify whether the abundance trends discussed in Section 4 could be explained by 3D effects.

Caffau et al. (2011) determined solar abundances from a 3D non-LTE analysis using the CO⁵BOLD code, providing 3D abundance corrections for several elements. For Fe, Caffau et al. (2011) find 3D corrections to be about 0.03 dex using the solar spectrum. The C abundance reported by Caffau et al. (2011) has a −0.02 dex 3D correction, while for K the authors obtain a correction of 0.07 dex. From Caffau et al. (2009), the solar N abundance is reported to have a 3D correction smaller than 0.01 dex.

The previous work by Bergemann et al. (2012b) studied the 3D deviations for stars in different evolutionary stages at different metallicities. They find 3D effects in the iron abundance for the Sun to be very small: 3D corrections ∼0.01 dex. More recently, Bergemann et al. (2017) studied the Mg abundances in the Sun and found 3D corrections to be ∼0.02 dex. Amarsi & Asplund (2017) studied non-LTE 3D Si abundances in the Sun and found corrections to be lower than 0.01 dex. Amarsi et al. (2016), analyzing the O i forbidden line at 630 nm, find 3D corrections to the O abundance to be between 0.05 and 0.20 dex, negative in the Sun and reaching higher values for turnoff stars.

All 3D corrections discussed above are smaller than 0.05 dex (except for K), which is at the limit of the measurement uncertainties of this work. Given the small 3D corrections found for main-sequence stars, as well as the lack of studies in the literature for turnoff, subgiants, and red giant stars at solar metallicity, we conclude that deviations from 3D modeling are not enough to explain the abundance trends observed in this work.

The study of the relation between stellar rotation and abundance variations in late-type stars is often motivated by the investigation of lithium depletion. Several authors have found correlations between stellar rotation and the lithium abundance depletion, such as Balachandran (1990, 1995), King et al. (2000), da Silva et al. (2009), Canto Martins et al. (2011), and Delgado Mena et al. (2014). None of the spectra analyzed in this study exhibit measurable rotational broadening (v sin(i)) above the limits set by the APOGEE spectral resolution of ∼7–8 km s⁻¹.

We computed our mixing and atomic diffusion models using solar models (solar metallicity and solar age 4 Gyr) to calibrate the degree of gravitational settling precisely (using the surface solar He as a proxy); this gives a predicted reduction in the efficiency of the settling of 15%, or an effective coefficient of 0.85. The methodology adopted in the modeling of mixing and atomic diffusion is described in detail in Bahcall et al. (2001) and Delahaye & Pinsonneault (2006). We note that, overall, our models agree with the ones from MIST (Choi et al. 2016, Dotter et al. 2017); however, our models cover all of the species studied in this work, while the MIST models are not available for Al, K, V, Cr, Mn, and Ni.

When a low-mass star, such as a ∼1.2M_⊙ M67 star that is currently evolving off of the main sequence and across the subgiant branch, reaches the base of the RGB, the outer convective envelope reaches its largest extent in mass. At this point in the H-R diagram (where T_eff ∼ 5000 K and log g ∼ 3.5 in M67), the base of the convective envelope ingests material that has been exposed previously to H burning via the CN cycle. As a consequence of CN-cycle H burning, this nuclear-processed material contains an enhanced abundance of ¹⁴N and a decreased abundance of ¹²C. The convective envelope will carry this mixture to the surface, resulting in a lower surface abundance of ¹²C and a larger abundance of ¹⁴N for stars evolving onto the RGB; this phase of stellar evolution is referred to as first dredge-up, or FDU (Iben 1965; for a more recent overview of the various red giant dredge-up episodes, see Karakas & Lattanzio 2014). In the case of dredge-up in M67 red giants, the ¹⁴N abundance is predicted to be enhanced by roughly ∼+0.30 to +0.40 dex, while the ¹²C abundance is predicted to be depleted by ∼−0.10 to −0.20 dex. The magnitudes of the abundance changes in C and N are a function of red giant mass (Iben 1965), with larger-mass stars having deeper convective envelopes that dredge up more nuclear-processed material, resulting in larger ¹⁴N enhancements and larger ¹²C depletions, producing lower C/N ratios.

The expected relationship between red giant mass and C/N ratio has been exploited by a number of recent studies using APOGEE data and results (e.g., Martig et al. 2016; Ness et al. 2016; see also D. Feuillet et al. 2019, in preparation) to produce age–mass relations as a function of red giant [C/N] abundances, while Masseron & Gilmore (2015) have analyzed [C/N] to study the possible formation of the thin and thick disk.

In addition to standard convection in 1D, other physical processes can modify the interior abundance profiles in stars as they evolve from the main sequence, across the subgiant branch, and onto the RGB, with two important processes being rotation and the inversion of the mean molecular weight gradient in a small region outside the H-burning shell created by ³He burning via ³He(³He, 2p)α (Eggleton et al. 2006; Charbonnel & Zahn 2007); this last process is referred to as thermohaline mixing. The inclusion of rotation-induced mixing and thermohaline mixing produces larger carbon depletions and larger nitrogen enhancements as a result of FDU. In this section, we use ¹²C and ¹⁴N abundances derived here to compare with predictions from various models of first dredge-up mixing.

As shown in Figure 8, the M67 red giants display clear evidence of the first dredge-up through the behavior of the C and N abundances as functions of both T_eff and log g (which map the position of a star along the subgiant and RGB); observed APOGEE abundances are plotted as the various symbols, while models are plotted as the continuous lines and are models from this study, along with those from Lagarde et al. (2012). The left panels of Figure 8 plot the [C/N] values versus T_eff (top, (a)) and log g (bottom, (b)), with the [C/N] values decreasing rapidly at T_eff ∼ 5000 K and log g ∼ 3.5, right at the base of the RGB as predicted by FDU. The right panels plot the individual abundances of ¹²C (as [C/H]) and ¹⁴N (as [N/H]) versus log g. Carbon and nitrogen abundance differences between red giants on the RGB relative to those in the RC were found to agree with results from Tautvaišiene et al. (2000) and Masseron et al. (2017), who found slightly lower values of C/N in RC stars compared to those on the RGB. Our values for M67 stars are $\langle$¹²C/¹⁴N${\rangle }_{\mathrm{RGB}}=1.86$ and $\langle$¹²C/¹⁴N${\rangle }_{\mathrm{RC}}=1.40$, excluding the two evolved stars with log g < 2.1 dex, which places them on the upper RGB or possibly in an early-AGB phase of evolution.

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Figure 8 also highlights differences in the C and N abundance variations predicted from mixing models when compared to those abundances derived in this study. In the left panels of Figure 8 (a) and (b), we show the [C/N] ratio as a function of T_eff and log g, respectively, and we note that the overall observational results follow the model predictions, although the observed [C/N] values are systematically lower. Such a difference can be a consequence of an overestimated nitrogen abundance in our analysis (as pointed out by Bertelli Motta et al. 2017 using ASPCAP data), due to a subestimated log g. In the right panels of Figure 8, we present the [C/H] (panel (c)) and [N/H] (panel (d)) abundances as a function of log g. For nitrogen, the abundances are in agreement with the models; however, the observational carbon abundances differ from the models by ∼−0.15 dex. We conclude that the abundance variations observed for ¹²C and ¹⁴N in the subgiant and red giant stars can be explained well by FDU mixing models. The mixing models here (as well as from Lagarde et al. 2012) predict changes for the other elemental abundances to be smaller than 0.01 dex as the star evolves. Therefore, mixing models cannot explain their abundance variations.

Atomic diffusion is a likely explanation for most of the observed abundance variations across the H-R diagram in M67, thus adding members of this old open cluster to those stars in which diffusion has been observed. Evidence of diffusion in the Sun is found both in its surface helium abundance, which is lower than the initial value, and in the solar sound speed profile being best fit by models that include diffusion (Bahcall et al. 1995; Chaboyer et al. 1995a). Lithium abundances settle at a rate similar to He, and the flatness of the Spite Li plateau is likely set by diffusion (Chaboyer et al. 1992). The diffusion signature can be altered or erased by mixing, for example, mixing driven by rotation and dredge-up (see Section 6.2), thus complicating the detection and interpretation of diffusion patterns. Such mixing processes are likely at work in the Sun, which has a smoother composition profile than that predicted by diffusion alone, with the magnitude of diffusion being overestimated by about 25%. This is also confirmed by looking at A-type stars: if they rotate fast enough, they are not chemically peculiar (Michaud 1970, see also Michaud et al. 2015). The interplay between diffusion creating abundance signatures that various mixing processes can then modify means that there are not necessarily firm theoretical predictions about the amplitude of the diffusion signature and its mass or metallicity dependence. Reasonably well-motivated trends can be expected, though, and Chaboyer et al. (1995b, 1995c), Choi et al. (2016), Dotter (2016), and Dotter et al. (2017) have approximated limits on how efficient diffusion can be in thin surface convective zones.

A few previous studies have probed atomic diffusion in cluster stars, with most of them focused on low-metallicity globular clusters: Korn et al. (2007), Lind et al. (2008), and Nordlander et al. (2012). The latter analyzed stars belonging to the globular cluster NGC 6397, with a metallicity [Fe/H] = −2.00 and age 13.5 Gyr, with their sample containing stars from the turnoff point (TOP) up to the RGB. The abundances of Li, Mg, Ca, Ti, Cr, and Fe in those studies were derived from high-resolution optical spectroscopy, and they found that changes in the stellar abundances for different evolutionary phases are in good agreement with predictions from diffusion models from the literature; see Richard et al. (2002, 2005). In particular, Nordlander et al. (2012) found abundance differences of 0.06 and 0.18 dex between TOP and RGB stars in NGC 6397, with the largest difference for Mg, and which is a much smaller variation than we see in M67, for example for Mg or Al. Of course, the chemical abundance of NGC 6397 is rather distinct from that of M67.

Önehag et al. (2014) found some evidence of atomic diffusion operating in M67. This was later supported by the abundance results in Blanco-Cuaresma et al. (2015). As previously mentioned, abundance differences of up to ∼0.20 dex between the turnoff and red giant stars were observed by Souto et al. (2018) for the elements O, Na, Mg, Al, Si, Ca, V, Mn, and Fe. A comparison of the results in Souto et al. (2018) with the diffusion patterns from the models by Choi et al. (2016) and Dotter et al. (2017) indicated good agreement. In addition, Bertelli Motta et al. (2018) found >0.15 dex abundance differences from main-sequence to red giant stars for the elements Al, Si, Mn, and Ni. Gao et al. (2018) found a good match between the abundance variations for Al and Si with diffusion models.

We find significant abundance differences (up to ∼0.50 dex) for most of the studied species between main-sequence, turnoff, subgiant, and red giant stars in M67. Using the K-S test (Figure 4), we obtained clear evidence of abundance differences between stars in different evolutionary stages in M67. In addition, we showed (Section 4) that the abundances of stars belonging to the same evolutionary class are indistinguishable.

In Figure 9 we present the mass–Δ[X/H] ([X/H]_Current–[X/H]_Initial) diagram for the 12 studied elements. Similar diagrams with abundances as a function of surface gravity and effective temperature are presented in Figures 5 and 6. In all three figures, we show the atomic diffusion models computed in this work as solid black lines and the MIST models as brown dashed lines. The pristine Fe abundance in the models is assumed to be the mean Fe abundance for the red giants, which is used as the fiducial point (i.e., δ[Fe/H] = 0.00) for the initial cluster value. We note that all other abundance ratios are assumed to be solar, that is, [X/Fe] = 0.00. The abundance variations across the H-R diagram indicate that atomic diffusion is operating in most of the studied elements. The models for all of the elements display similar trends driven by atomic diffusion, except for C and N, which include mixing signatures.

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The complex trend observed in the carbon abundances is a consequence of diffusion operating in the main-sequence and turnoff stars (smaller convective envelopes), and mixing at the first dredge-up being responsible for the carbon abundance variation in subgiant and red giant stars (Section 6.2; Figure 8). These results suggest that atomic diffusion dominates over mixing in the main-sequence and turnoff stars, while mixing processes control the abundance changes in subgiant and red giant stars.

The nitrogen abundance variation can be explained as a signature of first dredge-up (Section 6.2). For oxygen, the scattered abundance results for red giant and subgiant stars, combined with the lack of results for main-sequence and turnoff stars, impedes detection of signatures of diffusion. Due to the weakness of CN and OH molecular lines in the APOGEE spectra of main-sequence and turnoff stars, it is not possible to derive N and O abundances in such stars.

The comparison of the abundance patterns for all elements with the model predictions indicates an overall good match between the atomic diffusion models and the derived abundances across the H-R diagram. However, the derived abundances exhibit a more significant dip across the main-sequence—turnoff stars when compared to what is expected from the atomic diffusion models, in particular for Mg, Al, Ti, and Mn.

For Al, Mg, Si, and to a lesser degree V, the relative dip across the main-sequence—turnoff stars is more significant because the red giant abundances are higher than those predicted by the models. (The Na abundances of red giants are also higher than the models, but there are no abundances for turnoff and main-sequence stars.) On the one hand, for Ti and Mn, the dip is more considerable because the abundances of turnoff and main-sequence stars are lower. As discussed in Section 5 (Figure 7), non-LTE corrections for Mg and Si (as well as Fe) would reduce the abundance dip by a factor of ∼0.03 dex, while for Ti, the dip would be reduced by roughly 0.05 dex. The non-LTE corrections for Mn, on the other hand, would systematically change the red giant abundances and increase the abundance difference between turnoff and red giant stars by ∼0.14 dex, which would worsen the comparison with the models.