The Presolar Grain Database. I. Silicon Carbide

In our evaluation of the old PGD, we found that many grains had multiple entries in the database, often because they had been analyzed in various measurement sessions, sometimes in different laboratories, and often using inconsistent grain labels. We therefore introduced a new specific label for each grain in the database called PGD ID, which is not meant to replace a grain label defined by the original authors, but to prevent confusion from inconsistent nomenclatures used. The PGD ID (e.g., SiC-2001-AMA-000001) consists of four sections separated by hyphens. The first section denotes the phase (e.g., SiC). The second part is the year data for the grain appeared for the first time in a publication (0000 for data not related to any publication). In general, we try to avoid citing conference abstracts and refer to refereed publications. However, there are some exceptions where data have never appeared in any refereed literature, so some extended abstracts had to be cited. In the third section, the first three letters of the first author's surname are given (e.g., AMA stands for Amari). Should the first author have a two-letter surname, the first letter of the first name is added after the surname. The fourth part is a six-digit number denoting the individual grain. If a first author has published more than one paper with grain data in one year, we skip numbers to the next 100,000 in the fourth part of the PGD ID (e.g., SiC-2001-AMA-100001 denotes the first grain of a series described by Amari et al. in a second paper from 2001). A specific ID will only be assigned once. If it turns out in the future that one grain has more than one ID, the data will be consolidated, and the obsolete IDs will be retired. A list of such obsolete PGD IDs will be provided as part of the PGD, and all changes to the PGD will be documented. PGD IDs will only be assigned and approved by the person managing the PGD (currently Thomas Stephan) and not by any researcher submitting data to be included in the PGD. We follow this rule strictly to avoid any confusion or use of unauthorized IDs.

SiC grains have been classified into many types, also referred to as populations or subgroups, such as mainstream, X, Y, Z, etc. (e.g., Anders & Zinner 1993; Davis 2011; Zinner 2014). However, over the last three decades, type definitions have changed, and new types have been discovered. We also came to the realization that different researchers have used slightly different definitions for the various types. Therefore, we tried to unify these different definitions and come up with an automated type assignment based on carbon, nitrogen, silicon, and aluminum isotope ratios. We call this the PGD Type as it should not be confused with the type assigned in the original study. Since there are many grains for which only partial data are available, we have also set rules about how to deal with missing data. The exact type definitions used and the probable stellar sources they are associated with are discussed in Section 3.

As will be discussed in Section 3, some types (X, AB, C) have been further subdivided into subtypes (e.g., X0, X1, and X2). Analogous to the PGD Type, we therefore also assign a PGD Subtype wherever possible. If no subtype is defined or cannot be determined from the available data, this field remains empty.

The type, and in some cases subtype, assignments made in the original studies are given in the field labeled Type. For approximately 95% of all SiC grains in the PGD, the original type assignment agrees with the PGD Type. However, we encourage the community to use the PGD Type in future studies as it unifies previous, often inconsistent definitions, which were typically based on limited data sets.

The probabilities that a grain is consistent with the definition of a given PGD Type, p(M), p(X), p(Y), p(Z), p(AB), p(C), p(D), and p(N), are provided in the PGD. They are calculated according to mathematical rules laid out in Section 3 and the Appendix.

It is important to mention that the probabilities for being consistent with the various types do not add up to 1, because these terms describe the probability of a grain being consistent with a specific grain type but not the probability of being of that grain type. A grain can be consistent with more than one grain type due to overlap in their definitions and uncertainties in the grain data.

The Grain Label is the name a grain was given in the original study or published article. In cases where different names have been given to the very same grain, all names are provided. While the grain label is usually the name a grain should be referred to when mentioning a grain from a specific study, these labels are in some cases not unique. Therefore, when grain data from different studies are discussed, the PGD ID can be used as a unique identifier.

Reference refers to the main publication where data for the grain has usually been reported for the first time. It is also the reference that has been used for parts two and three of the PGD ID. We provide in this field the reference in an abbreviated format. In addition, we provide a table where the full references are given as part of the PGD. In cases where additional isotope data have been provided in other publications, these are mentioned as part of the Notes (see Section 2.12).

The field Data Published has three possible entries. Here, Full means that the complete data shown in the PGD for the specific grain have been published. No is used in the case where no tabulated grain data have been published. This could still mean that the data is associated with a specific publication, e.g., only used for a plot, but none of the individual grain data has been tabulated in the paper nor is any supplementary material available to the reader. Partial stands for the case where only part of the data, i.e., data only for some elements but not for others, have been published. Unpublished data were provided directly by the authors to the PGD as spreadsheets. Even where data are labeled Full, we often relied on data from unpublished spreadsheets, as they contain more digits than are available in the published literature. This became especially important in cases where only rounded numbers had been published, not providing all significant digits.

Source usually names the meteorite the presolar grain was found in or extracted from. There are a few exceptions, e.g., for grains from the Hayabusa2 mission, for which the source is the asteroid Ryugu, and grains from interplanetary dust particles, given as IDP.

Technique names the laboratory or laboratories and instrument(s) where the data have been collected. In the main spreadsheet of the PGD, we use an abbreviated form, which is further explained in a separate spreadsheet also provided.

The size of a grain is given in micrometers in those cases where this information is available. This could be a single number, given as Size a, referring to the longest dimension or an average dimension. In some cases, two dimensions are given, Size a and Size b, which usually refer to the longest dimension and the dimension perpendicular to it. There is no real standard for how these were determined. Therefore, users of the PGD should exercise caution in using this information and refer to the original publication if needed.

The Notes give additional references if part of the data was published separately. Other additional information about any specific grain is also mentioned here, including information about typographical errors found in the literature.

The following fields contain the Isotope Data for each grain. The number of SiC grains with isotope data for a specific element in the PGD are shown in Figure 1. Isotope data that have been measured on SiC bulk samples instead of single grains have not been included in the PGD. Some isotope data on single grains for elements like hafnium, tungsten, and europium have not been included as they have been measured only in very few grains, which showed in almost all cases no significant deviation from solar isotope ratios for these elements (Ávila et al. 2012, 2013).

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For ⁷Li/⁶Li, ¹¹B/¹⁰B, ¹²C/¹³C, ¹⁴N/¹⁵N, ²⁶Al/²⁷Al, ⁴⁴Ti/⁴⁸Ti, and ⁵¹V/⁴⁸Ti, data are provided as atomic ratios. For all other ratios, δ-values are used, giving the permil deviations of an isotope ratio ⁱ A/^j A measured in a grain from that measured in a standard typically representing terrestrial isotope ratios:

$$\begin{eqnarray}&&{\delta }^{i}{\rm{A}}=\delta \left({}^{i}{\rm{A}}{/}^{j}{\rm{A}}\right)=\left[\displaystyle \frac{{\left({}^{i}{\rm{A}}{/}^{j}{\rm{A}}\right)}_{\mathrm{grain}}}{{\left({}^{i}{\rm{A}}{/}^{j}{\rm{A}}\right)}_{\mathrm{standard}}}-1\right]\times 1000{\%} .\end{eqnarray} \tag{ 1 }$$

While different terrestrial standards are used in different laboratories, isotopic variations among terrestrial materials are considered negligible when compared to isotopic anomalies in presolar grains and taking into account typical precisions in grain analysis. Therefore, grain data are not renormalized to compensate for any differences in standards.

Wherever original data were available, we did not round the data to any specific number of digits but provide all digits reported.

Errors are given as 1σ uncertainties as provided in the original studies, again with all digits reported. For carbon and nitrogen isotope ratios, we provide asymmetric errors σ⁺ and σ⁻ if these are available. Such asymmetric uncertainties can include infinity, which occurs when the measured intensity of the denominator isotope is less than 1σ from zero. In such cases, we use 10⁶ as the positive uncertainty and mention this in the notes. In some grains from two studies (Hoppe et al. 1996a, 1996b), zero counts were detected for ¹⁵N. In these cases, Hoppe et al. (1996b) assumed 0.69 counts for ¹⁵N, calculated the ¹⁴N/¹⁵N ratio, and reported uncertainties of zero (positive and negative). We reported these numbers as in the original studies but mentioned in the notes that zero counts were detected for ¹⁵N.

In some other cases, uncertainties were reported as zero in the original studies due to rounding the data to a certain number of digits. If a number reported with zero uncertainty is rounded to n digits behind the decimal point, we assumed an uncertainty of 0.5 × 10⁻ⁿ . For example, a ¹²C/¹³C ratio of 3.0 ± 0.0 reported for grain SiC-1993-ALE-000092 (grain C23 from Alexander 1993) is now listed as 3.0 ± 0.05, which can be taken as an upper limit for the uncertainty. This avoids mistakes that could arise from applying zero uncertainty, which would have no physical meaning in this context. However, to be fully transparent here, we also provide notes specifying how the errors were originally reported.

For some elements, currently silicon and molybdenum, error correlation coefficients for isotope δ-values are provided for some grains. These correlation coefficients, designated as rho (ρ) in the PGD, were typically not given in the original publication but could be retrieved from laboratory records. In the case of silicon, this error correlation typically stems from variations in instrumental mass fractionation leading to correlated errors. Molybdenum isotope ratios have substantial error correlation due to the normalization to ⁹⁶Mo, whose uncertainty contributes significantly to the overall uncertainties of these ratios. Error correlation leads to error bars or error ellipses that are not parallel to the major axes of a three-isotope plot. The details of error correlation in three-isotope plots have been discussed by Stephan & Trappitsch (2023b).

Aluminum isotope data are inferred ²⁶Al/²⁷Al ratios, as there is no live ²⁶Al (half-life 0.717 Ma) found in these grains. The ²⁶Al concentration at the time the SiC grain formed is inferred from the concentration of its decay product ²⁶Mg. There is usually a negligible amount of ²⁴Mg and ²⁵Mg in presolar SiC grains, and any detection of these isotopes is typically attributed to terrestrial or asteroidal contamination. Magnesium-26 is corrected for such contamination by assuming solar magnesium isotopic ratios for the contaminant. Large variations in ²⁵Mg/²⁴Mg have been reported in presolar silicates (Hoppe et al. 2022), but such variations have not been found in presolar SiC, for which contamination is the major source of magnesium. Another contaminant that could affect inferred ²⁶Al/²⁷Al ratios is stable ²⁷Al, leading to underestimated ²⁶Al/²⁷Al ratios (Groopman et al. 2015). To minimize contributions from contamination, ion imaging and/or depth profiling can be used to avoid sampling any meteoritic or other material not part of the SiC grains (Groopman et al. 2015; Liu et al. 2021; Hoppe et al. 2023). To assess the role contamination could have played in calculating inferred ²⁶Al/²⁷Al ratios, users of the PGD are encouraged to consult the original literature, which could provide further information on data corrections applied. However, one major caveat remains, since, for calculating an inferred ²⁶Al/²⁷Al ratio, relative sensitivities between magnesium and aluminum have to be determined. This would ideally require measuring a SiC standard with a significant and known Mg/Al ratio. As such standards do not exist, and oxygen-rich standards are used instead, there remains significant uncertainty about the accuracy of reported ²⁶Al/²⁷Al ratios (Liu et al. 2021; Hoppe et al. 2023).

Similar to the case of ²⁶Al measured as ²⁶Mg, the presence of ⁴⁴Ti (half-life 59.1 yr) in presolar grains is inferred from its stable decay product ⁴⁴Ca and reported as ⁴⁴Ti/⁴⁸Ti ratios in the PGD.

Vanadium data are reported as ⁵¹V/⁴⁸Ti ratios as the low-abundance isotope ⁵⁰V has not been measured in such grains. The reported ratio might be helpful for assessing the presence of ⁴⁹V (half-life 330 days) as a possible source of some ⁴⁹Ti in some grains (Hoppe & Besmehn 2002; Lin et al. 2010; Liu et al. 2018a).

The vast majority of grains are so-called mainstream grains, labeled with M in the PGD. They are defined by having ¹²C/¹³C ratios between 13.5 and 100, ¹⁴N/¹⁵N ratios above 200, inferred ²⁶Al/²⁷Al ratios lower than 0.02, and silicon isotopes that fall along the so-called mainstream line in the silicon three-isotope plot. These areas and mainstream grain data are shown in Figure 2.

We chose a ¹²C/¹³C ratio of 13.5 to separate mainstream grains from AB grains since it represents a minimum in a histogram of ¹²C/¹³C ratios of mainstream, AB, and Y grains (Figure 3). However, since the distribution of ¹²C/¹³C ratios is rather flat between 13.5 and 25 (Figure 3), grains with ratios up to 25 could still be classified as AB grains, if nitrogen or inferred aluminum isotope ratios exclude them from being identified as mainstream grains. The upper limit of 100 for the ¹²C/¹³C ratio, which separates mainstream grains from Y grains, cannot be inferred from the distribution shown in Figure 3 but was arbitrarily defined and has been used consistently throughout the literature (e.g., Hoppe & Ott 1997; Amari et al. 2001c; Nittler & Alexander 2003; Zinner 2014). As discussed in Section 3.3, there seems to be a more gradual difference between mainstream and Y grains. Adopting a value of 100, significantly above the terrestrial ¹²C/¹³C ratio of 89, ensures that mainstream grains compromised by terrestrial or meteoritic carbon contamination are not falsely classified as Y grains. Such contamination is probably responsible for the minor peak around the terrestrial value in Figure 3. While there is some indication that the solar ¹²C/¹³C ratio is slightly higher, ∼93.5, than the terrestrial value (Lyons et al. 2018), such subtle differences play no role here in the context of presolar grains. For the mainstream grains, we observe a maximum in the distribution of ¹²C/¹³C ratios at ∼52 (Figure 3).

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Parameters for the mainstream line in the silicon three-isotope diagram have been determined iteratively, calculating the weighted regression line (Stephan & Trappitsch 2023b) through all grains initially classified as mainstream grains and then reclassifying the grains using the newly calculated regression line. After a few iteration steps, the line parameters converged to

$$\begin{eqnarray}&&{\delta }^{29}\mathrm{Si}=-(19.0\pm 0.2){\%} +(1.342\pm 0.004)\times {\delta }^{30}\mathrm{Si},\end{eqnarray} \tag{ 11 }$$

which is based on data of 15,606 mainstream grains.

These parameters are in good agreement with previously reported values (Hoppe et al. 1994; Lugaro et al. 1999; Nittler & Alexander 2003; Zinner et al. 2007; Zinner 2014).

The MSWD value of the regression line is 2.433. An MSWD value significantly greater than 1 shows that silicon isotopes in mainstream grains do not form a simple line but might be interpreted as a band, with a certain width.

We define the distance in δ³⁰Si from the mainstream regression line as

$$\begin{eqnarray}&&{{\rm{\Delta }}}^{30}\mathrm{Si}={\delta }^{30}\mathrm{Si}-\displaystyle \frac{\left({\delta }^{29}\mathrm{Si}+19.0{\%} \right)}{1.342}.\end{eqnarray} \tag{ 12 }$$

Most mainstream grains plot in a band defined by –100‰ < Δ³⁰Si < +100‰ about the mainstream regression line (Figures 2(b) and 4). However, there are 58 grains with more negative Δ³⁰Si values that we also include in our definition of mainstream grains, because they seem to be part of the distribution shown in Figure 4. We, therefore, define –250‰ < Δ³⁰Si < +100‰ as the range for mainstream grains in Δ³⁰Si. The corresponding borders are shown as lines M1 and M2 in Figure 2(b). Grains that plot to even more negative Δ³⁰Si values are not considered part of the distribution and are classified as type D grains (see Section 3.8). The upper limit of Δ³⁰Si < +100‰, which separates mainstream grains from Z grains (see Section 3.4), cannot be inferred from the distribution shown in Figure 4 but was arbitrarily defined. It ensures that a large fraction of grains previously classified as Z grains are still classified as such but also ensures that many grains whose classification as Z grains seemed questionable are now classified as mainstream grains.

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There are two lines perpendicular to the regression line that mark the lower and upper limits for mainstream grains along the band (Figure 2(b)). They cross the regression line (M0) at δ³⁰Si = +200‰ (M3) and δ³⁰Si = –75‰ (M4). Parameters for lines M0–M4 and details on how to calculate probabilities are given in the Appendix.

As discussed extensively in the literature, carbon, nitrogen, and inferred aluminum isotope ratios in mainstream grains can be roughly explained by an origin in low-mass, carbon-rich asymptotic giant branch (AGB) stars with approximately solar metallicity (e.g., Nittler & Alexander 2003; Zinner 2014). The silicon isotopic signature of mainstream grains, however, cannot be explained by nuclear processes in AGB stars (Zinner 2014). The spread along the mainstream line is therefore often explained by Galactic chemical evolution (Gallino et al. 1994; Clayton & Timmes 1997). Other explanations include mixing of different regions of the Galaxy (Clayton 1997), local heterogeneities in the Galaxy caused by contributions from Type Ia and Type II supernovae (Lugaro et al. 1999; Nittler 2005), and a merger of the Milky Way with a satellite galaxy (Clayton 2003).

Low-mass AGB star models predict ¹⁴N/¹⁵N ratios that are typically more than 10 times higher than the value of 200 (Palmerini et al. 2011) the lower limit adopted for mainstream grains in this work. While values <200 reported previously (e.g., Zinner 2014) were due to errors in earlier versions of the PGD, the corrected database suggests a lower limit of ∼200. One reason for low ¹⁴N/¹⁵N values is contamination with terrestrial or asteroidal nitrogen as demonstrated by Liu et al. (2021). However, this would not explain values below the terrestrial ¹⁴N/¹⁵N ratio of 272 as seen in Figure 2(a). Another scenario, suggested by Palmerini et al. (2011), involves contamination with ¹⁵N produced by cosmic-ray spallation.

Type X grains (Figure 5) span a wide range in carbon isotopic ratios, and their ¹⁴N/¹⁵N ratios are typically significantly below the terrestrial ratio of 272. Using the terrestrial ¹⁴N/¹⁵N ratio as an upper limit ensures that nitrogen contamination could not lead to a misclassification of X grains. Inferred ²⁶Al/²⁷Al ratios are usually higher than 0.01. However, most diagnostic for X grains are their highly negative δ³⁰Si and δ²⁹Si values. Since there are other types, e.g., mainstream, Y, and AB grains, that could also show slightly negative δ³⁰Si and δ²⁹Si values, distinguishing these grains from some X grains can sometimes be difficult. In such cases, carbon, nitrogen, and aluminum isotopic information can be useful for a proper identification. To prevent grains from being misclassified as X grains, we lower their probability of being consistent with a classification as X grains if they plot in the region where they could also be identified as mainstream, Y, or AB grains (lighter colored area in Figure 5(b); for details, see the Appendix).

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Type X grains are attributed to core-collapse (Type II) supernovae (SN II; Amari et al. 1992; Hoppe et al. 1996c, 2000; Nittler et al. 1996).

Based on silicon isotopes, X grains have been subdivided into three subtypes (Lin et al. 2010). Most X grains (61%) plot along a line with a slope of ∼2/3 in the three-isotope plot and are called X1 grains. X0 grains (6%) plot significantly above and X2 grains (33%) significantly below that line. Strontium and barium isotopic analyses suggest that X1 and X2 grains represent two different populations among SN II grains, representing different timing and mixing conditions (Stephan et al. 2018; Ott et al. 2019).

For our classification, we used the following two lines (Figure 5(b)) as dividers between the subtypes:

$$\begin{eqnarray}&&{\delta }^{29}\mathrm{Si}=\pm 30{\%} +\left(\displaystyle \frac{2}{3}\mp 0.05\right)\times {\delta }^{30}\mathrm{Si}.\end{eqnarray} \tag{ 13 }$$

Further details are given in the Appendix.

Type Y grains (Figure 6) are defined based on their high ¹²C/¹³C ratios and are separated from mainstream grains by having ratios above 100. In nitrogen and inferred aluminum isotopes, they are indistinguishable from mainstream grains. In the silicon three-isotope diagram, Y grains mostly plot within the area defined by the mainstream grains but also reach into the area below and to the right of the band for mainstream grains, which is mostly populated by Z grains (see Section 3.4). Therefore, the definition in silicon isotopes used for Y grains combines the areas of those two types. Further details are given in the Appendix.

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It has been suggested that Y grains form their own correlation line with a smaller slope than the mainstream line: Hoppe et al. (1994) gave a slope of 0.35 and an intercept of −22.5 for a best fit through data from five Y grains, while Nittler & Alexander (2003) calculated a slope of 0.64 for 113 Y grains. Based on data from 759 Y grains currently available in the PGD, we calculate the following line:

$$\begin{eqnarray}&&{\delta }^{29}\mathrm{Si}=-(26.5\pm 1.2){\%} +(0.644\pm 0.028)\times {\delta }^{30}\mathrm{Si},\end{eqnarray} \tag{ 14 }$$

which seems to be in good agreement with the number from Nittler & Alexander (2003). However, a high MSWD of 13.923 clearly indicates that the distribution of Y grains does not follow a simple regression line.

The exact ¹²C/¹³C ratio used as a divider between mainstream and Y grains has little influence on the parameters of the calculated regression line. A ¹²C/¹³C ratio of 100 is an arbitrary number, as there is no physical justification for it, but has long been used as a divider and was therefore also adopted for the PGD. It means that Y grains have clearly higher ¹²C/¹³C ratios than the terrestrial ratio, while mainstream grains usually have ratios that are lower than the terrestrial ratio. Contamination with terrestrial or asteroidal carbon, which has always to be considered when analyzing presolar grains, would move the ¹²C/¹³C ratio toward ∼89. However, the data clearly show that grains with higher ¹²C/¹³C ratios tend to form a distribution in the silicon three-isotope plot with a slope smaller than the one for the mainstream grains. Cluster analysis of PGD data also showed that there is not a clear separation between mainstream and Y grains (Boujibar et al. 2021). This should not come as a surprise if the prevailing model for Y grains is correct: they come from similar stellar sources (AGB stars) as mainstream grains but with lower metallicities (∼1/2 Z_⊙; Amari et al. 2001c). A range in metallicities of their parent stars plausibly leads to a continuum in their isotopic properties.

Type Z grains (Figure 7) have ¹²C/¹³C ratios between 13.5 and 100, the same range as mainstream grains. In nitrogen and inferred aluminum isotopes, they are indistinguishable from mainstream and Y grains. In the silicon three-isotope plots, they populate a region that is below and to the right of the band for mainstream grains:

$$\begin{eqnarray}&&{\delta }^{29}\mathrm{Si}\lt -19.0{\%} +1.342\times \left({\delta }^{30}\mathrm{Si}-100\right).\end{eqnarray} \tag{ 15 }$$

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In addition, Z grains are confined by –200‰ < δ²⁹Si < 200‰ and δ³⁰Si > 0.

Previously, Z grains have typically been identified by their distance in units of σ from the mainstream regression line (e.g., Nittler & Alexander 2003), which leads to the issue that the classification of a grain as mainstream or Z grain depends more on the uncertainty and less on the actual value itself. This seems unsatisfactory, especially when keeping in mind that mainstream grains clearly do not lie on a simple line but form a band about the regression line, as can be seen from an MSWD value significantly greater than 1. The new grain type definition led to a reassignment of many grains previously classified as Z to other types, mainly mainstream grains, and only a few grains formerly classified as mainstream grains are now identified as Z grains. Therefore, the total number of Z grains in the PGD has decreased from 639 to 284 according to the new classification (Table 1).

Cluster analysis also showed that Z grains are not clearly separated from mainstream and Y grains (Boujibar et al. 2021), which probably reflects the gradual differences in metallicity among the parent stars of these types of grains. Type Z grains are argued to have originated in the winds of low-mass AGB stars with even lower metallicity (∼1/3 Z_⊙) than Y grains (Hoppe et al. 1997).

Type AB grains (Figure 8) have low ¹²C/¹³C and were originally defined as two different types, A and B, based on carbon isotopes, with (¹²C/¹³C)_A < 3.5 < (¹²C/¹³C)_B < 10, where a dividing value of 3.5 was chosen based on it being the equilibrium composition of CNO-cycle H burning in low-mass stars (Hoppe et al. 1994). Later, the two types were combined into one type AB (e.g., Amari et al. 2001b). More recently, Liu et al. (2017a, 2017b, 2018b) suggested redividing AB grains into two subtypes based on their nitrogen isotopes: AB1 with low ¹⁴N/¹⁵N and AB2 with high ¹⁴N/¹⁵N ratios, using the solar value of ¹⁴N/¹⁵N = 441 (Marty et al. 2011) as a divider.

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Cluster analysis showed that AB grains consist of two clusters, which can be described as one with relatively low ¹²C/¹³C and ¹⁴N/¹⁵N ratios and the other with relatively high ¹²C/¹³C and ¹⁴N/¹⁵N ratios (Boujibar et al. 2021), confirming earlier observations that AB grains with lower ¹²C/¹³C ratios tend to have lower ¹⁴N/¹⁵N ratios (Amari et al. 2001b). However, these clusters show significant overlap, especially in nitrogen isotope ratios, and the solar ¹⁴N/¹⁵N ratio seems poorly suited as a divider (Boujibar et al. 2021).

From our assessment of the data (see Section 3.1 and Figure 3), we adopt a value ¹²C/¹³C = 13.5 as the cutoff value between AB and mainstream or Z grains, but grains with ratios up to 25 could still be classified as AB grains, if nitrogen or inferred aluminum isotope ratios exclude those other types. Furthermore, we define the two subtypes AB1 and AB2, using ¹²C/¹³C < 4.5 and ¹⁴N/¹⁵N < 441 (solar value) as upper limits for AB1, and ¹²C/¹³C > 4.5 and ¹⁴N/¹⁵N > 272 (terrestrial value) as lower limits for AB2 grains. AB grains that do not match the definition of subtypes AB1 or AB2 are simply designated as AB grains.

Inferred aluminum isotope ratios cannot be used to identify AB grains or their subtypes, as they cover a wide range, but can be useful for excluding other grain types. However, AB1 grains have on average higher inferred ²⁶Al/²⁷Al ratios than AB2 grains, a trend that has been reported earlier (Amari et al. 2001b), but there is no clear separation.

In silicon isotopes, AB grains lie within the area defined by the mainstream grains. However, they form their own line in the silicon three-isotope plot with a slightly steeper slope than mainstream grains,

$$\begin{eqnarray}&&{\delta }^{29}\mathrm{Si}=-(23.2\pm 0.7){\%} +(1.494\pm 0.015)\times {\delta }^{30}\mathrm{Si},\end{eqnarray} \tag{ 16 }$$

and an MSWD of 3.895 based on data from 1229 AB grains. Such a steeper slope for AB than for mainstream grains has also been previously reported by Nittler & Alexander (2003).

Among the proposed stellar origins of AB grains are J-type carbon stars (Abia & Isern 2000; Liu et al. 2017b), born-again AGB stars (Asplund et al. 1999; Herwig et al. 2011), and SN II (Liu et al. 2017a; Hoppe et al. 2019).

Type C grains (Figure 9) are very rare and characterized by high δ²⁹Si and δ³⁰Si values, substantially higher than those in other types of grains. We use lower limits of δ²⁹Si > +200‰ and δ³⁰Si > +200‰ to separate C grains from other grain types. In carbon, nitrogen, and inferred aluminum isotopes, they cover the same areas as X grains.

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Liu et al. (2016) introduced a distinct subtype C2, characterized by ¹²C/¹³C < 10, which are clearly separated from other C grains, now named C1, that have much higher ¹²C/¹³C ratios. We adopted the same subtype assignment using ¹²C/¹³C < 10 as the upper limit for C2 grains.

Type C grains have been suggested to form in SN II (Croat et al. 2010; Zinner 2014).

A few grains, often called putative nova grains, have been attributed to a possible origin in novae (Amari et al. 2001a) and are labeled with N in the PGD (Figure 10). They are defined by having ¹²C/¹³C < 13.5 like AB grains, and, as for the AB grains, we also allow for ratios up to 25. The main distinction between AB and N grains is that, in silicon three-isotope diagrams, the latter populate a region that is below and to the right of the band for mainstream and AB grains, following Equation (15) like Z grains. Like Z grains, N grains are confined by δ²⁹Si < +200‰ and δ³⁰Si > 0 but have no lower limit in δ²⁹Si. As for X grains, ¹⁴N/¹⁵N ratios are significantly below the terrestrial ratio of 272 for N grains, and inferred ²⁶Al/²⁷Al ratios are higher than 0.01. Details are provided in the Appendix.

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Whether all N grains are really connected to novae or rather some originated in SN II is still debated (Nittler & Hoppe 2005; José & Hernanz 2007; Liu et al. 2016; Hoppe et al. 2018). Schulte et al. (2021), who also favored an origin in SN II, suggested renaming this type to D grains in order to disassociate them from a proposed origin in novae. However, this seemed not to have been widely accepted in the literature, and we continue to refer to these grains as N grains. To avoid confusion when referring to two decades of literature, where these grains are called N grains, nova grains, or putative nova grains, we strongly advise against using the term D grains for these grains.

Instead, we use "D" to define a new type of SiC grain, of which there are currently only 10, that populate a region in the silicon three-isotope diagram mostly to the left and above the band for mainstream and AB grains and are indistinguishable from X grains in nitrogen and inferred aluminum isotopes (Figure 11):

$$\begin{eqnarray}&&{\delta }^{29}\mathrm{Si}\gt -19.0{\%} +1.342\times \left({\delta }^{30}\mathrm{Si}+250{\%} \right).\end{eqnarray} \tag{ 17 }$$

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Grains that are below the upper border of the mainstream band but still above the mainstream regression line can also classified as D grains if nitrogen or inferred aluminum isotope ratios exclude other grain types. In addition, D grains are confined to δ²⁹Si > 0 and δ³⁰Si < +200‰. Details are given in the Appendix.

From their similarities in nitrogen and inferred aluminum isotope ratios with X and C grains, we suggest that D grains also form in SN II.

Grains that do not fit into other types have been named U grains, which could stand for ungrouped, unknown, unique, or unusual. With our introduction of the new type D, all grains for which either carbon, nitrogen, or silicon data are available can now be assigned to a specific type.

All grains currently shown as U grains in the PGD are grains for which neither carbon, nitrogen, nor silicon isotope data are available. Some U grains had been initially classified as mainstream or X grains based on isotopic data of trace elements. However, as we base our PGD Type assignments solely on carbon, nitrogen, aluminum, and silicon data, the lack of such data consequently leads to classifying them as U grains.

While Table 1 provides absolute numbers of grains for a given grain type in the PGD, this is not an unbiased sample, as many studies contributed data focused on specific grain types. However, there are several studies that have performed unbiased surveys of large numbers of grains, allowing determination of relative abundances of grain types in primitive solar system material (Alexander 1993; Hoppe et al. 1994, 1996b, 2010, 2012, 2018; Huss et al. 1997; Nittler & Alexander 2003; Nittler & Hoppe 2005; Orthous-Daunay et al. 2012; Xu et al. 2015; Gyngard et al. 2018; Nguyen et al. 2018). Based on data for 17,529 grains from these unbiased studies, we calculated the relative abundances given in Table 2.

The probability that the measured ¹²C/¹³C ratio is consistent with a given grain type is calculated using the following equations:

$$\begin{eqnarray}\begin{array}{rcl}{p}_{{\rm{C}}}\left({\rm{M}}\right) & = & {p}_{{\rm{C}}}\left({\rm{Z}}\right)=p\left({}^{12}{\rm{C}}{/}^{13}{\rm{C}}\lt 100\right)\\ & & -\,p\left({}^{12}{\rm{C}}{/}^{13}{\rm{C}}\lt 13.5\right),\end{array}\end{eqnarray} \tag{ A1 }$$

$$\begin{eqnarray}&&{p}_{{\rm{C}}}\left({\rm{Y}}\right)=p\left({}^{12}{\rm{C}}{/}^{13}{\rm{C}}\gt 100\right),\end{eqnarray} \tag{ A2 }$$

$$\begin{eqnarray}\begin{array}{rcl}{p}_{{\rm{C}}}\left(\mathrm{AB}\right) & = & {p}_{{\rm{C}}}\left({\rm{N}}\right)=0.8\times p\left({}^{12}{\rm{C}}{/}^{13}{\rm{C}}\lt 13.5\right)\\ & & +\,0.2\times p\left({}^{12}{\rm{C}}{/}^{13}{\rm{C}}\lt 25\right).\end{array}\end{eqnarray} \tag{ A3 }$$

Types X, C, and D do not rely in their classification on carbon isotopes. Therefore,

$$\begin{eqnarray}&&{p}_{{\rm{C}}}\left({\rm{X}}\right)={p}_{{\rm{C}}}\left({\rm{C}}\right)={p}_{{\rm{C}}}\left({\rm{D}}\right)=1.\end{eqnarray} \tag{ A4 }$$

When applying limits to the ranges of isotope ratios or δ-values for a given grain type, we distinguish between hard and soft limits. A hard limit in this context means that the rule has to be applied strictly as shown in Equations (4), (6), or (10). For soft limits, however, we weaken the criterion by applying a prefactor <1 and adding a minimum value to Equations (4), (6), or (10), simultaneously making sure that the probability cannot be >1. For example, while AB and N grains typically have a ¹²C/¹³C ratio below 13.5, we allow for somewhat higher numbers up to 25 if, e.g., nitrogen isotopes exclude the grain to be classified as a mainstream or Z grain. This is achieved by using coefficients of 0.8 and 0.2 in the definitions of ${p}_{{\rm{C}}}\left(\mathrm{AB}\right)$ and ${p}_{{\rm{C}}}\left({\rm{N}}\right)$ according to Equation (A3), making ¹²C/¹³C < 25 a hard limit and ¹²C/¹³C < 13.5 a soft limit for those grains. The limits of mainstream, Y, and Z grains, as defined in Equations (A1) and (A2), are hard.

Finally, we also had to take into account the case that no carbon isotope data are available for a grain. As identification of a grain as type Y, AB, or N mainly relies on carbon isotopes, we assume the following probabilities if no carbon data are available:

$$\begin{eqnarray}&&{p}_{{\rm{C}}}\left({\rm{M}}\right)={p}_{{\rm{C}}}\left({\rm{X}}\right)={p}_{{\rm{C}}}\left({\rm{Z}}\right)={p}_{{\rm{C}}}\left({\rm{C}}\right)={p}_{{\rm{C}}}\left({\rm{D}}\right)=1,\end{eqnarray} \tag{ A5 }$$

$$\begin{eqnarray}&&{p}_{{\rm{C}}}\left({\rm{Y}}\right)={p}_{{\rm{C}}}\left(\mathrm{AB}\right)={p}_{{\rm{C}}}\left({\rm{N}}\right)=0.\end{eqnarray} \tag{ A6 }$$

The probability that a measured ¹⁴N/¹⁵N ratio is consistent with a given grain type is calculated using the following equations:

$$\begin{eqnarray}&&{p}_{{\rm{N}}}\left({\rm{M}}\right)={p}_{{\rm{N}}}\left({\rm{Y}}\right)={p}_{{\rm{N}}}\left({\rm{Z}}\right)=p\left({}^{14}{\rm{N}}{/}^{15}{\rm{N}}\gt 200\right),\end{eqnarray} \tag{ A7 }$$

$$\begin{eqnarray}\begin{array}{rcl}{p}_{{\rm{N}}}\left({\rm{X}}\right) & = & {p}_{{\rm{N}}}\left({\rm{C}}\right)={p}_{{\rm{N}}}\left({\rm{D}}\right)={p}_{{\rm{N}}}\left({\rm{N}}\right)\\ & = & p\left({}^{14}{\rm{N}}{/}^{15}{\rm{N}}\lt 272\right).\end{array}\end{eqnarray} \tag{ A8 }$$

Type AB does not rely in its classification on nitrogen isotopes. Therefore,

$$\begin{eqnarray}&&{p}_{{\rm{N}}}\left(\mathrm{AB}\right)=1.\end{eqnarray} \tag{ A9 }$$

Since no grain type identification relies mainly on nitrogen isotopes, we assume a probability of 1 for all types if no nitrogen data exist:

$$\begin{eqnarray}&&\begin{array}{l}{p}_{{\rm{N}}}\left({\rm{M}}\right)={p}_{{\rm{N}}}\left({\rm{X}}\right)={p}_{{\rm{N}}}\left({\rm{Y}}\right)\\ \ =\ {p}_{{\rm{N}}}\left({\rm{Z}}\right)={p}_{{\rm{N}}}\left(\mathrm{AB}\right)={p}_{{\rm{N}}}\left({\rm{C}}\right)={p}_{{\rm{N}}}\left({\rm{D}}\right)={p}_{{\rm{N}}}\left({\rm{N}}\right)=1.\end{array}\end{eqnarray} \tag{ A10 }$$

For silicon isotopes, an area around the mainstream line (M0) has to be considered, which is defined by four lines M1–M4 with the parameters given in Table A1. As a shorthand for a given pair of parameters a and b from Table A1 applied to Equation (10), we use the following nomenclature:

$$\begin{eqnarray}&&{p}_{\mathrm{Si}}\left({\rm{M}}0\ldots {\rm{M}}4\right)\,:=\,{p}_{\mathrm{Si}}\left({a}_{{\rm{M}}0\ldots {\rm{M}}4},{b}_{{\rm{M}}0\ldots {\rm{M}}4}\right).\end{eqnarray} \tag{ A11 }$$

Table A1. Parameters of Lines in Silicon Three-isotope Plots

Line	a	b
M0	−19.0‰	1.342
M1	−19.0‰ + 250‰ × 1.342	1.342
M2	−19.0‰ − 100‰ × 1.342	1.342
M3	$-19.0{\%} +200{\%} \times \left(1.342+1/1.342\right)$	−1/1.342
M4	$-19.0{\%} -75{\%} \times \left(1.342+1/1.342\right)$	−1/1.342

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With this, the probability that a grain is consistent with a given grain type based on silicon isotope ratios is calculated using the following equations:

$$\begin{eqnarray}\begin{array}{rcl}{p}_{\mathrm{Si}}\left({\rm{M}}\right) & = & {p}_{\mathrm{Si}}\left(\mathrm{AB}\right)=\left[{p}_{\mathrm{Si}}\left({\rm{M}}1\right)-{p}_{\mathrm{Si}}\left({\rm{M}}2\right)\right]\\ & & \times \,\left[{p}_{\mathrm{Si}}\left({\rm{M}}3\right)-{p}_{\mathrm{Si}}\left({\rm{M}}4\right)\right],\end{array}\end{eqnarray} \tag{ A12 }$$

$$\begin{eqnarray}\begin{array}{rcl}{p}_{\mathrm{Si}}\left({\rm{X}}\right) & = & p\left({\delta }^{29}\mathrm{Si}\lt 0\right)\times p\left({\delta }^{30}\mathrm{Si}\lt 0\right)\\ & & \times \,\left[0.2+0.8\times {p}_{\mathrm{Si}}\left({\rm{M}}4\right)\right],\end{array}\end{eqnarray} \tag{ A13 }$$

$$\begin{eqnarray}&&\begin{array}{l}{p}_{\mathrm{Si}}\left({\rm{Y}}\right)={p}_{\mathrm{Si}}\left({\rm{M}}1\right)\\ \times \,\left\{1-\left[1-{p}_{\mathrm{Si}}\left({\rm{M}}3\right)\right]\times p\left({\delta }^{29}\mathrm{Si}\gt +200{\%} \right)\right\}\\ \times \,\left[1-{p}_{\mathrm{Si}}\left({\rm{M}}4\right)\times p\left({\delta }^{30}\mathrm{Si}\lt 0\right)\right]\times p\left({\delta }^{29}\mathrm{Si}\gt -200{\%} \right),\end{array}\end{eqnarray} \tag{ A14 }$$

$$\begin{eqnarray}&&\begin{array}{l}{p}_{\mathrm{Si}}\left({\rm{Z}}\right)={p}_{\mathrm{Si}}\left({\rm{M}}2\right)\times \left[p\left({\delta }^{29}\mathrm{Si}\lt +200{\%} \right)\right.\\ -\left.\,p\left({\delta }^{29}\mathrm{Si}\lt -200{\%} \right)\right]\times p\left({\delta }^{30}\mathrm{Si}\gt 0\right),\end{array}\end{eqnarray} \tag{ A15 }$$

$$\begin{eqnarray}&&\begin{array}{l}{p}_{\mathrm{Si}}\left({\rm{C}}\right)=p\left({\delta }^{29}\mathrm{Si}\gt +200{\%} \right)\\ \times \ p\left({\delta }^{30}\mathrm{Si}\gt +200{\%} \right)\times \left[1-{p}_{\mathrm{Si}}\left({\rm{M}}3\right)\right],\end{array}\end{eqnarray} \tag{ A16 }$$

$$\begin{eqnarray}&&\begin{array}{l}{p}_{\mathrm{Si}}\left({\rm{D}}\right)=p\left({\delta }^{29}\mathrm{Si}\gt 0\right)\times p\left({\delta }^{30}\mathrm{Si}\lt +200{\%} \right)\\ \times \,\left[1-0.8\times {p}_{\mathrm{Si}}\left({\rm{M}}1\right)-0.2\times {p}_{\mathrm{Si}}\left({\rm{M}}0\right)\right],\end{array}\end{eqnarray} \tag{ A17 }$$

$$\begin{eqnarray}&&{p}_{\mathrm{Si}}\left({\rm{N}}\right)={p}_{\mathrm{Si}}\left({\rm{M}}2\right)\times p\left({\delta }^{29}\mathrm{Si}\lt +200{\%} \right)\times p\left({\delta }^{30}\mathrm{Si}\gt 0\right).\end{eqnarray} \tag{ A18 }$$

For a proper grain type assignment, ideally, all three silicon isotopes should have been measured. However, for some grains, only one δ-value has been measured. If only δ²⁹Si data exist, we use the following equations:

$$\begin{eqnarray}&&\begin{array}{l}{p}_{\mathrm{Si}}\left({\rm{M}}\right)={p}_{\mathrm{Si}}\left(\mathrm{AB}\right)=p\left({\delta }^{29}\mathrm{Si}\lt +200{\%} \right)\\ \,\,-\ p\left({\delta }^{29}\mathrm{Si}\lt -120{\%} \right),\end{array}\end{eqnarray} \tag{ A19 }$$

$$\begin{eqnarray}&&{p}_{\mathrm{Si}}\left({\rm{X}}\right)=p\left({\delta }^{29}\mathrm{Si}\lt -120{\%} \right),\end{eqnarray} \tag{ A20 }$$

$$\begin{eqnarray}&&{p}_{\mathrm{Si}}\left({\rm{Y}}\right)=p\left({\delta }^{29}\mathrm{Si}\lt +200{\%} \right)-p\left({\delta }^{29}\mathrm{Si}\lt -200{\%} \right),\end{eqnarray} \tag{ A21 }$$

$$\begin{eqnarray}&&{p}_{\mathrm{Si}}\left({\rm{Z}}\right)={p}_{\mathrm{Si}}\left({\rm{C}}\right)={p}_{\mathrm{Si}}\left({\rm{D}}\right)={p}_{\mathrm{Si}}\left({\rm{N}}\right)=0.\end{eqnarray} \tag{ A22 }$$

Similarly, if only δ³⁰Si data exist, we use the following equations:

$$\begin{eqnarray}&&\begin{array}{l}{p}_{\mathrm{Si}}\left({\rm{M}}\right)={p}_{\mathrm{Si}}\left(\mathrm{AB}\right)=p\left({\delta }^{30}\mathrm{Si}\lt +200{\%} \right)\\ \,\,-\ p\left({\delta }^{30}\mathrm{Si}\lt -100{\%} \right),\end{array}\end{eqnarray} \tag{ A23 }$$

$$\begin{eqnarray}&&{p}_{\mathrm{Si}}\left({\rm{X}}\right)=p\left({\delta }^{30}\mathrm{Si}\lt -100{\%} \right),\end{eqnarray} \tag{ A24 }$$

$$\begin{eqnarray}&&{p}_{\mathrm{Si}}\left({\rm{Y}}\right)=p\left({\delta }^{30}\mathrm{Si}\gt -100{\%} \right),\end{eqnarray} \tag{ A25 }$$

$$\begin{eqnarray}&&{p}_{\mathrm{Si}}\left({\rm{Z}}\right)={p}_{\mathrm{Si}}\left({\rm{C}}\right)={p}_{\mathrm{Si}}\left({\rm{D}}\right)={p}_{\mathrm{Si}}\left({\rm{N}}\right)=0.\end{eqnarray} \tag{ A26 }$$

Finally, if no silicon data exist, we apply the following equations:

$$\begin{eqnarray}&&{p}_{\mathrm{Si}}\left({\rm{M}}\right)={p}_{\mathrm{Si}}\left({\rm{Y}}\right)={p}_{\mathrm{Si}}\left(\mathrm{AB}\right)=1,\end{eqnarray} \tag{ A27 }$$

$$\begin{eqnarray}&&{p}_{\mathrm{Si}}\left({\rm{X}}\right)=0.2,\end{eqnarray} \tag{ A28 }$$

$$\begin{eqnarray}&&{p}_{\mathrm{Si}}\left({\rm{Z}}\right)={p}_{\mathrm{Si}}\left({\rm{C}}\right)={p}_{\mathrm{Si}}\left({\rm{D}}\right)={p}_{\mathrm{Si}}\left({\rm{N}}\right)=0.\end{eqnarray} \tag{ A29 }$$

Identification of grain types Z, C, D, and N mainly relies on the measurement of all three silicon isotopes. If no or only partial silicon data exist, we therefore assign 0 probability for these types. For type X, partial silicon data could be used for identification, and we use a probability of 0.2 if no silicon data is available, as they might be still identifiable by their distinctively low ¹⁴N/¹⁵N or high inferred ²⁶Al/²⁷Al ratios.

Inferred ²⁶Al/²⁷Al ratios can also be useful for the identification of some grain types:

$$\begin{eqnarray}&&{p}_{\mathrm{Al}}\left({\rm{M}}\right)={p}_{\mathrm{Al}}\left({\rm{Y}}\right)={p}_{\mathrm{Al}}\left({\rm{Z}}\right)=p\left({}^{26}\mathrm{Al}{/}^{27}\mathrm{Al}\lt 0.02\right),\end{eqnarray} \tag{ A30 }$$

$$\begin{eqnarray}&&\begin{array}{l}{p}_{\mathrm{Al}}\left({\rm{X}}\right)={p}_{\mathrm{Al}}\left({\rm{C}}\right)={p}_{\mathrm{Al}}\left({\rm{D}}\right)={p}_{\mathrm{Al}}\left({\rm{N}}\right)\\ =\,0.05+0.95\times p\left({}^{26}\mathrm{Al}{/}^{27}\mathrm{Al}\gt 0.01\right),\end{array}\end{eqnarray} \tag{ A31 }$$

$$\begin{eqnarray}&&{p}_{\mathrm{Al}}\left(\mathrm{AB}\right)=1.\end{eqnarray} \tag{ A32 }$$

While mainstream, Y, and Z grains have inferred ²⁶Al/²⁷Al ratios <0.02, X, C, D, and N grains usually have ratios >0.01. However, we account for the possibility that ²⁷Al contamination might have compromised the grains by slightly softening the limit for X, C, D, and N grains. For AB grains, inferred ²⁶Al/²⁷Al ratios are not indicative.

We assume a probability of 1 for all types if no aluminum data exist:

$$\begin{eqnarray}&&\begin{array}{l}{p}_{\mathrm{Al}}\left({\rm{M}}\right)={p}_{\mathrm{Al}}\left({\rm{X}}\right)={p}_{\mathrm{Al}}\left({\rm{Y}}\right)={p}_{\mathrm{Al}}\left({\rm{Z}}\right)\\ \ =\ {p}_{\mathrm{Al}}\left(\mathrm{AB}\right)={p}_{\mathrm{Al}}\left({\rm{C}}\right)={p}_{\mathrm{Al}}\left({\rm{D}}\right)={p}_{\mathrm{Al}}\left({\rm{N}}\right)=1.\end{array}\end{eqnarray} \tag{ A33 }$$

For X grains, we use subtype assignments X0, X1, and X2 according to Table A2.

Table A2. X Grain Subtype Assignment

Subtype	Condition
X0	${\delta }^{29}\mathrm{Si}\gt 30{\%} +\left(\tfrac{2}{3}-0.05\right)\times {\delta }^{30}\mathrm{Si}$
X1	$30{\%} +\left(\tfrac{2}{3}-0.05\right)\times {\delta }^{30}\mathrm{Si}\geqslant {\delta }^{29}\mathrm{Si}\geqslant -30{\%} +\left(\tfrac{2}{3}+0.05\right)\times {\delta }^{30}\mathrm{Si}$
X2	${\delta }^{29}\mathrm{Si}\lt -30{\%} +\left(\tfrac{2}{3}+0.05\right)\times {\delta }^{30}\mathrm{Si}$
No subtype	δ29Si or δ30Si data do not exist

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For AB grains, we use subtype assignments AB1 and AB2 according to Table A3.

Table A3. AB Grain Subtype Assignment

Subtype	Condition
AB1	$p\left({}^{12}{\rm{C}}{/}^{13}{\rm{C}}\lt 4.5\right)\times p\left({}^{14}{\rm{N}}{/}^{15}{\rm{N}}\lt 441\right)\gt p\left({}^{12}{\rm{C}}{/}^{13}{\rm{C}}\gt 4.5\right)\times p\left({}^{14}{\rm{N}}{/}^{15}{\rm{N}}\gt 272\right)$ and ${}^{12}{\rm{C}}{/}^{13}{\rm{C}}-{\sigma }_{{}^{12}{\rm{C}}{/}^{13}{\rm{C}}}^{-}\leqslant 4.5$ and ${}^{14}{\rm{N}}{/}^{15}{\rm{N}}-{\sigma }_{{}^{14}{\rm{N}}{/}^{15}{\rm{N}}}^{-}\leqslant 441$
AB2	$p\left({}^{12}{\rm{C}}{/}^{13}{\rm{C}}\lt 4.5\right)\times p\left({}^{14}{\rm{N}}{/}^{15}{\rm{N}}\lt 441\right)\leqslant p\left({}^{12}{\rm{C}}{/}^{13}{\rm{C}}\gt 4.5\right)\,\times p\left({}^{14}{\rm{N}}{/}^{15}{\rm{N}}\gt 272\right)$ and ${}^{12}{\rm{C}}{/}^{13}{\rm{C}}+{\sigma }_{{}^{12}{\rm{C}}{/}^{13}{\rm{C}}}^{+}\geqslant 4.5$ and ${}^{14}{\rm{N}}{/}^{15}{\rm{N}}+{\sigma }_{{}^{14}{\rm{N}}{/}^{15}{\rm{N}}}^{+}\geqslant 272$
No subtype	Nitrogen data do not exist or conditions for AB1 and AB2 are not met

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For C grains, we use subtype assignment C1 and C2 according to Table A4.