Equilibrated Gas and Carbonate Standard-Derived Dual (Δ47 and Δ48) Clumped Isotope Values

Carbonate clumped isotope geochemistry has primarily focused on mass spectrometric determination of m/z 47 CO2 for geothermometry, but theoretical calculations and recent experiments indicate paired analysis of the m/z 47 (13C18O16O) and m/z 48 (12C18O18O) isotopologues (referred to as Δ47 and Δ48) can be used to study non-equilibrium isotope fractionations and refine temperature estimates. We utilize 5,448 Δ47 and 3,400 Δ48 replicate measurements of carbonate samples and standards, and 183 Δ47 and 195 Δ48 replicate measurements of gas standards from 2015 to 2021 from a multi-year and multi-instrument data set to constrain Δ47 and Δ48 values for 27 samples and standards, including Devils Hole cave calcite, and study equilibrium Δ47–Δ48, Δ47-temperature, and Δ48-temperature relationships. We compare results to previously published findings and calculate equilibrium regressions based on data from multiple laboratories. We report acid digestion fractionation factors, Δ*63−47 and Δ*64−48, and account for their dependence on the initial clumped isotope values of the mineral.

The abundance of the 13 C 18 O 16 O and 12 C 18 O 18 O isotopologues is denoted with δ 47 , δ 48 , Δ 47 , and Δ 48 notation . These are defined as: where Ri sample is the measured ratio of i/44 CO 2 isotopologues in the sample, Ri stochastic is the ratio of i/44 CO 2 isotopologues that would be expected in a random distribution, and Ri ref. gas is the ratio of i/44 CO 2 in a reference gas of known isotopic composition (Eiler, 2007;Schauble et al., 2006). The values are given in permil (‰). The most abundant m/z 48 CO 2 isotopologue ( 12 C 18 O 18 O) has two 18 O substitutions and is therefore in extremely low abundance at 4.1 ppm in air, which is an order of magnitude lower than m/z 47 isotopologues at 45 ppm . The minor m/z 48 CO 2 isotopologue ( 13 C 18 O 17 O) has an abundance of 16.7 ppb .
The precise measurement of Δ 47 was enabled by modification of the Thermo MAT 253, specially configured for the digestion of carbonate minerals, purification of liberated CO 2 , and m/z 47-49 Faraday cups Ghosh et al., 2006). On this instrument, m/z 48 isotopologues were used only to screen for contaminants. More precise measurements of Δ 48 have recently emerged due to the use of 10 13 Ω resistors for m/z 47-49 Faraday cups in the Thermo MAT 253 Plus (Bajnai et al., 2020;Fiebig et al., 2019Fiebig et al., , 2021Swart et al., 2021), and secondary electron suppression in the Nu Perspective IS. These advances contribute to increased accuracy and precision for determination of Δ 48 values, and paired Δ 47 and Δ 48 values.
A unique attribute of carbonate clumped isotope thermometry based on Δ 47 or Δ 48 is that it does not depend on the bulk oxygen isotope composition (δ 18 O) of the water the mineral precipitated from , unlike the more widely used oxygen isotope thermometer (Urey, 1947). Measurements of Δ 47 have been used for the reconstruction of numerous paleo-environmental parameters, including but not limited to land (Passey & Henkes, 2012) and ocean (Henkes et al., 2018;Tripati et al., 2015) paleotemperatures, paleoelevation (Huntington et al., 2010;Lechler et al., 2013), and dinosaur body temperature (Eagle et al., 2010), while simultaneously estimating water δ 18 O. Previous research has shown that kinetic isotope effects observed in abiotic and biogenic carbonate minerals, including speleothems (Affek et al., 2008;Daëron et al., 2011) and coral (Bajnai et al., 2020;Kimball et al., 2016;Saenger et al., 2012;Thiagarajan et al., 2011), may affect the accuracy of Δ 47 -based temperature reconstructions. However, the paired analysis of Δ 47 and Δ 48 has been shown by theory (Guo, 2020;Hill et al., 2014Hill et al., , 2020Schauble et al., 2006;Tripati et al., 2015) and experimentation (Bajnai et al., 2020;Fiebig et al., 2019Fiebig et al., , 2021Swart et al., 2021) to have a characteristic equilibrium relationship to temperature which may be used to identify and study kinetic effects in carbonate minerals.
Several studies have proposed the use of new methods to advance the consistency of Δ 47 measurements between laboratories. Interlaboratory reproducibility of Δ 47 values was advanced by using accurately determined carbonate standard values that are anchored to the absolute reference frame, using a reference frame constructed using primary gas standards, secondary carbonate standards, or a mixture of gas and carbonate standards, detailed by Dennis et al. (2011). Recent work from Bernasconi et al. (2021) has proposed nominal carbonate standard Δ 47 values and the use of carbonate standards for data normalization in the 90°C reference frame. These advances form the foundation for the potential application of carbonate-based data normalization to yield reproducible Δ 48 values, and paired Δ 47 and Δ 48 values, on the absolute reference frame.
Here, we utilize data collected over multiple years on multiple instruments to determine if carbonate-based data normalization produces reproducible Δ 48 values, and examine if widely used carbonate standards, in-house standards, and a suite of both biogenic and abiogenic samples of varying minerology deviate significantly from equilibrium. We used both equilibrated gas and carbonate-based data normalization to report the isotopic composition of 27 samples of varying mineralogy, including standards and 4 Devils Hole calcite samples. We determine acid digestion fractionation factors, Δ* 63-47 and Δ*  , that account for the dependence on the mineral Δ 63 and Δ 64 values, respectively.

Samples
In total, 27 different samples were analyzed for clumped and bulk isotope compositions on mass spectrometers in the Tripati Lab at University of California, Los Angeles. Table 1 contains a description of the mineralogy and origin of all samples. These materials were chosen for analysis because many of them are standards used widely among clumped isotope laboratories, such as ETH-1, ETH-2, ETH-3, ETH-4, Carrara Marble, IAEA-C1, IAEA-C2, and Mallinckrodt. Others are used commonly in a certain region or country, such as ISTB-1, TB-1, and Note. Uranium-series ages for Devils Hole calcite were determined by Winograd et al. (2006).  (Bernasconi et al., 2018;Upadhyay et al., 2021;Chang et al., 2020)

Instrumentation
Standards and samples were analyzed on three mass spectrometers using five configurations (Table 2), including Nu Perspective-EG, Nu Perspective-1, Nu Perspective-1a, Nu-Perspective-2, and MAT 253. Nu Perspective-EG is the only configuration that analyzed equilibrated gases. On both the MAT 253 and Nu Perspective mass spectrometers, the detectors for m/z 44, 45, and 46 are registered through 3 × 10 8 , 3 × 10 10 , and 10 11 Ω resistors, respectively, while detectors for m/z 47 through 49 are registered through 10 12 Ω resistors.
The most notable difference between the Nu Instruments Perspective and the more widely used older generation Thermo Fisher MAT 253 is the implementation in the former of electrostatic analyzers (ESAs) before the m/z 47-49 detectors. These ESAs consist of two curved plates with a voltage difference placed directly in front of each of the Faraday collectors. The addition of the ESAs as well as ion lenses following the magnetic sector of the flight tube removes secondary ion and electron signals from the mass detection. This removal results in a drastic reduction in the interfering signals on all masses (m/z 44-49) during operation, producing flatter and more stable baselines, relative to the older MAT 253 ( Figure S2 in Supporting Information S1). In addition, the lowered interference, which is largely comprised of signals from secondary electrons, in the Nu Perspectives results in greater intensities and lowered noise in the signals from the higher masses, especially m/z 48 and 49. This advancement has contributed to a Δ 47 non-linearity slope for the Nu Perspective (median slope observed was −0.00005) that ranges from one to two orders of magnitude less than the MAT 253 (median slope observed was −0.007), and a Δ 48 non-linearity slope for the Nu Perspective (median slope observed was −0.004) that is an order of magnitude less than the MAT 253 (median slope observed was −0.013).

Mass spectrometer model
Acid digestion temperature  The Thermo Fisher MAT 253 used an autosampler similar to what is described in Passey et al. (2010) with a 105 weight % phosphoric acid bath held at 90°C. After calcite samples of 5 mg were digested, CO 2 (g) was cryogenically purified through traps containing dry ice-cooled ethanol and liquid nitrogen, which remove low vapor pressure gases such as H 2 O (g). The CO 2 passed through elemental silver wool (Sigma Aldrich) to remove sulfur compounds, followed by a −20°C gas chromatograph (GC) that contains Porapak Type-Q™ 50/80 mesh column pack material with He carrier gas. The m/z 44 beam intensity is 16 V. Data are acquired in 9 blocks of 10 cycles, with each consisting of 8 s of integration and 16 s of changeover delay, for a total integration time of 720 s.
Nu Perspective-EG, Nu Perspective-1, and Nu Perspective-1a used the same mass spectrometer and a similar autosampler setup as the MAT 253. Samples weighing 5 mg were analyzed in bellows on the Nu Perspective-EG and Nu Perspective-1 in 4 blocks of 20 cycles, including 8 s of changeover delay and 20 s of integration per cycle, with a total integration time of 1,600 s. The m/z 44 ion beam intensity was 24 V before 6/2017 and 18 V thereafter. Nu Perspective-1a analyzed 0.5 mg samples, with sample and working gas volumes depleted in microvolume mode at precisely matched rates, with m/z 44 ranging from 24-9 V during sample acquisition. Microvolume mode allows for a full hour-long measurement per sample. Data were taken in 3 blocks of 20 cycles, including 8 s of changeover delay and 20 s of integration per cycle, for a total integration time of 1,200 s. The sample preparation system was operated by software in Labview that controls the sampler, GC column, cryogenic dewar lifters, and valves. The Labview software is integrated with the Perspective Stable Gas Control software interface that controls the Nu Perspective mass spectrometer.
Nu Perspective-2 used a Nu Carb Sample Digestion System instead of a common acid bath, where 0.5 mg of carbonate mineral was digested at 70°C in individual glass vials with 105 wt% phosphoric acid. The sample gas was cryogenically purified in liquid nitrogen-cooled tubes called coldfingers before passing through a relatively short GC column packed with Porapak Type-Q™ 50/80 and silver wool. This instrument operates under vacuum pressure and does not use a carrier gas. The sample and working gas volumes were matched precisely during depletion into the mass spectrometer, with m/z 44 ranging from 24-9 V. Sample data were analyzed in 3 blocks of 20 cycles, with each cycle integrating for 20 s, for a total integration time of 1,200 s.

Equilibrated Gas Standards
We analyzed equilibrated gas standards on Nu Perspective-EG (Table 2). We utilized two gases with differing bulk isotope values, with a ∼60 ‰ difference in δ 47 values, prepared using standard procedures (Dennis et al., 2011;Ghosh et al., 2006). The heavy isotope depleted δ 47 gas was from an Airgas CO 2 gas cylinder and was equilibrated with 5-10 mL of 25°C deionized (DI) water. The heavy isotope enriched δ 47 gas was produced by phosphoric acid digestion of Carrara Marble. The resulting CO 2 was equilibrated with evaporated DI water held at 25°C. Aliquots of the two 25°C gases were re-equilibrated at 1000°C by heating the gases in quartz tubes inside a muffle furnace for >1 hr, and then flash cooling the tubes, to produce gases with near stochastic clumped isotope values.

Data Processing and Normalization
Raw data files from all instrument configurations were transferred into Easotope (John & Bowen, 2016) Figure S3 in Supporting Information S1). We do not perform pressure baseline corrections; however, a background correction is performed for all masses (m/z 44-49) on all instruments before any further data normalization. The background is measured (in amps on the Nu Perspective instruments; mV on the MAT 253) at the start of an analysis and is subtracted from the measurement. For the nonlinearity slope correction, a slope was determined over a ±10-replicate moving average for the regression lines between δ 47 raw and Δ 47 raw , and δ 48 raw and Δ 48 raw values for CO 2 gas standards equilibrated at 25 and 1000°C and/or ETH-1 and ETH-2 ( Figure S3 in Supporting Information S1). Nonlinearity slope corrections were applied to all analyses using Equations 5 and 6:

Use of Statistical Methods for Determination of Δ 47 and Δ 48 Values
To streamline data processing and ensure all replicate data were handled identically, we developed an R script that automated outlier identification, calculation of sample replicate pool average Δ 47 , Δ 48 , δ 18 O, and δ 13 C values, total number of replicates (N), replicate pool standard deviation (SD), replicate pool standard error (SE), and normality of the replicate data distribution. A density function was determined for each sample and standard replicate pool on every instrumental configuration after an initial removal of very large outliers ( Figure 2b). A 3σ or 5σ (3 SD or 5 SD from the mean) cut was then made for each density function ( Figure 2c) to yield the final replicate pool. This method is particularly useful for datasets with a large number of replicates where data processing can be time intensive; it also helps reduce potential human bias. We do not recommend this method for samples with less than 12 replicates, as this was the smallest number of replicates we successfully tested the method on. The sample error reported here as SD and SE, which is typical for clumped isotope measurements, does not fully account for additional error associated with

Note.
All data in this table was normalized using only 25 and 1,000°C equilibrated gases.  Bernasconi et al. (2021), Fiebig et al. (2019), Bajnai et al. (2020), and Swart et al. (2021) standardizing raw data into the final Δ 47 values, described as "allogenic" errors by Daëron (2021). These errors likely play a larger role for Δ 48 given larger measurement uncertainties. However, we report the minimum error contribution from standardization to Δ 47 (Daëron, 2021) and Δ 48 values for each instrument configuration.
For data pooling between instrumental configurations, the Δ 47 and Δ 48 replicate distributions for standards and samples run on multiple instrument configurations (consistency standards) were directly compared. If no statistically significant differences were observed between configurations, replicates were pooled to calculate a combined average. The Δ 48 replicate values from the MAT 253 were not pooled with replicate values from the Nu Perspective instruments.
In the Supporting Information S1, we provide a detailed description of this method for replicate-level outlier identification and data pooling from multiple instruments. The R script is publicly available at https://doi.org/10.5281/zenodo.7311624.

Calculation of Δ 47 -T and Δ 48 -T Equilibrium Relationships Using Acid Fractionation Factors
When carbonate minerals are digested in phosphoric acid, the removal of oxygen atoms from CO 2− 3 depends on the temperature of the reaction and the clumped isotope composition of the reactant mineral ). This removal of oxygen atoms results in a significant increase of the Δ 47 and Δ 48 values of the liberated CO 2 versus the initial Δ 63 and Δ 64 values of the mineral . To account for this difference and its dependence on the clumped isotope composition of the reactant mineral, we determined regression-form AFFs, Δ* 63-47 and Δ* 64-48 , for when calcite is digested in phosphoric acid at 90°C. The AFFs were determined by first calculating the difference between measured Δ 47 and Δ 48 values for samples with known precipitation temperatures at 600 and 33.7°C and theoretical equilibrium Δ 63 and Δ 64 values for calcite at 600 and 33.7°C (Hill et al., 2014;Tripati et al., 2015), respectively. The dependence of the AFFs on the initial clumped isotope composition of the mineral was determined by calculating linear regressions between the calculated Δ* 63-47 and Δ* 64-48 values for 600 and 33.7°C and the corresponding theoretically predicted Δ 63 and Δ 64 values for 600 and 33.7°C (Hill et al., 2014;Tripati et al., 2015), respectively. The measured Δ 47 and Δ 48 values used for 600°C were the pooled replicate values for ETH-1 and ETH-2 (Bernasconi et al., 2018), and the values used for 33.7°C were the pooled replicate values for Devils Hole calcite (Coplen, 2007).
The temperature-dependent equilibrium Δ 47 and Δ 48 values were then calculated using Equations 9 and 10, where Δ 63 and Δ 64 values are theoretical equilibrium values for calcite from 0 to 1,000°C (Hill et al., 2014;Tripati et al., 2015), and Δ* 63-47 and Δ* 64-48 are the AFFs determined here. A detailed description of this calculation is in Section S3 in Supporting Information S1.

Statistical Methods
We found no evidence of statistically significant differences in the Δ 47 or Δ 48 values of samples analyzed on multiple configurations ( Figure S4 in Supporting Information S1; Tables S1 and S2 in Supporting Information S2), thus, replicate analyses from the Nu Perspective instruments were pooled. However, due to higher error, lower precision, and offsets in Δ 48 values for ETH-1 and ETH-2 that did not exist in data from the Nu Perspective instruments (Table S3 in Supporting Information S2), Δ 48 replicate data from the MAT 253 was not pooled with Nu Perspective replicate data. Additionally, we have not combined replicate values produced using equilibrated gas-based data normalization with replicate values produced using carbonate-based data normalization.
We found there was a negligible difference in the number of replicates removed when a 3σ versus 5σ cutoff was used for outliers due to narrow peak widths for sample replicate distributions ( Figure S5 in Supporting information S1; Table S4 in Supporting Information S2). To further ensure the accuracy of the data presented here, we compared our final Δ 47 values to Upadhyay et al. (2021) which presented a subset of the data reported here using other methods for outlier removal and data processing (Table S5 in Supporting Information S2). The datasets are in good agreement, with an average offset of 0.011 ‰, despite the Δ 47 data from their study being normalized differently than the data here, and then being transferred into the I-CDES reference frame using an Equation from Appendix A in Bernasconi et al. (2021).

Δ 47 and Δ 48 Results
The Δ 47 and Δ 48 values were determined for 7 standards using equilibrated gas-based data normalization, with replicate analyses performed from May 2015 to May 2017 (Table 3) Figure S4 in Supporting Information S1). The δ 18 O and δ 13 C results are presented in Table S8 in Supporting Information S2.

Combined Average Δ 47 and Δ 48 Values for All Samples and Standards Analyzed in This Study
For the Nu Perspective mass spectrometers, ∼9 replicates were required to reach a Δ 48 mean value within the shot noise limit bounds ( Figure S6 in Supporting Information S1). For the MAT 253, ∼25 replicates were required to reach a Δ 48 mean value within the shot noise limit bounds ( Figure S6 in Supporting Information S1). Additionally, the minimum error contribution to Δ 47 values from standardization (Daëron, 2021) were 0.000 ‰, 0.002 ‰, 0.001 ‰, 0.001 ‰ for Nu Perspective-EG, Nu Perspective-1, Nu Perspective-2, and MAT 253, respectively. The minimum error contribution to Δ 48 values from standardization were 0.005 ‰, 0.004 ‰, 0.002 ‰, 0.004 ‰ for Nu Perspective-EG, Nu Perspective-1, Nu Perspective-2, and MAT 253, respectively.
All Δ 47 and Δ 48 values used to calculate Equation 11 can be found in Table 4. Of the 21 total samples in Figure 3a, all lie within 1 SE of the 95% confidence interval of the regression, with the exception of Merck, Carmel Chalk, and 47407 Coral. 47407 Coral was the only sample excluded from Equation 11 due to the apparent influence of kinetic isotope effects on the Δ 47 and Δ 48 values, which resulted in an offset of >1 SD from the regression.
The AFFs for the compositionally-dependent fractionation of O isotopes during phosphoric acid digestion of carbonate minerals ( Figure S1 in Supporting Information S1) are represented by Equations 15 and 16, where Δ* 63-47 and Δ* 64-48 are the AFFs.

Comparison of Δ 47 and Δ 48 Values Determined With Equilibrated Gas-Based Data Normalization to Previously Published Results
Since the accurate determination of Δ 48 is a relatively new method, the development of robust standard values is of the utmost importance to ensure intra-and inter-laboratory reproducibility.  Bajnai et al. (2020), while the Δ 48 shot noise limit for the Nu Instruments in this study range from 0.027-0.044 ‰. Further, the average interlaboratory Δ 48 offset was 0.019 ‰ (taken as the average of the absolute value of offsets of replicated samples in Table 3). These offsets are likely from random error, considering that the m/z 48 isotopologue is an order of magnitude lower in abundance than the m/z 47 isotopologue , and the offsets are within the shot noise limits.
The use of equilibrated gases for data normalization has been shown to be a potential source of error and interlaboratory offsets since the sample undergoes acid digestion and the gas standard does not, different laboratories use different setups to produce gas standards, and fractionations may occur from quenching during the production of heated gas standards (Bernasconi et al., 2018). However, interlaboratory Δ 47 offsets up to 0.024 ‰ in Bernasconi et al. (2021) were determined to be the result of random error which may be amplified during data normalization. The range in Δ 47 offsets observed here are smaller than what was observed between laboratories reported in Bernasconi et al. (2021), possibly from overall high replication.

Carbonate-Based Data Normalization of Δ 47 -Δ 48 Measurements
Previously, important contributions have demonstrated that carbonate standard-based data normalization that uses readily available materials can produce robust Δ 47 values and yield interlaboratory discrepancies that are consistent with analytical uncertainties (Bernasconi et al., 2018(Bernasconi et al., , 2021Meckler et al., 2014). We applied this approach, using ETH-1, ETH-2, and ETH-3 as carbonate standards on multiple instruments in our laboratory for the paired analysis of Δ 47 -Δ 48 . The combined instrument average from this study (Table 4) and Bernasconi et al. (2021) had excellent agreement between Δ 47 values for samples used as unknowns in both studies, with offsets of 0.005‰, 0.003‰, 0.003‰, 0.001‰ for ETH-4, IAEA-C1, IAEA-C2, and Merck, respectively. This is likely because the nominal Δ 47 values determined in Bernasconi et al. (2021) for ETH-1, ETH-2, and ETH-3 were used here in transfer functions for data normalization, adding supporting evidence for the importance of laboratories using common standard values to improve reproducibility.
We also present Δ 48 data determined on the older generation Thermo MAT 253. We decided to include these data due to the large amount of clumped isotope data produced on this instrument going back to 2014 and given comments from J. Eiler (pers. comm.) indicating these instruments may produce useable Δ 48 data. We sought to test as to whether this instrument, with sufficient replication and quality control, could yield reproducible Δ 48 values. The MAT 253 produced similar sample average Δ 48 values when compared to the Nu Perspective Instruments for the majority of samples (Table S3 in Supporting Information S2). The decision was made to not pool the Δ 48 values produced on the MAT 253 due to lower external precision relative to the Nu Instruments (average 1 SD error for MAT 253 = 0.105 ‰; average 1 SD error for Nu Perspective instruments = 0.056 ‰), more noise and smaller overall peaks observed in the Δ 48 peak-shapes relative to the Nu Perspective instruments ( Figure S2 in Supporting Information S1), and the large offset (0.017 ‰) between the Δ 48 values for ETH-1 and ETH-2 determined on the MAT 253 (Table S3 in Supporting Information S2), which was not observed on the Nu Perspective instruments. However, it may be worth mining past MAT 253 datasets to examine Δ 48 depending on the reproducibility of measurements, although newer generation instrumentation is preferable for the measurement of Δ 48 values due to significantly improved precision.

Δ 47 -Δ 48 Equilibrium Regression Using Samples and Standards
We report a Δ 47 -Δ 48 regression (Equation 11) for 20 carbonate standards and samples (combined average values in Table 4). To have a constraint as to whether the materials included in the regression achieved quasi-equilibrium clumped isotope values, we compared the experimental regression to a regression based on theoretical calcite equilibrium (Figure 3a). The theoretical regression for Δ 63 -Δ 64 equilibrium was transferred into Δ 47 -Δ 48 space using AFFs (Equations 15 and 16). When the experimental regression was compared to the theoretically based regression, they were found to be statistically indistinguishable (P = 0.39; F = 1.03; Table S10 in Supporting Information S2). This supports the assumption that the materials used in the experimental regression have achieved quasi-equilibrium clumped isotope values.
All sample and standard Δ 47 and Δ 48 values are within 1 SE of the 95% confidence interval of the regression (Equation 11; Figure 3a), with the exception of Merck, Carmel Chalk, and 47407 Coral. The 47407 Coral was the only sample not included in the regression. The possibility that Merck, an ultra-pure synthetic calcite, and Carmel Chalk, a natural calcite chalk, are exhibiting subtle clumped isotope disequilibrium cannot be excluded. However, 47407 Coral is a deep-sea coral of the genus Desmophyllum with an estimated growth temperature of 4.2°C (Thiagarajan et al., 2011). Guo (2020) used model estimates to predict a negative correlation between Δ 47 and Δ 48 values for cold-water corals, with kinetic effects causing enrichments in Δ 47 values and depletions in Δ 48 values. We determined that the 47407 Coral exhibits an enrichment of 0.030 ‰ in Δ 47 and depletion of −0.018 ‰ in Δ 48 by defining nominal equilibrium as the regression through the remaining samples, and the offsets were determined by using a kinetic slope for CO 2 absorption in corals of −0.6 (Bajnai et al., 2020;Guo, 2020). Bajnai et al. (2020) also measured Δ 47 and Δ 48 values for a coral of the same genus (Desmophyllum) and a brachiopod (Magellania venosa) and observed similar enrichments in Δ 47 (0.038-0.069 ‰) and depletions in Δ 48 (−0.0004 to −0.095 ‰).

Constraining Equilibrium Δ 47 -Δ 48
The equilibrium Δ 47 -Δ 48 relationship is of recent interest due to the potential for use in identifying kinetic effects in biotic and abiogenic carbonate minerals that are or could be used for paleotemperature reconstructions. A study (Bajnai et al., 2020) used a kinetic slope calculated relative to a proposed equilibrium Δ 47 -Δ 48 regression to recover temperature signals in kinetically controlled samples. To further develop the use of Δ 47 -Δ 48 equilibrium as a proxy to identify kinetic effects, the Δ 47 -Δ 48 equilibrium relationship must be well constrained. Thus, we compared the experimentally determined Δ 47 -Δ 48 regressions for quasi-equilibrium materials determined here (Equation 11) to those from Swart et al. (2021) and Fiebig et al. (2021) using a sum-of-squares F test (Table S10 in Supporting Information S2). This compares the fit of a regression through all datasets to the fit of individual regressions for each data set, and tests whether the datasets differ sufficiently from each other to warrant separate regressions. The data set from Swart et al. (2021) contains 7 calcite precipitations in 5°C increments from 5-65°C and carbonate standards ETH-1, ETH-2, ETH-3, and ETH-4. The data set from Fiebig et al. (2021) includes 16 samples, some of which are combined into averages, yielding 10 samples that are used for comparison here, including lake calcite, Devils Hole calcite, calcite precipitations, and calcite equilibrated at high temperatures, with crystallization temperatures for all samples ranging from 8-1,100°C. We found no evidence of statistically significant differences between the individual regressions (P = 0.86; F = 0.43; Table S10 in Supporting Information S2), and we therefore produced a combined regression, described by Equation 17, which is composed of 41 samples that are believed to have achieved quasi-equilibrium clumped isotope values (Figure 3b). Of the 41 samples used in Equation 17, 35 are within 1 SE of the 95% confidence interval. The samples outside of this threshold include Carmel Chalk, ETH-4, and Merck from this study; ETH-2 and ETH-4 from Swart et al. (2021); and a cave calcite sample from Fiebig et al. (2021). It is unlikely that ETH-2 is exhibiting kinetic effects since it has an equilibration temperature of 600°C (Bernasconi et al., 2018), and has near stochastic isotopic values (Müller et al., 2017). The cave calcite sample from Fiebig et al. (2021) is from Laghetto Basso, Italy with a precipitation temperature of 7.9 ± 0.2°C. Fiebig et al. (2021) and Daëron et al. (2019) argued that this sample precipitated close to equilibrium due to long residence times of water in the lake, low calcite saturation index (<0.3), slow precipitation rate (0.3 μm/yr), and consistent δ 18 O values for contemporaneously deposited calcite layers. It cannot be ruled out that ETH-4, the same commercially available calcite as ETH-2 but unheated (Bernasconi et al., 2018), exhibits subtle kinetic effects. The ETH-4 sample from this study is much closer to the equilibrium regression than the ETH-4 sample from Swart et al. (2021), mostly due to offsets in the Δ 48 value (0.030 ‰) between the studies, which is larger than the offset for Δ 47 (0.014 ‰). Both the Δ 47 and Δ 48 offsets between the studies are within the threshold of observed scatter from random error (Bernasconi et al., 2021). It is also possible that different data normalization methods, carbonate-based here and equilibrated gas-based in Swart et al. (2021), contribute to the difference in ETH-4 values. As discussed above in Section 4.3, it also cannot be ruled out that Carmel Chalk from this study exhibits subtle kinetic effects. However, the scatter for all samples and standards around the equilibrium line are well within what is expected from random error (Bernasconi et al., 2021). Further, the lack of statistical differences between the combined experimental regression (Equation 17) and the theoretically based equilibrium regression (Equation 12), support that Equation 17 is a robust experimental representation of Δ 47 -Δ 48 equilibrium. Samples with Δ 47 -Δ 48 values that deviate significantly from this relationship may have non-equilibrium clumped isotope signatures.

Acid Digestion Fractionation Factors
The values for AFFs, Δ* 63-47 and Δ* 64-48 , for when O atoms are cleaved from CO 2− 3 during phosphoric acid digestion at 90°C, are useful for comparison of measured Δ 47 and Δ 48 values and theoretical Δ 63 and Δ 64 values. The direct measurement of carbonate mineral Δ 63 and Δ 64 is currently not possible. The AFFs can be used to estimate calcite Δ 63 and Δ 64 values via Equations 9 and 10. Model calculations from Guo et al. (2009) predicted that these AFFs should depend on the Δ 63 and Δ 64 values of the reactant carbonate mineral. Our data indicates that the use of a regression-form AFF versus a constant AFF may be important for Δ 47 and has only a negligible effect on Δ 48 , as there is a ∼0.009 ‰ difference in Δ* 63-47 from 0-600°C, while there is only a ∼0.001 ‰ difference in Δ* 64-48 over the same temperature range (Table S9 in Supporting Information S2).

Constraining Equilibrium Δ 47 -T and Δ 48 -T
To date, 3 groups have published relationships for both Δ 47 -T and Δ 48 -T. The regressions from Swart et al. (2021) and Fiebig et al. (2021) are based on measured values from calcite precipitated/equilibrated at quasi-equilibrium, while the regressions from this study and Bajnai et al. (2020) are based on a combination of theoretical calcite mineral Δ 63 -Δ 64 equilibrium values, which were transformed into Δ 47 -Δ 48 space using AFFs (see Methods 2.7). The Δ 47 -T and Δ 48 -T regressions from Bajnai et al. (2020) were calculated for 0-40°C, while the experimentally based regressions from Swart et al. (2021) are for 0-65°C. In this study we calculated Δ 47 -T and Δ 48 -T values from 0-1,000°C, and Fiebig et al. (2021) has experimentally constrained values from 8-1,100°C. Due to the regressions from this study and Bajnai et al. (2020) being theoretically based and therefore difficult to accurately provide an error calculation, we were unable to perform the same type of statistical analysis to compare regressions, as we did for the experimental Δ 47 -Δ 48 regressions. Instead, we have compared the absolute difference between the regressions over a wide temperature range, at 0 and 600°C, and compared this difference to measurement error observed in standards replicated between the laboratories, as well as offsets expected from random error. We used these metrics to determine if it was appropriate to determine a combined regression. For Δ 47 -T, the largest offset at 0°C was 0.002 ‰ between this study and Bajnai et al. (2020). The offset at 600°C was 0.005 ‰ between this study and Fiebig et al. (2021). For Δ 48 -T, the largest offset at 0°C was 0.012 ‰ between this study and Bajnai et al. (2020). The offset at 600°C was 0.007 ‰ between this study and Fiebig et al. (2021).
These offsets are well within the bounds of what we observed when comparing differences between ETH standard Δ 47 and Δ 48 values between laboratories (Figure 4). This is a good metric for interlaboratory analytical error due to large numbers of replicates of ETH standards in all groups. The offsets are also within the bounds expected from random error in Δ 47 measurements (Bernasconi et al., 2021). Therefore, we determined combined regressions for Δ 47 -T and Δ 48 -T (Figure 6 where T is in Kelvin. We also report the inverse of the relationships for ease of use for samples with unknown precipitation temperature in Equations 20 and 21.
It is unlikely that the offsets are the result of Devils Hole samples exhibiting kinetic effects from CO 2 degassing from groundwater, which is observed in other speleothems (Affek et al., 2008;Affek & Zaarur, 2014;Daëron et al., 2011;Guo, 2020;Guo & Zhou, 2019;Kluge & Affek, 2012). Clumped isotope values that exhibit kinetic effects from degassing result in decreased Δ 47 values and increased Δ 48 values, with an approximately linear early departure from equilibrium that has a slope of ∼−0.793 (Bajnai et al., 2020;Guo, 2020). The samples from Devils Hole do not follow this trend, as was concluded in Bajnai et al. (2021) and here (red arrow in Figure 7). Although we cannot preclude the possibility there are small, yet resolvable differences in Devils Hole clumped isotope values from samples of different ages given that these studies did not measure the same samples, the evidence here does not provide sufficient support for such a conclusion. It is noteworthy that the average Δ 47 and Δ 48 values from each study are within error of the interlaboratory Δ 47 -Δ 48 equilibrium regression presented in Equation 17 (Figure 7).
The combined average Δ 47 value from all replicates from samples in this study, Bajnai et al. (2021), , yielded a Δ 47 value of 0.571 ± 0.001 ‰, which yields a temperature value of 33.9 ± 0.3°C from Equation 20. The combined average Δ 48 value of 0.238 ± 0.007 ‰ yields a temperature value of 30.8 ± 6.8°C when input into Equation 21. Both the Δ 47 and Δ 48 reconstructed temperatures are consistent with measured temperature values from Devils Hole ranging from 32.8 to 34.3°C (Dudley & Larson, 1976;Hoffman, 1988;

Conclusions
This study contributes to establishing Δ 48 standard values that can be used in carbonate standard-based data normalization; however, further analyses of carbonate standard Δ 48 values may increase interlaboratory agreement. Our data supports previous research (Bernasconi et al., 2018(Bernasconi et al., , 2021Dennis et al., 2011;Upadhyay et al., 2021) that carbonate-based data normalization is a robust technique for Δ 47 , and demonstrates that it also produces statistically indistinguishable Δ 48 data on varying instrumentation. Carbonate-based standardization allows workers to use routinely analyzed standards for both Δ 47 and Δ 48 analyses and applies similar correction schemes to raw Δ 47 and Δ 48 values, reducing standardization error.
We have further constrained the Δ 47 -Δ 48 , Δ 47 -T, and Δ 48 -T equilibrium relationships with experimental values for standards and samples, and theoretical equilibrium values, and formed regressions using data from this study and previously published work. These regressions are useful for determining if unknown samples precipitated at isotopic equilibrium and can therefore be used in accurate temperature reconstructions, or potentially to recover primary temperatures by using Δ 47 -Δ 48 slopes determined for various kinetic and mixing processes.  Bajnai et al. (2021) were taken from their Supporting Information data, which provided values determined using carbonate standard based data normalization.  Bajnai et al. (2021),

Data Availability Statement
All code used in analyses are available for review at https://doi.org/10.5281/zenodo.7311624. All replicate data are available in Supporting Information S2.