Michael A Antonelli
email: mantonelli@berkeley.edu

Justin I Simon

Calcium isotopes in high-temperature terrestrial processes

Keywords: Ca isotopes, igneous petrology, metamorphic petrology, volcanology, hydrothermal systems

Introduction

Over the last two decades it has become increasingly clear that Ca isotopes can be used to trace a wide variety of high-temperature solid Earth processes. Ca isotopes have been used to gain insight into diverse topics relevant to high-temperature geochemistry, including crustal recycling and contamination, rates and mechanisms of metamorphic and igneous processes (such as mantle metasomatism, metamorphic mineral paragenesis, and igneous crystal growth), and Ca transport in otherwise inaccessible regions of the solid Earth. The increasing body of work that has investigated Ca isotope fractionation has also yielded fundamental information on the behavior of redox-insensitive cations in high-temperature systems. In this fashion, understanding Ca isotope fractionation serves as a baseline for understanding the behavior of more complex, redox-sensitive, isotopic systems that undergo changes in oxidation-state (e.g. Fe, Cr, S, Ce), but which must also succumb to similar fractionation effects during chemical reaction and diffusive transport.

There are six stable isotopes of calcium with a total relative mass range of ~20%: 40 Ca (96.94%), 42 Ca (0.65%), 43 Ca (0.14%), 44 Ca (2.09%), 46 Ca (~40 ppm), and 48 Ca (0.19%). Due to the higher abundances of 40 Ca and 44 Ca, stable Ca isotope variations are commonly reported as δ 44 Ca:

= -1 (Eq. 1)
This formalism results in positive δ 44 Ca values when a sample is enriched in 44 Ca relative to a known standard, and negative values when a sample is depleted in 44 Ca. It is important to note, however, that the most abundant isotope, 40 Ca, can also be produced by the radioactive decay of 40 K (half-life of ~1.25 Ga). Stable Ca isotope variations (δ 44 Ca) must therefore be corrected for the potential radiogenic ingrowth of 40 Ca in analyzed samples. This is commonly accomplished by either measuring 44 Ca/ 42 Ca and then converting to 44 Ca/ 40 Ca by multiplying by 2.05 [START_REF] Fantle | Calcium isotopes in the global biogeochemical Ca cycle: Implications for development of a Ca isotope proxy[END_REF], which is the canonical mass-law for kinetic Ca isotope fractionation (the equilibrium mass law is ~2.10), or by measuring 44 Ca/ 40 Ca directly and then correcting for radiogenic 40 Ca ingrowth using additional (unspiked) measurements. Reporting Ca isotope data as δ 44/42 Ca instead of δ 44 Ca would serve to separate stable from radiogenic isotope variations, but double-spike techniques used to correct for instrumental mass fractionation (IMF) during thermal ionization mass spectrometry (TIMS) often include 42 Ca and 48 Ca (or 43 Ca) spikes. Although 43 Ca- 46 Ca double spikes would be the most useful to explore variations in 40 Ca, 42 Ca, 44 Ca, and 48 Ca simultaneously, the rarity of these isotopes greatly increases the cost of analysis, rendering this combination much less common.

The application of δ 44 Ca measurements to high-temperature systems began to flourish starting with the observation that large δ 44 Ca fractionations (up to ~10‰) were possible during chemical diffusion in silicate liquids [START_REF] Richter | Isotope fractionation by chemical diffusion between molten basalt and rhyolite[END_REF][START_REF] Richter | Isotope fractionation by diffusion in molten oxides[END_REF]. For the next decade, research focused mainly on exploring the various mechanisms responsible for diffusive Ca isotope fractionations in silicate liquids, which were shown to depend on: (i) the magnitude of Ca concentration gradients, (ii) the direction and magnitude of chemical gradients for other species, relative to those of Ca, (iii) the strength of imposed temperature gradients, and (iv) the degree of melt polymerization (F. Huang et al., 2010;[START_REF] Richter | Isotopic fractionation of the major elements of molten basalt by chemical and thermal diffusion[END_REF][START_REF] Richter | Isotope fractionation by chemical diffusion between molten basalt and rhyolite[END_REF][START_REF] Richter | Isotope fractionation by diffusion in molten oxides[END_REF][START_REF] Watkins | Kinetic Fractionation of Non-Traditional Stable Isotopes by Diffusion and Crystal Growth Reactions[END_REF][START_REF] Watkins | Diffusion of multi-isotopic chemical species in molten silicates[END_REF][START_REF] Watkins | Influence of liquid structure on diffusive isotope separation in molten silicates and aqueous solutions[END_REF][START_REF] Watkins | Liquid composition-dependence of calcium isotope fractionation during diffusion in molten silicates[END_REF].

The first δ 44 Ca measurements in natural igneous rocks served as a baseline for ecological studies

[where food-chain δ 44 Ca variations were found to depend on trophic levels [START_REF] Skulan | Biological control of calcium isotopic abundances in the global calcium cycle[END_REF]],

and were not further explored until the late 2000's [START_REF] Amini | Calcium isotopes (d44/40Ca) in MPI-DING reference glasses, USGS rock powders and various rocks: Evidence for Ca isotope fractionation in terrestrial silicates[END_REF][START_REF] Jochum | MPI-DING reference glasses for in situ microanalysis: New reference values for element concentrations and isotope ratios[END_REF][START_REF] Simon | Stable calcium isotopic composition of meteorites and rocky planets[END_REF][START_REF] Wombacher | Separation of Mg, Ca and Fe from geological reference materials for stable isotope ratio analyses by MC-ICP-MS and double-spike TIMS[END_REF]. In the last ten years, however, there have been an increasing number of Ca isotope studies in natural high-temperature rocks and minerals, partially motivated by the initial observation that coexisting orthopyroxene and clinopyroxene in mantle xenoliths have significant δ 44 Ca variations (up to ~1 ‰) and that δ 44 Ca variations in Hawaiian basalts could be related to subduction of carbonates with low δ 44 Ca [START_REF] Huang | Stable calcium isotopic compositions of Hawaiian shield lavas: Evidence for recycling of ancient marine carbonates into the mantle[END_REF][START_REF] Huang | Calcium isotopic fractionation between clinopyroxene and orthopyroxene from mantle peridotites[END_REF]. Subsequent studies have mainly focused on understanding δ 44 Ca variations in mantle whole-rock samples (and rock standards) but have also explored δ 44 Ca variations in whole-rocks and mineral separates from mafic, silicic, metamorphic, and hydrothermal samples. These studies highlight the potential for Ca isotopes to be used as proxies for mantle metasomatism, carbonate recycling, crustal contamination, Ca diffusion rates/mechanisms in the continental crust, partial melting, equilibration temperature, metamorphic mineral paragenesis, K-loss during anatexis and high-temperature metamorphism, mineral precipitation rates in hydrothermal systems, and crystal growth rates in magmatic systems. Here, we present an overview of available Ca isotope data in high-temperature samples and explore the potential strengths and limitations of their use as constraints for natural hightemperature processes.

Ca isotope nomenclature

Similar to many other stable isotope systems, Ca isotope values are generally reported relative to a known standard in lowercase delta-notation (δ 44 Ca, Eq. 1) and followed by a per mil symbol (‰), which indicates that the value has been multiplied by 1,000. Although it is redundant (and technically incorrect) to include both a per mil symbol after reported values and a factor of 1,000 in the definition of δ 44 Ca, many authors and peer-reviewers still push for both, which unfortunately renders isotope nomenclature more complex than it needs to be, especially for students and scientists outside of our community. IUPAC has suggested to either (i) remove the factor of 1,000 from the definition (as we have done in Equation 1) if using a per mil symbol after reported values, or (ii) to omit the per mil symbol after values, if including a factor of 1,000 in the definition of δ 44 Ca [START_REF] Coplen | Guidelines and recommended terms for expression of stable-isotope-ratio and gasratio measurement results[END_REF]. Other than common delta-notation, formalisms typically used in the isotope community include (i) epsilon (ε) notation, which reports deviations from the standard in parts per ten-thousand (commonly used for radiogenic 40 Ca variations), and

(ii) mu (μ) notation, which reports values as parts per million (often used for particularly small variations).

In this review, we have chosen to include either per mil symbols (‰, for delta-notation) or indicate parts-per-million (ppm, for μ-notation) after reported values, in order to omit the factors of 1,000 and 1,000,000 from our definitions. For epsilon notation, we could potentially omit the factor of 10,000 from our definition of εCa by reporting values with a per myriad (‱) or partsper-ten-thousand (pptt) symbol, but to our knowledge, neither have been previously used. We therefore include the multiplicative factor in our definition of εCa (in Section 7) and do not use any symbols after reporting ε-values. When comparing two related samples, isotopic differences are generally reported using capital delta notation (e.g. Δ 44 Caa-b, = δ 44 Caa -δ 44 Cab, for samples a and b), which can also be used to report deviations from an expected value [e.g. Δ 48 Ca' = μ 48 Cameasured -μ 48 Caequilibrium]. All these notations, however, are further introduced and discussed in the most pertinent sections of this review.

Observations of mass-dependent Ca isotope variations in igneous and metamorphic rocks

Since the first measurements of stable Ca isotope distributions in volcanic rocks, there have been over 60 studies measuring δ 44 Ca in natural high-temperature samples. One of the common goals for many of these studies is the definition of δ 44 Ca for the bulk silicate Earth (BSE). Although analytical uncertainties and potential sample-biases initially hindered the precise definition of BSE δ 44 Ca, a growing number of studies based on mantle rocks and meteorites are converging towards a relatively well-constrained δ 44 Ca value of +0.95‰ relative to carbonate standard SRM915a, which is approximately the same as initially estimated from early studies. The use of BSE as a reference value for Ca isotope measurements [which comes from the first exploration of stable Ca isotope variations in terrestrial/biological materials [START_REF] Skulan | Biological control of calcium isotopic abundances in the global calcium cycle[END_REF]], however, predates the use of SRM915a. Nevertheless, the analyzed igneous rocks and minerals in this early study gave the same values (within uncertainty, ~3-4 times larger than today) as the most up-to-date high-precision estimates e.g. [START_REF] Kang | Calcium isotope sources and fractionation during melt-rock interaction in the lithospheric mantle: Evidence from pyroxenites, wehrlites, and eclogites[END_REF]. In order to best illustrate potential differences and complementarity between Earth reservoirs, therefore, δ 44 Ca data from UC-Berkeley have since been reported relative to an estimated BSE value (which has corresponded to +0.95‰ relative to SRM915a for the last decade, but was slightly higher in earlier studies).

More than four different reference materials, however, are commonly used to define δ 44 Ca (including SRM915a, SRM915b, BSE, and modern seawater) and they must be converted to a common scale in order to allow for comparison between studies. Although SRM915a is no longer commercially available and seawater δ 44 Ca is, on average, controlled by the extent of carbonate precipitation in the oceans (which may have changed over geologic time) and can also vary based on collection depth and location [START_REF] Chakrabarti | Large stable Ca isotopic (δ44/40 Ca) variation in open ocean samples from the Bay of Bengal[END_REF], it is hard to argue for the use of one standard over another. In Table Additional complications result from small radiogenic 40 Ca enrichments in standard SRM915a

[and in modern carbonates/seawater [START_REF] Antonelli | Radiogenic 40Ca in Seawater[END_REF]] relative to basaltic igneous rocks (Antonelli et al., 2019a;[START_REF] He | Mass-Independent and Mass-Dependent Ca Isotopic Compositions of Thirteen Geological Reference Materials Measured by Thermal Ionisation Mass Spectrometry[END_REF][START_REF] Mills | Calcium and neodymium radiogenic isotopes of igneous rocks: Tracing crustal contributions in felsic magmas related to super-eruptions and continental rifting[END_REF][START_REF] Simon | Calcium Isotope Composition of Meteorites, Earth, and Mars[END_REF][START_REF] Zhao | Coupled extremely light Ca and Fe isotopes in peridotites[END_REF], meteorites [START_REF] Simon | Calcium Isotope Composition of Meteorites, Earth, and Mars[END_REF][START_REF] Yokoyama | Extreme early solar system chemical fractionation recorded by alkali-rich clasts contained in ordinary chondrite breccias[END_REF], and modern hydrothermal fluids (Antonelli et al., 2019a). These radiogenic enrichments can lead to small discrepancies in (Antonelli et al., 2019a[START_REF] Antonelli | Radiogenic 40Ca in Seawater[END_REF][START_REF] Mills | Calcium and neodymium radiogenic isotopes of igneous rocks: Tracing crustal contributions in felsic magmas related to super-eruptions and continental rifting[END_REF]. The apparent radiogenic enrichments in SRM915a and seawater, however, would require a negative shift of ~0.1‰ for TIMS δ 44 Ca data from samples with lesser amounts of radiogenic 40 Ca than the normalizing standards.

Bulk silicate Earth composition: measurements and models

Early literature data for oceanic basalts suggest relatively large (>0.5‰) variations in δ 44 Ca [START_REF] Depaolo | Calcium Isotopic Variations Produced by Biological, Kinetic, Radiogenic and Nucleosynthetic Processes[END_REF][START_REF] Skulan | Biological control of calcium isotopic abundances in the global calcium cycle[END_REF][START_REF] Zhu | Calcium isotopes in the marine environment and the oceanic calcium cycle[END_REF]. Likewise, measurement of igneous minerals, especially the principal Ca-bearing minerals found in mantle peridotites (clinopyroxene, orthopyroxene), exhibit significant isotopic heterogeneity (e.g. S. Huang et al., 2010). Based on these observations and the upper mantle mineral assemblage, Huang et al.

(2010) estimated that the upper mantle and thus Bulk Silicate Earth (BSE) δ 44 Ca is 1.05±0.04‰, relative to Ca isotope standard SRM915a. This mantle estimate is slightly higher (by ~0.1) than the average δ 44 Ca of the basalt samples measured by Huang et al. (2010), and the average value reported by [START_REF] Amini | Calcium isotopes (d44/40Ca) in MPI-DING reference glasses, USGS rock powders and various rocks: Evidence for Ca isotope fractionation in terrestrial silicates[END_REF] and more recent studies. Using the published data on igneous rocks from [START_REF] Skulan | Biological control of calcium isotopic abundances in the global calcium cycle[END_REF][START_REF] Depaolo | Calcium Isotopic Variations Produced by Biological, Kinetic, Radiogenic and Nucleosynthetic Processes[END_REF] the approach of Huang et al. (2010) would indicate that δ 44 Ca of BSE should be close to ~1.00±0.15‰, relative to standard SRM915a. The work of [START_REF] Simon | Stable calcium isotopic composition of meteorites and rocky planets[END_REF] found that the average δ 44 Ca value representing a large diversity of differentiated materials and ordinary chondritic meteorites (n=12), including samples from the Moon and Mars, is indistinguishable from terrestrial basalts and mantle peridotites (n=5) and used this value to define both the Ca isotopic composition of BSE as well as the terrestrial planets and planetary bodies that formed within the inner Solar System (their value for BSE is 0.97±0.07‰ after re-normalization to the δ 44 Ca of SRM915a). More recently, [START_REF] Kang | Calcium isotopic fractionation in mantle peridotites by melting and metasomatism and Ca isotope composition of the Bulk Silicate Earth[END_REF] reported that a subset of 28 peridotites xenoliths, those which experienced little or no melting and metasomatism (n=14) exhibited limited δ 44 Ca variation from 0.90 to 0.99‰ yielding an average value of 0.94±0.05‰ (relative to SRM915a). Following a similar approach to Simon and DePaolo (2010), [START_REF] Kang | Calcium isotopic fractionation in mantle peridotites by melting and metasomatism and Ca isotope composition of the Bulk Silicate Earth[END_REF] used their new data and a compilation of available Ca isotope literature data to define the Ca isotopic composition of BSE (δ 44 Ca = +0.95 ± 0.05‰, relative to SRM915a). Given the precision and accuracy of Ca isotope measurements in the published literature it appears that both approaches, those that compute BSE values from mineral compositions and those who define BSE from representative whole rock samples (from Earth and other planetary bodies) yield similar values. As more studies of carefully selected mantlederived samples are analyzed, the Ca isotopic composition of the upper mantle will be more fully characterized.

Observations from igneous rocks

Since the first observation of systematic δ 44 Ca variations in Hawaiian basalts [START_REF] Huang | Stable calcium isotopic compositions of Hawaiian shield lavas: Evidence for recycling of ancient marine carbonates into the mantle[END_REF] 5% Cpx). Reactions between percolating pyroxene-undersaturated basaltic melts and lherzolites can then lead to formation of replacive dunites (> 90% olivine), while simultaneous meltpercolation and crystallization can lead to formation of websterites (< 5% olivine), clinopyroxenites (> 90% Cpx), orthopyroxenites (> 90% Opx), and wehrlites (<5% Opx), depending on the saturation state of the percolating melt with respect to the various minerals (e.g. Chen et al., 2019b). Further classification of major ultramafic rock types depends on the presence of either spinel or garnet as an aluminous phase, with garnet-bearing rocks indicative of higher pressures. Mafic to felsic rocks are also distinguished through modal mineralogy but are more commonly classified by their SiO2 and associated alkali element concentrations (Na2O and K2O).

In Fig. 1, we have compiled >650 igneous whole rock δ 44 Ca measurements and have grouped them by major rock type, increasing from ultramafic (bottom) to felsic compositions (top).

Within each major group the categories are further broken down by whether they are intrusive or extrusive, and then by their dominant mineralogies (if originally described).

Although there is scatter in the data, the first observation is that there is a rough inverse relationship between SiO2 content and δ 44 Ca, where increasingly felsic rocks have decreasing δ 44 Ca (Fig. 1). This observation is in agreement with recent work regarding the average δ 44 CaBSE compositions of mid-ocean ridge basalts (MORB, ~ -0.1‰) (Chen et al., 2019a) and estimated average continental crust (~ -0.2‰) [START_REF] Wang | Calcium isotope fractionation during crustal melting and magma differentiation: Granitoid and mineral-pair perspectives[END_REF] relative to non-metasomatized peridotite samples, e.g. (Chen et al., 2019b). By combining data from different labs, figure 1 compensates for potential inter-lab biases (e.g. ~0.2‰ variation in published values for BHVO and BCR-2 standards), but additional work is necessary in order to reach agreement between labs and move forward with higher precision Ca isotope measurements as a community. Earth (BSE). δ 44 CaBSE is +0.95‰ relative to standard SRM915a, +0.23‰ relative to SRM915b, and -0.90‰ relative to seawater (see Table 1). Igneous rock data are from [START_REF] Amini | Calcium isotopes (d44/40Ca) in MPI-DING reference glasses, USGS rock powders and various rocks: Evidence for Ca isotope fractionation in terrestrial silicates[END_REF][START_REF] Amsellem | Evolution of the Ca isotopic composition of the mantle[END_REF][START_REF] Amsellem | Testing the chondrule-rich accretion model for planetary embryos using calcium isotopes[END_REF]Antonelli et al., 2019b;Banerjee andChakrabarti, 2019, 2018;Chen et al., 2019aChen et al., , 2019b[START_REF] Chen | Calcium isotope evidence for subduction-enriched lithospheric mantle under the northern North China Craton[END_REF][START_REF] Colla | Calcium-isotope fractionation between solution and solids with six, seven or eight oxygens bound to Ca(II)[END_REF][START_REF] Dai | Calcium isotope compositions of mantle pyroxenites[END_REF][START_REF] Depaolo | Calcium Isotopic Variations Produced by Biological, Kinetic, Radiogenic and Nucleosynthetic Processes[END_REF][START_REF] Feng | A rapid and simple single-stage method for Ca separation from geological and biological samples for isotopic analysis by MC-ICP-MS[END_REF][START_REF] Feng | Calcium Isotopic Compositions of Sixteen USGS Reference Materials[END_REF][START_REF] He | Mass-Independent and Mass-Dependent Ca Isotopic Compositions of Thirteen Geological Reference Materials Measured by Thermal Ionisation Mass Spectrometry[END_REF][START_REF] Holmden | Ca isotope cycling in a forested ecosystem[END_REF][START_REF] Huang | Stable calcium isotopic compositions of Hawaiian shield lavas: Evidence for recycling of ancient marine carbonates into the mantle[END_REF][START_REF] Huang | Calcium isotopic fractionation between clinopyroxene and orthopyroxene from mantle peridotites[END_REF][START_REF] Ionov | Calcium isotopic signatures of carbonatite and silicate metasomatism, melt percolation and crustal recycling in the lithospheric mantle[END_REF][START_REF] Jacobson | Silicate versus carbonate weathering in Iceland: New insights from Ca isotopes[END_REF][START_REF] Jochum | MPI-DING reference glasses for in situ microanalysis: New reference values for element concentrations and isotope ratios[END_REF][START_REF] John | Volcanic arcs fed by rapid pulsed fluid flow through subducting slabs[END_REF][START_REF] Kang | Calcium isotopic fractionation in mantle peridotites by melting and metasomatism and Ca isotope composition of the Bulk Silicate Earth[END_REF][START_REF] Lehn | Optimization of a 48Ca-43Ca double-spike MC-TIMS method for measuring Ca isotope ratios (δ44/40Ca and δ44/42Ca): limitations from filament reservoir mixing[END_REF]Liu et al., 2017bLiu et al., , 2017a;;[START_REF] Lu | Behavior of calcium isotopes during continental subduction recorded in meta-basaltic rocks[END_REF][START_REF] Magna | The calcium isotope systematics of Mars[END_REF][START_REF] Peterson | Rates and Mechanisms of Magmatic Processes: Isotopic and Geochronological Evidence[END_REF][START_REF] Richter | Isotope fractionation by chemical diffusion between molten basalt and rhyolite[END_REF][START_REF] Ryu | The major ion, d44/40Ca, d44/42Ca, and d26/24Mg geochemistry of granite weathering at pH=1 and T=25 C: Power-law processes and the relative reactivity of minerals[END_REF][START_REF] Schiller | Calcium isotope measurement by combined HR-MC-ICPMS and TIMS[END_REF][START_REF] Simon | Calcium and titanium isotope fractionation in refractory inclusions : Tracers of condensation and inheritance in the early solar protoplanetary disk[END_REF][START_REF] Simon | Stable calcium isotopic composition of meteorites and rocky planets[END_REF][START_REF] Skulan | Biological control of calcium isotopic abundances in the global calcium cycle[END_REF][START_REF] Valdes | The nature of Earth's building blocks as revealed by calcium isotopes[END_REF]Wang et al., 2017b[START_REF] Wang | Calcium isotope fractionation during crustal melting and magma differentiation: Granitoid and mineral-pair perspectives[END_REF][START_REF] Wombacher | Separation of Mg, Ca and Fe from geological reference materials for stable isotope ratio analyses by MC-ICP-MS and double-spike TIMS[END_REF][START_REF] Zhang | No measurable calcium isotopic fractionation during crystallization of Kilauea Iki lava lake[END_REF][START_REF] Zhao | Coupled extremely light Ca and Fe isotopes in peridotites[END_REF][START_REF] Zhu | Calcium isotopic fractionation during plate subduction: Constraints from back-arc basin basalts[END_REF][START_REF] Zhu | Calcium Isotopic Compositions of Normal Mid-Ocean Ridge Basalts From the Southern Juan de Fuca Ridge[END_REF][START_REF] Zhu | Calcium isotopes in the marine environment and the oceanic calcium cycle[END_REF]. The approximate ranges for 95% of carbonates and rivers come from [START_REF] Fantle | Calcium isotopes in the global biogeochemical Ca cycle: Implications for development of a Ca isotope proxy[END_REF].

Ultramafic samples have the highest δ 44 Ca variability of all analyzed igneous rocks, though this may be partially due to the greater number of analyzed samples (Fig. 1). The ultramafic samples with the highest δ 44 Ca are dunites, which is likely due to their high percentages of olivine (which is predicted to favor heavy isotopes at equilibrium, discussed in Section 3). The lightest ultramafic δ 44 Ca values are found in komatiites, foidites, carbonatites, and spinel lherzolites and harzburgites. The spinel-facies sub-types of lherzolites, harzburgites, and websterites tend to have more negative δ 44 Ca than their garnet-facies counterparts. This observation may be explained by the presence of garnet as an isotopically heavy residual phase (discussed in Section 3) but requires further exploration. Heavy δ 44 Ca compositions in certain komatiites have been argued to reflect mantle-heterogeneities inherited from Ca-perovskite crystallization during the early magma ocean period of Earth history, while light δ 44 Ca may reflect incorporation of serpentinized oceanic crust into komatiite sources [START_REF] Amsellem | Evolution of the Ca isotopic composition of the mantle[END_REF]; only one study, however, suggests that subducting slabs preferentially lose fluids enriched in heavy Ca isotopes [START_REF] John | Volcanic arcs fed by rapid pulsed fluid flow through subducting slabs[END_REF]. This conclusion was reached primarily because blueschists adjacent to eclogites in Western Tianshan had slightly heavier Ca isotopic compositions (suggesting the influx of fluids enriched in heavy Ca isotopes), but it is uncertain whether this reflects (i) a primary signature of the fluids lost from the eclogites or (ii) that these fluids were subsequently enriched due to precipitation of isotopically light carbonates [START_REF] John | Volcanic arcs fed by rapid pulsed fluid flow through subducting slabs[END_REF]. A recent study suggests that prograde dehydration metamorphism in subduction zones does not lead to significant Ca isotope fractionation [START_REF] Lu | Behavior of calcium isotopes during continental subduction recorded in meta-basaltic rocks[END_REF] Although there is significantly less data for intermediate to felsic rocks, the largest variations are observed in granites (Fig. 1). Granites, along with intermediate igneous rocks, have δ 44 Ca almost exclusively more negative than bulk silicate Earth, but their higher variability (> 1‰) compared to other groups suggests that they are (i) affected by a greater number of (potentially kinetic)

processes and/or (ii) that they incorporate a greater variety of isotopically-distinct source materials. A larger number of studies on intermediate to felsic rocks is required to understand this variability.

Given the greater number of data for ultramafic samples, partial melting and fractional crystallization processes (as shown in Fig. 2) are better understood in mantle rocks. The general observation is that partial melting of fertile mantle generally leaves behind isotopically heavier mafic minerals (e.g. Opx, Ol) and higher Mg#, defined 10 2 [Mg/(Mg+Fe)], as isotopically lighter phases (e.g. Cpx) are preferentially lost to the melt. Fractional crystallization of mafic phases leads to decreasing Mg# in the residual liquid, but the isotopic effects are negligible as a probable consequence of their low Ca concentrations (e.g. olivine, orthopyroxene). Zhang et al.

(2018) modeled melting of spinel facies peridotite (Fig. 2c,d) at pressures between 1 and 2 GPa using the pMELTS program. They found that isotopic variations generated from both primitive and depleted mantle compositions lead to less than ~0.1‰ variation in the melt fractions.

Residual material, on the other hand, can show larger variations, as isotopically-heavy orthopyroxene becomes the dominant Ca bearing phase, with potential δ 44 Ca increases of the order ~0.2 to 0.3‰ [at a corresponding melt-fraction (F) of 0.3] when using a fractional melting model [START_REF] Zhang | No measurable calcium isotopic fractionation during crystallization of Kilauea Iki lava lake[END_REF]. Mantle melting at higher-pressure, amenable to the stabilization of garnet as a residual phase, leads to larger effects (up to ~0.4‰) for the generated melts (Kang et al., 2019), while generation of adakitic melts may succumb to similar processes [START_REF] Wang | Calcium isotope fractionation during crustal melting and magma differentiation: Granitoid and mineral-pair perspectives[END_REF]. Fractionation factors still remain poorly constrained, however, and the large δ 44 Ca variations observed in metasomatized mantle samples, along with their relationships to a wide variety of percolating mantle fluids [potentially causing kinetic isotope fractionations (e.g. [START_REF] Zhao | Coupled extremely light Ca and Fe isotopes in peridotites[END_REF]], still merit further study. come from studies published before 2019 [START_REF] Amini | Calcium isotopes (d44/40Ca) in MPI-DING reference glasses, USGS rock powders and various rocks: Evidence for Ca isotope fractionation in terrestrial silicates[END_REF][START_REF] Colla | Calcium-isotope fractionation between solution and solids with six, seven or eight oxygens bound to Ca(II)[END_REF][START_REF] Depaolo | Calcium Isotopic Variations Produced by Biological, Kinetic, Radiogenic and Nucleosynthetic Processes[END_REF][START_REF] Fantle | Calcium isotopes in the global biogeochemical Ca cycle: Implications for development of a Ca isotope proxy[END_REF][START_REF] Feng | Calcium Isotopic Compositions of Sixteen USGS Reference Materials[END_REF][START_REF] He | Mass-Independent and Mass-Dependent Ca Isotopic Compositions of Thirteen Geological Reference Materials Measured by Thermal Ionisation Mass Spectrometry[END_REF][START_REF] Holmden | Ca isotope cycling in a forested ecosystem[END_REF][START_REF] Huang | Calcium isotopic fractionation between clinopyroxene and orthopyroxene from mantle peridotites[END_REF][START_REF] Huang | Stable calcium isotopic compositions of Hawaiian shield lavas: Evidence for recycling of ancient marine carbonates into the mantle[END_REF][START_REF] Jacobson | Silicate versus carbonate weathering in Iceland: New insights from Ca isotopes[END_REF][START_REF] Jochum | MPI-DING reference glasses for in situ microanalysis: New reference values for element concentrations and isotope ratios[END_REF][START_REF] John | Volcanic arcs fed by rapid pulsed fluid flow through subducting slabs[END_REF][START_REF] Kang | Calcium isotopic fractionation in mantle peridotites by melting and metasomatism and Ca isotope composition of the Bulk Silicate Earth[END_REF][START_REF] Kang | Calcium isotopic composition of mantle xenoliths and minerals from Eastern China[END_REF][START_REF] Lehn | Optimization of a 48Ca-43Ca double-spike MC-TIMS method for measuring Ca isotope ratios (δ44/40Ca and δ44/42Ca): limitations from filament reservoir mixing[END_REF]Liu et al., 2017aLiu et al., , 2017b;;[START_REF] Magna | The calcium isotope systematics of Mars[END_REF][START_REF] Richter | Isotope fractionation by chemical diffusion between molten basalt and rhyolite[END_REF][START_REF] Ryu | The major ion, d44/40Ca, d44/42Ca, and d26/24Mg geochemistry of granite weathering at pH=1 and T=25 C: Power-law processes and the relative reactivity of minerals[END_REF][START_REF] Schiller | Calcium isotope measurement by combined HR-MC-ICPMS and TIMS[END_REF][START_REF] Simon | Calcium and titanium isotope fractionation in refractory inclusions : Tracers of condensation and inheritance in the early solar protoplanetary disk[END_REF][START_REF] Simon | Stable calcium isotopic composition of meteorites and rocky planets[END_REF][START_REF] Skulan | Biological control of calcium isotopic abundances in the global calcium cycle[END_REF][START_REF] Valdes | The nature of Earth's building blocks as revealed by calcium isotopes[END_REF][START_REF] Wombacher | Separation of Mg, Ca and Fe from geological reference materials for stable isotope ratio analyses by MC-ICP-MS and double-spike TIMS[END_REF][START_REF] Zhang | No measurable calcium isotopic fractionation during crystallization of Kilauea Iki lava lake[END_REF][START_REF] Zhao | Coupled extremely light Ca and Fe isotopes in peridotites[END_REF][START_REF] Zhu | Calcium Isotopic Compositions of Normal Mid-Ocean Ridge Basalts From the Southern Juan de Fuca Ridge[END_REF][START_REF] Zhu | Calcium isotopes in the marine environment and the oceanic calcium cycle[END_REF].

Observations from metamorphic rocks

Metamorphic rocks are typically classified based on mineral compositions and modal mineralogy, which are related to the starting materials (either igneous or sedimentary protoliths)

and to the temperature and pressure conditions of metamorphism. In a general sense, high temperature metamorphism (e.g. granulite-facies, T > ~750 ○ C, ~0.4-1.4 GPa) favors the transformation of hydrous minerals to anhydrous assemblages, whereas high-pressure metamorphism [e.g. blueschist-facies (< ~500 ○ C, ~0.5-1.5 GPa) and eclogite-facies (> ~400 ○ C, > ~1.5 GPa)] favors the formation of high-density mineral assemblages, though dehydration can also occur. Metamorphic facies intermediate to the others (e.g. greenschist and amphibolitefacies) are also defined by their mineral assemblages and result from the variable influences of metasomatic fluids, temperature (300-700 ○ C), and pressure (~0.2 to ~1 GPa).

Although Ca isotopes in metamorphic rocks are not as extensively studied as in igneous rocks, the limited number of studies (Antonelli et al., 2019c;[START_REF] John | Volcanic arcs fed by rapid pulsed fluid flow through subducting slabs[END_REF][START_REF] Kang | Calcium isotope sources and fractionation during melt-rock interaction in the lithospheric mantle: Evidence from pyroxenites, wehrlites, and eclogites[END_REF]Liu et al., 2017b;[START_REF] Lu | Behavior of calcium isotopes during continental subduction recorded in meta-basaltic rocks[END_REF][START_REF] Peterson | Rates and Mechanisms of Magmatic Processes: Isotopic and Geochronological Evidence[END_REF][START_REF] Valdes | The effects of high-temperature fractional crystallization on calcium isotopic composition[END_REF][START_REF] Wang | Calcium isotope fractionation during crustal melting and magma differentiation: Granitoid and mineral-pair perspectives[END_REF][START_REF] Wombacher | Separation of Mg, Ca and Fe from geological reference materials for stable isotope ratio analyses by MC-ICP-MS and double-spike TIMS[END_REF] suggest that large fractionation effects occur during metamorphism. Whole rocks span a δ 44 Ca range of ~6‰ (-2.5 to +3.3‰ relative to BSE, Fig. 3). The δ 44 Ca variations appear to get larger with increasing metamorphic grade, with the largest fractionations in rocks that have experienced upper amphibolite-facies [START_REF] Peterson | Rates and Mechanisms of Magmatic Processes: Isotopic and Geochronological Evidence[END_REF] and granulite-facies metamorphism (Antonelli et al., 2019c;[START_REF] Valdes | The effects of high-temperature fractional crystallization on calcium isotopic composition[END_REF]. Data are from (Antonelli et al., 2019c;[START_REF] John | Volcanic arcs fed by rapid pulsed fluid flow through subducting slabs[END_REF][START_REF] Kang | Calcium isotope sources and fractionation during melt-rock interaction in the lithospheric mantle: Evidence from pyroxenites, wehrlites, and eclogites[END_REF]Liu et al., 2017b;[START_REF] Lu | Behavior of calcium isotopes during continental subduction recorded in meta-basaltic rocks[END_REF][START_REF] Peterson | Rates and Mechanisms of Magmatic Processes: Isotopic and Geochronological Evidence[END_REF][START_REF] Valdes | The effects of high-temperature fractional crystallization on calcium isotopic composition[END_REF][START_REF] Wombacher | Separation of Mg, Ca and Fe from geological reference materials for stable isotope ratio analyses by MC-ICP-MS and double-spike TIMS[END_REF], retrograde greenschist data are from Valdes et al. (2018).

Given that inter-mineral fractionations in a closed system should not result in whole rock δ 44 Ca variations, the observed variability indicates that Ca isotope fractionations occur in open systems and require either addition or removal of isotopically distinct components during metamorphism.

The larger fractionations at higher temperatures suggest that the δ 44 Ca effects are likely to be the result of kinetic fractionation effects during Ca diffusion, melting, and/or dehydration reactions, which are further discussed in Section 6.

Equilibrium Ca isotope fractionation

As with many isotopic systems, the initial goal for inter-mineral Ca isotope analyses was to use them as constraints on equilibration temperature ('Ca isotope geothermometry'). Using Ca isotopes as a proxy for temperature in natural samples, however, requires (i) knowledge of equilibrium isotope fractionations between minerals at different temperatures, and (ii) that isotopic equilibrium was reached between minerals. The following sections evaluate the potential use of Ca isotopes for geothermometry and describe current constraints on equilibrium Δ 44 Caa-b, defined:

= - (Eq. 2)
Where subscripts 'a' and 'b' denote two minerals 'a' and 'b'.

General background principles

Isotope partitioning between mineral phases at equilibrium depends mainly on vibrational frequency differences for heavy vs. light isotopes in mineral structures. Tighter bonds (with higher vibrational frequencies) are preferentially stabilized by substitution of a heavy isotope.

This effect is a consequence of quantum mechanical factors, where tighter bonds have greater relative decreases in zero-point energy (ZPE, the quantized vibrational energy of a chemical bond at absolute zero), which stems from the Heisenberg uncertainty principle, and in higherlevel vibrational frequencies upon substitution of a heavier isotope [START_REF] Bigeleisen | Calculation of Equilibrium Constants for Isotopic Exchange Reactions[END_REF][START_REF] Chacko | Equilibrium Oxygen, Hydrogen and Carbon Isotope Fractionation Factors Applicable to Geologic Systems[END_REF][START_REF] Urey | The thermodynamic properties of isotopic substances[END_REF][START_REF] Young | Kinetic and equilibrium mass-dependant isotope fractionation laws in nature and their geochemical and cosmochemical significance[END_REF]. Consequently, heavier isotopes are preferentially placed into mineral sites with stronger bonds in order to decrease the Gibbs free energy (ΔG) of a system. For Ca isotopes, this leads to higher δ 44 Ca in minerals with stronger/shorter Ca-O bonds. When bond lengths are unknown, a general rule is that sites with higher coordination numbers have longer bonds and therefore have less preference for heavy isotopes than sites with lower coordination numbers [e.g. Ca-perovskite has 12-fold coordination and is thus predicted to have low reduced-partition function ratios (RPFR, discussed in Section 3.2) and concentrate lighter Ca isotopes [START_REF] Amsellem | Evolution of the Ca isotopic composition of the mantle[END_REF]]. At a given Ca-O bond-length, however, it is important to note that sites with higher coordination numbers have stronger overall bonding environments and therefore favor heavier isotopes.

Equilibrium constants for isotopic substitution reactions (denoted α, as opposed to K for chemical equilibrium constants) are highly dependent on temperature due to the increasing population of higher quantum vibrational states with increasing temperatures [START_REF] Chacko | Equilibrium Oxygen, Hydrogen and Carbon Isotope Fractionation Factors Applicable to Geologic Systems[END_REF]. As a result, isotopic differences between mineral pairs are often used as constraints on equilibration temperature ('isotope geothermometry'), where higher temperatures are expected to decrease inter-mineral fractionations.

Equilibrium fractionation factors (α), which must be constrained in order to derive temperatures from isotopic data, can be obtained through various methods. These generally include (i) direct measurements from experiments, such as mineral-mineral or mineral-carbonate exchange experiments [START_REF] Chacko | Equilibrium Oxygen, Hydrogen and Carbon Isotope Fractionation Factors Applicable to Geologic Systems[END_REF][START_REF] Shahar | Equilibrium high-temperature Fe isotope fractionation between fayalite and magnetite: An experimental calibration[END_REF], and/or determination of vibrational frequencies through (ii) advanced spectroscopic techniques [e.g. nuclear resonant inelastic x-ray scattering [START_REF] Dauphas | A general moment NRIXS approach to the determination of equilibrium Fe isotopic fractionation factors: Application to goethite and jarosite[END_REF]], or (iii) ab-initio simulations (e.g. [START_REF] Widanagamage | Stable strontium isotope fractionation in synthetic barite[END_REF].

The determination of vibrational frequencies using spectroscopic methods is not currently possible for Ca because there are no Mossbauer-sensitive isotopes. Common experimental approaches (e.g. [START_REF] Kueter | Experimental carbonatite/graphite carbon isotope fractionation and carbonate/graphite geothermometry[END_REF][START_REF] Shahar | Equilibrium high-temperature Fe isotope fractionation between fayalite and magnetite: An experimental calibration[END_REF] would technically allow for the determination of equilibrium Ca isotope fractionation factors, but at present we are not aware of any reliable experimental constraints. Thus, our current understanding of equilibrium Ca isotope fractionation between various minerals relies principally on ab-initio estimates, which are discussed in the following section.

Ab-initio Predictions

The first Ca isotope measurements in mineral separates revealed large δ 44 Ca differences (~1 ‰) in orthopyroxene and clinopyroxene from mantle xenoliths (S. Huang et al., 2010), suggesting that Ca isotope fractionations could potentially be used for geothermometry. The first ab-initio estimates for equilibrium Ca isotope fractionation between minerals, published four years later, confirmed the initial observation that orthopyroxene concentrates heavy isotopes of Ca over clinopyroxene at equilibrium. Ab-initio techniques generally rely on advanced computational methods that serve to predict vibrational frequency shifts for different isotopes in a given molecule or crystal lattice, which can then be used to predict equilibrium isotope effects.

Predicted equilibrium Δ 44 Caopx-cpx values were found to range from ~0.3‰ at 1200 ○ C to ~1‰ at 500 ○ C [START_REF] Feng | First-principles investigations of equilibrium calcium isotope fractionation between clinopyroxene and Ca-doped orthopyroxene[END_REF], which has also been corroborated by more recent ab-initio studies (Antonelli et al., 2019c;Wang et al., 2017b). Equilibrium predictions for minerals other than pyroxenes have only recently been investigated (Antonelli et al., 2019c;[START_REF] Huang | First-principles calculations of equilibrium Ca isotope fractionation : Implications for oldhamite formation and evolution of lunar magma ocean[END_REF][START_REF] Song | First-principles investigation of the concentration effect on equilibrium fractionation of Ca isotopes in forsterite[END_REF], but find that olivine is expected to have the heaviest Ca isotope compositions, followed by garnet, orthopyroxene, clinopyroxene, apatite, anorthite, labradorite (An50), and oldhamite (CaS).

In Fig. 4 we show the reduced partition function ratios (formalized as 1000lnβ, which quantifies the preference for heavy vs. light isotopes within a single phase) for various minerals vs.

temperature [START_REF] Huang | First-principles calculations of equilibrium Ca isotope fractionation : Implications for oldhamite formation and evolution of lunar magma ocean[END_REF][START_REF] Song | First-principles investigation of the concentration effect on equilibrium fractionation of Ca isotopes in forsterite[END_REF]Wang et al., 2017a). Fractionation factors (1000lnα), which represent the prediction for Δ 44 Caa-b between two minerals, can be approximated visually from these kinds of figures using 1000lnβa-1000lnβb, but should be calculated using αa-b=βa/βb (note: Δ 44 Caa-b ≈ 1000lnαa-b =1000lnβa-1000lnβb). The major-cation compositions of solid-solution minerals are also expected to affect Ca isotope fractionation (Antonelli et al., 2019c;[START_REF] Feng | First-principles investigations of equilibrium calcium isotope fractionation between clinopyroxene and Ca-doped orthopyroxene[END_REF][START_REF] Huang | First-principles calculations of equilibrium Ca isotope fractionation : Implications for oldhamite formation and evolution of lunar magma ocean[END_REF][START_REF] Song | First-principles investigation of the concentration effect on equilibrium fractionation of Ca isotopes in forsterite[END_REF]Wang et al., 2017b). For example, the preference for heavy Ca isotopes in orthopyroxene decreases with increasing Ca/Mg (Antonelli et al., 2019c;[START_REF] Feng | First-principles investigations of equilibrium calcium isotope fractionation between clinopyroxene and Ca-doped orthopyroxene[END_REF][START_REF] Huang | First-principles calculations of equilibrium Ca isotope fractionation : Implications for oldhamite formation and evolution of lunar magma ocean[END_REF]Wang et al., 2017b), as depicted in Figs. 4b and5. This effect is a general manifestation of increasing distortions in the mineral lattice with increasing Ca abundances (due to the relatively large size of Ca compared to Mg). The distortions lead to increased average Ca-O bond lengths and thus decrease the overall affinity for heavy Ca isotopes (Antonelli et al., 2019c;[START_REF] Feng | First-principles investigations of equilibrium calcium isotope fractionation between clinopyroxene and Ca-doped orthopyroxene[END_REF]Wang et al., 2017b). This dilution effect, however, becomes negligible below a Ca/(Ca+Mg) of ~1/32 in orthopyroxene, and larger Δ 44 CaOpx-Cpx are not predicted at lower values (Antonelli et al., 2019c;Wang et al., 2017b). Small dilution effects are also predicted for increasing orthopyroxene Fe/Mg, where increasing Ca-O bond lengths with increasing Fe lead to slightly decreased affinities for heavy Ca isotopes (Wang et al., 2017b). For pigeonite (a hightemperature pyroxene phase that commonly separates into orthopyroxene and clinopyroxene lamellae upon slow-cooling), RPFRs are generally found to be intermediate to clinopyroxene and orthopyroxene and also decrease with increasing pigeonite Ca/Mg [START_REF] Huang | First-principles calculations of equilibrium Ca isotope fractionation : Implications for oldhamite formation and evolution of lunar magma ocean[END_REF], shown in Fig. 4a.

Ca dilution effects (also known as "Ca concentration effects", depending on the author) have also been investigated in olivine, but have contradictory results. One study suggests that decreasing Ca/Mg in olivine leads to increased RPFRs, Fig. 4c [~0.3‰ difference between 1/16 and 1/64 at 1000 K [START_REF] Song | First-principles investigation of the concentration effect on equilibrium fractionation of Ca isotopes in forsterite[END_REF]], while another study suggests that Ca concentrations in olivine have no significant effect on its affinity for heavy Ca isotopes [(only ~0.1‰ difference between 1/16 and 1/64 at 1000 K (Antonelli et al., 2019c)]. Future experiments are required to confirm the veracity of dilution effects in olivine.

Garnet solid-solution end-members have decreasing RPFRs when the major cation is changed from Mg to Fe, Mn, and Ca [1000lnβ of pyrope > almandine ≈ spessartine > grossular (Antonelli et al., 2019c)], as shown in Fig. 5. Ca dilution effects in garnet have only been investigated in pyrope, but no significant difference was found between estimates with Ca/(Ca+Mg) of 1/12 and 1/24 (Antonelli et al., 2019c). The disagreement between predicted differences for Ca dilution effects in various minerals may be explained by methodological differences between the various studies, which use different approximations for defining the relevant electronic structures. Greater dilution effects due to decreasing Ca concentrations in both orthopyroxene and olivine are generally predicted when using local-density approximation (LDA) functionals [START_REF] Feng | First-principles investigations of equilibrium calcium isotope fractionation between clinopyroxene and Ca-doped orthopyroxene[END_REF][START_REF] Huang | First-principles calculations of equilibrium Ca isotope fractionation : Implications for oldhamite formation and evolution of lunar magma ocean[END_REF][START_REF] Song | First-principles investigation of the concentration effect on equilibrium fractionation of Ca isotopes in forsterite[END_REF]Wang et al., 2017b), while smaller effects are predicted when using generalized gradient approximations (GGA) such as PBE (Perdew, Burke, Ernzerhof) functionals (Antonelli et al., 2019c). LDA functionals are the simplest type of density functional, where the exchange-correlation energy density is assumed to be that of a uniform electron gas at a single point; GGA functionals (such as PBE) are formulas that use both the energy density and its gradient at each point, which gives more accurate estimates of bond dissociation energies and transition-state barriers [START_REF] Rappoport | Approximate Density Functionals: Which Should I Choose? Encycl[END_REF]. The slightly smaller 1000lnβ values and Ca dilution effects in Antonelli et al. (2019c) may thus be due the use of GGA vs. LDA exchangecorrelation functionals. However, this becomes less important when calculating equilibrium fractionation factors (α) because they depend only on the RPFR differences between phases.

Nevertheless, some of the differences between predicted equilibrium fractionation factors (1000lnα) are greater than average analytical uncertainties for δ 44 Ca (~0.1‰), even at high temperatures, and are worthy of further investigation.

Despite the different approaches used in the DFT estimates and small discrepancies between predicted 1000lnα for certain mineral pairs, first-order agreement is reached between the various studies regarding the general effects of mineral Ca concentration and average Ca-O bond length on predicted fractionation factors. In Figure 6, we show a summary of 44 Ca/ 40 Ca RPFRs from all published ab-initio studies versus (i) Ca concentrations, defined as Ca/(Ca+Mg+Fe+Mn+Na), in various minerals (Fig. 6a) and (ii) average Ca-O bond lengths (Fig. 6b). This figure demonstrates that minerals with less Ca (e.g. olivine, orthopyroxene, pyrope) generally have higher affinities for 44 Ca, while minerals with higher Ca concentrations (e.g. clinopyroxene & plagioclase) have lower affinities for 44 Ca. It also illustrates the dependence of 1000lnβ values on average Ca-O bond lengths and coordination numbers, where longer Ca-O bonds lead to lower affinities for 44 Ca and greater coordination numbers at a given bond-length lead to increased affinities for 44 Ca (Fig. 6b). Despite the general agreement for inter-mineral fractionation factors between studies, absolute values for 1000lnβ in single minerals can be dramatically different based on the approach [e.g.

~2.5 [START_REF] Song | First-principles investigation of the concentration effect on equilibrium fractionation of Ca isotopes in forsterite[END_REF] vs. ~1.8 (Antonelli et al., 2019c) for forsterite with Ca/(Ca+Mg) of 1/64 at 800 ○ C], so it is important to use RPFR values that were derived from the same method when calculating equilibrium fractionation factors (αa-b = βa/βb) to arrive at internally-consistent predictions for inter-mineral Δ 44 Ca. We also provide a compilation of RPFRs and fitting parameters for inter-mineral fractionations (Antonelli et al., 2019c;[START_REF] Feng | First-principles investigations of equilibrium calcium isotope fractionation between clinopyroxene and Ca-doped orthopyroxene[END_REF]Huang et al., 2019;[START_REF] Song | First-principles investigation of the concentration effect on equilibrium fractionation of Ca isotopes in forsterite[END_REF]Wang et al., 2017b), to serve as a reference for future studies (Table 2). + cx, where x = 10 6 /T 2 , The various labradorite structures (v1-v3) used in Antonelli et al. (2019c) are from [START_REF] Kaercher | Ab initio Calculations of Elastic Constants of Plagioclase feldespars[END_REF].

Observed inter-mineral Ca isotope fractionations

Unlike whole-rock analyses, δ 44 Ca in mineral separates from individual rock samples have only been reported in a limited number of studies (Antonelli et al., 2019c(Antonelli et al., , 2019b;;[START_REF] Bessey | Calcium Isotope Fractionation in Coexisting Minerals[END_REF]Chen et al., 2019b[START_REF] Chen | Calcium isotope evidence for subduction-enriched lithospheric mantle under the northern North China Craton[END_REF][START_REF] Dai | Calcium isotope compositions of mantle pyroxenites[END_REF][START_REF] Huang | Calcium isotopic fractionation between clinopyroxene and orthopyroxene from mantle peridotites[END_REF][START_REF] Kang | Calcium isotope sources and fractionation during melt-rock interaction in the lithospheric mantle: Evidence from pyroxenites, wehrlites, and eclogites[END_REF][START_REF] Kang | Calcium isotopic composition of mantle xenoliths and minerals from Eastern China[END_REF][START_REF] Ryu | The major ion, d44/40Ca, d44/42Ca, and d26/24Mg geochemistry of granite weathering at pH=1 and T=25 C: Power-law processes and the relative reactivity of minerals[END_REF][START_REF] Simon | Stable calcium isotopic composition of meteorites and rocky planets[END_REF][START_REF] Skulan | Biological control of calcium isotopic abundances in the global calcium cycle[END_REF][START_REF] Wang | Calcium isotope fractionation during crustal melting and magma differentiation: Granitoid and mineral-pair perspectives[END_REF][START_REF] Zhao | Coupled extremely light Ca and Fe isotopes in peridotites[END_REF] several of which analyze only one or two samples. In the following sections, we compare intermineral Δ 44 Ca measurements from igneous and metamorphic rocks to those predicted by abinitio estimates at relevant temperatures.

We generally find that predictions and measurements are often in disagreement. This observation suggests that inter-mineral Ca isotope fractionations are not likely to be reliable for geothermometry. Although experimental confirmation of the ab-initio predictions is still required, a recently published triple-isotope approach, which quantifies deviations from an equilibrium mass-law in 48 Ca/ 44 Ca vs. 42 Ca/ 44 Ca space (discussed in Section 5.2), suggests that they are approximately correct (Antonelli et al., 2019c). Thus, observed deviations from expected Ca isotope equilibrium are likely to be the result of kinetic effects in natural samples.

Igneous minerals

Mineral separates from ultramafic to felsic igneous rocks have a measured δ 44 CaBSE range of -1.2 to +1.3‰ (Fig. 7a) (Antonelli et al., 2019b;Chen et al., 2019b[START_REF] Chen | Calcium isotope evidence for subduction-enriched lithospheric mantle under the northern North China Craton[END_REF][START_REF] Huang | Calcium isotopic fractionation between clinopyroxene and orthopyroxene from mantle peridotites[END_REF][START_REF] Kang | Calcium isotopic composition of mantle xenoliths and minerals from Eastern China[END_REF][START_REF] Ryu | The major ion, d44/40Ca, d44/42Ca, and d26/24Mg geochemistry of granite weathering at pH=1 and T=25 C: Power-law processes and the relative reactivity of minerals[END_REF][START_REF] Simon | Stable calcium isotopic composition of meteorites and rocky planets[END_REF][START_REF] Skulan | Biological control of calcium isotopic abundances in the global calcium cycle[END_REF][START_REF] Zhao | Coupled extremely light Ca and Fe isotopes in peridotites[END_REF] . Orthopyroxene separates from ultramafic rocks tend to have the highest δ 44 Ca, but are highly variable and span the entire range observed in igneous minerals (~2.5‰). In mafic to felsic rocks there is a smaller δ 44 Ca range (~1‰), where plagioclase has the lowest δ 44 Ca (but is found to vary by ~0.5‰) and where ferromagnesian minerals (e.g. clinopyroxene, hornblende, chlorite, epidote, biotite) are generally heavier than non-ferromagnesian minerals (e.g. plagioclase, K-feldspar, melilite, apatite, titanite). Biotite has the most positive measured δ 44 Ca out of these minerals, but has only been analyzed in one sample [START_REF] Ryu | The major ion, d44/40Ca, d44/42Ca, and d26/24Mg geochemistry of granite weathering at pH=1 and T=25 C: Power-law processes and the relative reactivity of minerals[END_REF]. (Antonelli et al., 2019b;Chen et al., 2019b[START_REF] Chen | Calcium isotope evidence for subduction-enriched lithospheric mantle under the northern North China Craton[END_REF][START_REF] Dai | Calcium isotope compositions of mantle pyroxenites[END_REF][START_REF] Huang | Calcium isotopic fractionation between clinopyroxene and orthopyroxene from mantle peridotites[END_REF][START_REF] Kang | Calcium isotopic composition of mantle xenoliths and minerals from Eastern China[END_REF][START_REF] Ryu | The major ion, d44/40Ca, d44/42Ca, and d26/24Mg geochemistry of granite weathering at pH=1 and T=25 C: Power-law processes and the relative reactivity of minerals[END_REF][START_REF] Simon | Stable calcium isotopic composition of meteorites and rocky planets[END_REF][START_REF] Skulan | Biological control of calcium isotopic abundances in the global calcium cycle[END_REF][START_REF] Wang | Calcium isotope fractionation during crustal melting and magma differentiation: Granitoid and mineral-pair perspectives[END_REF][START_REF] Zhao | Coupled extremely light Ca and Fe isotopes in peridotites[END_REF], Fe-rich peridotite data, which are especially low in δ 44 Ca, are from [START_REF] Zhao | Coupled extremely light Ca and Fe isotopes in peridotites[END_REF].

Inter-mineral Δ 44 Ca values in igneous rocks span a range of ~2‰ (Fig. 7b). In mafic to felsic rocks, all ferromagnesian minerals have higher δ 44 Ca than coexisting feldspars (Δ 44 Camin-fsp up to +1‰). The largest inter-mineral fractionations (Δ 44 Caopx-cpx up to +1.5‰) are observed between orthopyroxene and clinopyroxene. In most rocks Δ 44 Caopx-cpx is positive, but a growing number of samples have negative Δ 44 Caopx-cpx (down to -0.5‰). To date, all analyzed clinopyroxeneorthopyroxene pairs in websterites are out of Ca isotopic equilibrium, with orthopyroxene too heavy, and all analyzed harzburgites, Fe-rich lherzolites, and a significant number of pyroxenites, are out of isotopic equilibrium with orthopyroxene too light (Fig. 8a). Due to the much higher concentration of Ca in clinopyroxene, however, imperfect mineral separations may push orthopyroxene δ 44 Ca towards the 1:1 line, and great care should be taken in future studies, including verification of Ca concentrations in mineral separates [which can then be used for eventual unmixing calculations e.g. (Antonelli et al., 2019c)]. (Antonelli et al., 2019b;Chen et al., 2019b[START_REF] Chen | Calcium isotope evidence for subduction-enriched lithospheric mantle under the northern North China Craton[END_REF][START_REF] Dai | Calcium isotope compositions of mantle pyroxenites[END_REF][START_REF] Huang | Calcium isotopic fractionation between clinopyroxene and orthopyroxene from mantle peridotites[END_REF][START_REF] Kang | Calcium isotopic composition of mantle xenoliths and minerals from Eastern China[END_REF][START_REF] Ryu | The major ion, d44/40Ca, d44/42Ca, and d26/24Mg geochemistry of granite weathering at pH=1 and T=25 C: Power-law processes and the relative reactivity of minerals[END_REF][START_REF] Simon | Stable calcium isotopic composition of meteorites and rocky planets[END_REF][START_REF] Skulan | Biological control of calcium isotopic abundances in the global calcium cycle[END_REF][START_REF] Zhao | Coupled extremely light Ca and Fe isotopes in peridotites[END_REF]. Ab-initio estimates for diopside, anorthite, orthopyroxene, and olivine are based on DFT estimates using PBE functionals (Antonelli et al., 2019c) and the lowest-Ca end-members for orthopyroxene and olivine.

Comparing the inter-mineral Δ 44 Ca data to ab-initio predictions, we find that a significant number of analyses do not agree with predictions at relevant temperatures. Additionally, a significant number of samples have negative Δ 44 Caopx-cpx (below the 1:1 line, Fig. 8a) that are not predicted at any temperature. The few analyzed Δ 44 Caol-plag and Δ 44 Caol-cpx fractionations are much smaller than expected for relevant temperatures in peridotites and gabbros (Fig. 8b). A small number of Δ 44 Cacpx-plag in basalts and gabbros, however, are in approximate agreement with predictions, but have much larger relative uncertainties (Fig. 8c), while the clinopyroxeneplagioclase fractionation in a dacitic volcanic rock is found to be significantly larger than expected (as a likely consequence of rapid plagioclase growth, discussed in Section 6.3).

Although a negative correlation is observed between Ca concentrations in orthopyroxene and

Δ 44 Caopx-cpx in igneous rocks (which is in the same direction as expected from dilution effects), the observed effects are greater than equilibrium predictions at relevant temperatures, and Ca dilution effects are negligible for the analyzed range of orthopyroxene Ca/(Ca+Mg).

The Ca concentration in orthopyroxene, however, is known to increase with increasing temperature [START_REF] Brey | Geothermobarometry in 4-phase lherzolites .2. New thermobarometers, and practical assessment of existing thermobarometers[END_REF], and is commonly used as a geothermometer ('Ca-in-Opx thermometry'). Thus, increased Ca contents in orthopyroxene should lead to smaller fractionations not only due to (i) increasing average Ca-O bond lengths with increasing Ca, but also due to (ii) decreasing equilibrium isotope fractionations with increasing temperatures.

Neither of these effects, however, can explain the range of observed data, which suggests that orthopyroxene-clinopyroxene mineral pairs are often out of isotopic equilibrium and that the rough inverse correlation between Ca in orthopyroxene and Δ 44 Caopx-cpx (Chen et al., 2019b) is likely due to disequilibrium effects (discussed in Section 6.2).

Metamorphic minerals

Ca isotope variability in metamorphic minerals is even larger than observed in igneous minerals, with a δ 44 CaBSE range of ~8‰ (-2.2 to +5.8‰, Fig. 9a), but minerals from only three metamorphic-facies (amphibolite, eclogite, and granulite-facies), have been analyzed in a total of only four studies (Antonelli et al., 2019c;[START_REF] Bessey | Calcium Isotope Fractionation in Coexisting Minerals[END_REF][START_REF] Kang | Calcium isotope sources and fractionation during melt-rock interaction in the lithospheric mantle: Evidence from pyroxenites, wehrlites, and eclogites[END_REF][START_REF] Wang | Calcium isotope fractionation during crustal melting and magma differentiation: Granitoid and mineral-pair perspectives[END_REF].

Mineral δ 44 Ca compositions in granulites are the most variable, with a δ 44 Ca range of ~3‰ in garnet, ~6‰ in orthopyroxene, ~5‰ in plagioclase, and ~2.5‰ in perthitic K-feldspar. The only mineral with a relatively limited δ 44 Ca range (< 0.5‰) is clinopyroxene, which appears to always have Ca isotope compositions close to BSE. In lower temperature metamorphic facies (eclogite and amphibolite), δ 44 Ca variability is more limited in all minerals, observed to be within ~1‰ of BSE (Fig. 9a).

Inter-mineral Δ 44 Ca values are the largest in granulite-facies rocks, ranging from -2 to +2.5‰ (Fig. 9b), but can be highly variable for a given mineral pair (e.g. Δ 44 Cagrt-fsp ranges from -1.5 to +1.5 and Δ 44 Caopx-fsp ranges from 0.0 to +2.5‰). In amphibolite-and eclogite-facies rocks, intermineral Δ 44 Ca values range from -0.5 to +1‰. The most negative Δ 44 Ca fractionation in these rocks is for an apatite-clinopyroxene pair from an eclogite sample, while the most positive values are for garnet-apatite, garnet-calcite, and garnet-clinopyroxene. Note that these statements are somewhat definitional, such that the most positive Δ 44 Ca values would be the most negative if defined in the opposite sense (i.e. Δ 44 Caap-grt instead of Δ 44 Cagrt-ap). (Antonelli et al., 2019c)]. Data are from (Antonelli et al., 2019c;[START_REF] Bessey | Calcium Isotope Fractionation in Coexisting Minerals[END_REF][START_REF] Kang | Calcium isotope sources and fractionation during melt-rock interaction in the lithospheric mantle: Evidence from pyroxenites, wehrlites, and eclogites[END_REF][START_REF] Wang | Calcium isotope fractionation during crustal melting and magma differentiation: Granitoid and mineral-pair perspectives[END_REF].

Comparing the measured inter-mineral Δ 44 Ca values to ab-initio predictions at relevant temperatures (Fig. 10), we find -similar to the igneous mineral pairs-that many represent isotopic disequilibrium. Garnet-mineral pairs in granulite-facies rocks are found to deviate furthest from ab-inito equilibrium predictions, with minerals from pelitic granulites having the greatest apparent disequilibrium effects, while lower-grade metamorphic garnet-mineral fractionations are closer to equilibrium predictions. Orthopyroxene-mineral fractionations in granulite-facies rocks also tend to suggest isotopic disequilibrium, with orthopyroxene δ 44 Ca compositions either too negative or too positive compared to predictions at relevant temperatures (Fig. 10c-e). Clinopyroxene-plagioclase fractionations, however, are in approximate agreement with equilibrium predictions, similar to observations from igneous mineral pairs. Indeed, the largest inter-mineral disequilibrium effects observed in granulite-facies rocks appear to be limited to rocks with less than ~15% modal clinopyroxene (Antonelli et al., 2019c). When clinopyroxene is present, the observed inter-mineral fractionations tend to better agree with equilibrium predictions, and thus, mafic (vs. pelitic) granulites tend to have fractionations closer to equilibrium. This could be a potential manifestation of (i) protolith-based differences in diffusive transport/paragenetic processes or may suggest that (ii) isotopic heterogeneities in pelitic precursor minerals are somehow carried over (discussed in Section 6.2).

It appears that the magnitude of disequilibrium effects increases with increasing temperatures in granulite-facies rocks, potentially due to the activation of otherwise subordinate diffusion pathways/mechanisms (Antonelli et al., 2019c). Garnet-mineral and garnet-whole-rock (WR) fractionations deviate further from equilibrium predictions with increasing WR and garnet Mg# (Fig. 11), and orthopyroxene-mineral fractionations increasingly deviate from equilibrium predictions with WR Mg#, feldspar anorthite component, modal orthopyroxene, and orthopyroxene Ca/Mg, all of which tend to increase with temperature (Antonelli et al., 2019c).

The trend of increasing Δ 44 Caopx-min with increasing orthopyroxene Ca/Mg, however, is the opposite of that observed in mantle mineral separates, suggesting a different underlying mechanism for the disequilibrium fractionations.

Fig. 10a-f. Ca isotope cross-plots for minerals from metamorphic rocks, grouped by rock-type, with temperature contours representing ab-initio equilibrium predictions from DFT estimates. Garnet-mineral predictions depend on the garnet end-member chosen, with pyrope having the highest predicted 1000lnβ and grossular having the lowest 1000lnβ. In order to represent the full range of fractionations expected at equilibrium, we report grossular-mineral predictions at high temperatures (1000 ○ C) and pyrope-mineral predictions at lower temperatures (700 ○ C). We also chose a similar approach for feldspar predictions, where mineral-anorthite is reported at high temperatures and mineral-labradorite (An50) are reported at lower temperatures. (Antonelli et al., 2019c). Data are from (Antonelli et al., 2019c;[START_REF] Bessey | Calcium Isotope Fractionation in Coexisting Minerals[END_REF][START_REF] Kang | Calcium isotope sources and fractionation during melt-rock interaction in the lithospheric mantle: Evidence from pyroxenites, wehrlites, and eclogites[END_REF][START_REF] Wang | Calcium isotope fractionation during crustal melting and magma differentiation: Granitoid and mineral-pair perspectives[END_REF]. Average 2sd uncertainties for δ 44 Ca (~ ±0.1‰) are smaller than the symbols except for in panels (e) and (f) where they are depicted in the lower right-hand corner. In sum, inter-mineral Δ 44 Ca variability in igneous and metamorphic rocks suggests that kinetic effects occur (and are subsequently preserved) at high-temperatures, however, the quantification of these effects relies mainly on the accuracy of ab-initio predictions for equilibrium fractionation and may be limited by various model assumptions. Although no experimental constraints are yet available for equilibrium Ca isotope fractionation (and represent important targets for future work), equilibrium and kinetic effects can potentially be distinguished using a triple-isotope approach (Antonelli et al., 2019c).

Distinguishing kinetic from equilibrium fractionations

The accurate quantification of equilibrium vs. kinetic effects and their applicability in different high-temperature systems is of paramount importance to Ca isotope geochemistry. This is because isotope fractionation factors are necessary inputs for most models (e.g. partial melting, crystallization, & diffusion models) that attempt to quantify high-temperature phenomena using δ 44 Ca and Δ 44 Ca data. Extents of melting and fractional crystallization based on Ca isotopes, for example, are often calibrated by either (i) assuming isotopic equilibrium (i.e. fractionation factors based on DFT models), or (ii) measuring inter-mineral Δ 44 Ca in coexisting minerals and assuming that these correspond to all of the fractionations occurring in a system. Both of these approaches have their own limitations, with the first approach assuming that DFT models are accurate for natural systems and that the various processes are happening at isotopic equilibrium, and the second approach assuming that, whether or not the fractionations are at equilibrium, that Δ 44 Ca is invariable in the system.

Chemical vs. isotopic equilibrium

Various lines of reasoning are often used in order to defend the applicability of equilibrium fractionation factors derived from DFT and/or to suggest that measured δ 44 Ca values are representative of isotopic equilibrium. These include observations of: (i) textural equilibrium and/or grain homogeneity, (ii) chemical equilibrium between minerals, paired with assumptions about the relative rates for isotopic vs. chemical equilibration, and/or (iii) assumptions regarding short length-scales for Ca diffusion (e.g. closed-system for Ca at the hand-sample scale). There are several issues with these approaches, however, with the most significant being that, for a given elementchemical-equilibrium is not a sufficient condition for isotopic equilibrium.

Potential confusion surrounding the relationship between isotopic and chemical equilibrium comes from the high-temperature experimental community, where the word equilibrate is used in the sense of isotopic homogenization across experimental diffusion couples [START_REF] Baker | Tracer versus trace element diffusion: Diffusional decoupling of Sr concentration from Sr isotope composition[END_REF][START_REF] Laan | Comparison of element and isotope diffusion of K and Ca in multicomponent silicate melts[END_REF][START_REF] Lesher | Kinetics of Sr and Nd exchange in silicate liquids: theory, experiments, and applications to uphill diffusion, isotopic equilibration, and irreversible mixing of magmas[END_REF][START_REF] Lesher | Decoupling of chemical and isotopic exchange during magma mixing[END_REF][START_REF] Watkins | Influence of liquid structure on diffusive isotope separation in molten silicates and aqueous solutions[END_REF], and not in the sense of free-energy minimization. In these diffusion experiments, it is observed that the initial pulse of isotopic fractionation (where 40 Ca diffuses faster than 44 Ca, discussed in Section 6) levels out faster than the chemical concentration gradient between the two starting materials (F. Huang et al., 2010;[START_REF] Richter | Isotopic fractionation of the major elements of molten basalt by chemical and thermal diffusion[END_REF][START_REF] Richter | Isotope fractionation by chemical diffusion between molten basalt and rhyolite[END_REF][START_REF] Richter | Isotope fractionation by diffusion in molten oxides[END_REF][START_REF] Watkins | Kinetic Fractionation of Non-Traditional Stable Isotopes by Diffusion and Crystal Growth Reactions[END_REF][START_REF] Watkins | Diffusion of multi-isotopic chemical species in molten silicates[END_REF][START_REF] Watkins | Influence of liquid structure on diffusive isotope separation in molten silicates and aqueous solutions[END_REF][START_REF] Watkins | Liquid composition-dependence of calcium isotope fractionation during diffusion in molten silicates[END_REF]. Thus, a not uncommon statement from these types of studies is that isotopic equilibration occurs faster than chemical equilibration, which has been taken at face value and used in several Ca isotope studies claiming isotopic equilibrium.

The thermodynamic drive for chemical reactions is several orders of magnitude larger than for isotopic substitution reactions (e.g. [START_REF] Chacko | Equilibrium Oxygen, Hydrogen and Carbon Isotope Fractionation Factors Applicable to Geologic Systems[END_REF]. In order to minimize free energy in a system, therefore, isotopic equilibration is largely subordinate to chemical equilibration.

Maintaining isotopic equilibrium during an ongoing chemical reaction depends mainly on the forward and backward exchange rates greatly exceeding the overall net reaction rate [START_REF] Depaolo | Surface kinetic model for isotopic and trace element fractionation during precipitation of calcite from aqueous solutions[END_REF]. Although one can envision a case where the net chemical reaction rate is so slow that the forward and backward exchange rates at the crystal surface are much larger (e.g. extremely slow carbonate precipitation), and could thus potentially maintain near isotopic equilibrium throughout the reaction, this is unlikely to apply in a majority of cases [START_REF] Depaolo | Surface kinetic model for isotopic and trace element fractionation during precipitation of calcite from aqueous solutions[END_REF]. As noted by many early experimental studies (e.g. [START_REF] Clayton | Oxygen Isotope Fractionation in the System Calcium Carbonate-Water[END_REF][START_REF] Matsuhisa | Mechanisms of hydrothermal crystallization of quartz at 250°C and 15 kbar[END_REF][START_REF] Matthews | Magnetite formation by the reduction of hematite with iron under hydrothermal conditions[END_REF]O'Neil andTaylor, 1969, 1967), mineral-synthesis, phase-change, and ion-exchange reactions are susceptible to kinetic isotope effects that are not likely to reach subsequent isotopic equilibrium, even over extended time-scales.

Ca isotope measurements in natural samples also suggest that chemical equilibrium generally precedes isotopic equilibrium, even at high temperatures. For Ca isotopes in granulite-facies metamorphic rocks, it has been shown that significant inter-mineral Δ 44 Ca disequilibrium can still exist, even when held at high-temperatures (~900 ○ C) over extended time periods (10 7 -10 8 years) and when Fe-Mg exchange and Ca-in-Opx thermometry indicate approximate chemical equilibrium (Antonelli et al., 2019c). These authors also observe chemical homogeneity across mineral grains (e.g. garnet) and equilibrium (granoblastic) textures in many samples, indicating that neither of these conditions are sufficient for Ca isotopic equilibrium.

Ca isotope distributions were also explored in samples from the Guelb el Azib Complex (a mafic intrusion metamorphosed to granulite-facies and subsequently retrograded through amphibolite and greenschist-facies), where mineral-specific δ 44 Ca values were estimated based on Monte Carlo simulations [combining whole-rock and mineral chemical compositions, whole-rock δ 44 Ca, and modal mineralogy estimates [START_REF] Valdes | The effects of high-temperature fractional crystallization on calcium isotopic composition[END_REF]]. Although not explicitly discussed by the authors, the development and preservation of disequilibrium isotopic compositions is evidenced by the large δ 44 Ca variations (up to ~ 3‰) estimated for the same mineral phases (e.g. amphibole, plagioclase) with otherwise similar chemical compositions across different samples.

Finally, the assumption that Ca isotopes behave as a closed-system on the hand-sample scale during high-temperature metamorphism, used in order to look-through potential metamorphic effects and quantify Ca isotope fractionation during fractional crystallization [START_REF] Valdes | The effects of high-temperature fractional crystallization on calcium isotopic composition[END_REF], is in disagreement with results from spatially-constrained granulite-facies samples that suggest diffusive Ca isotope fractionation can occur over several meters during metamorphism (Antonelli et al., 2019c). Ca diffusion in high-temperature systems with greater volatile element concentrations (and higher Ca diffusivities) could thus occur over even greater distances, given sufficiently large timescales.

The triple Ca isotope method (Δ 48 Ca')

Although no experimental constraints are available, in order to confirm ab-initio predictions, a recent study has exploited the predicted differences between kinetic and equilibrium massfractionation laws in order to (i) corroborate DFT estimates and (ii) confirm the kinetic origins of large (>1‰) fractionations in high-temperature whole-rocks and minerals (Antonelli et al., 2019c). In triple isotope space ( 48 Ca/ 44 Ca vs. 42 Ca/ 44 Ca) the predicted equilibrium mass law, based on vibrational frequency differences, is -1.751 (e.g. [START_REF] Young | Kinetic and equilibrium mass-dependant isotope fractionation laws in nature and their geochemical and cosmochemical significance[END_REF]. This means that measured 48 Ca/ 44 Ca should be equal to the measured 42 Ca/ 44 Ca to the exponent of -1.751, if fractionation occurred at equilibrium. The kinetic limit, based on diffusion in an ideal gas (Graham's Law), where the average speed of an isotope is inversely proportional to its mass, leads to a predicted 48 Ca/ 44 Ca that is related to 42 Ca/ 44 Ca by the exponent -1.872. Thus, the effects of equilibrium vs. kinetic fractionations should be distinguishable by measuring 48 Ca/ 44 Ca and comparing to 42 Ca/ 44 Ca. Although 42 Ca-44 Ca-48 Ca were used in this study due to their large relative mass differences, any combination of three Ca isotopes could theoretically be used in order to quantify deviations from an equilibrium mass law. The authors define a new variable (Δ 48 Ca') which quantifies 48 Ca/ 44 Ca deviations from isotopic equilibrium (in ppm):

= -( " + 1) %.'(% -1) (Eq. 3)
Where μ 4x Ca is defined as [( 4x Ca/ 44 Ca)sample/( 4x Ca/ 44 Ca)standard -1] and reported in ppm. A similar triple-isotope method has also been used to distinguish kinetic from equilibrium fractionations for Mg and Fe isotopes [START_REF] Mccoy-West | The Fe and Zn isotope composition of deep mantle source regions: Insights from Baffin Island picrites[END_REF][START_REF] Olsen | Magnesium isotope evidence for single stage formation of CB chondrules by colliding planetesimals[END_REF][START_REF] Young | The Isotope Geochemistry and Cosmochemistry of Magnesium[END_REF].

In order to test this method, two end-member samples were initially analyzed (Antonelli et al., 2019c). The kinetic end-member sample comes from rapid plagioclase crystallization in an experimental tonalite sample [also investigated in (Antonelli et al., 2019b)], where plagioclase and melt have a Δ 44 Ca difference greater than 2‰. The equilibrium end-member samples (plagioclase and orthopyroxene mineral separates) were taken from the Stillwater cumulate gabbro, which slowly crystallized over 10-100 Ka (DePaolo and Wasserburg, 1979). In Fig. 12, these end-member samples are demarcated with arrows, and can be seen to closely adhere to the mass-laws predicted by the equilibrium and kinetic limits for isotope fractionation: the Stillwater gabbro sample has no Δ 48 Ca' difference between minerals, whereas plagioclase from the experimental tonalite sample has a large Δ 48 Ca' depletion relative to glass. from (Antonelli et al., 2019c). The shaded purple region demarcates the compositional field that would be expected for a bulk-silicate Earth material that has not undergone any kinetic fractionations (taking into account the average uncertainty of the MC-ICP-MS measurements), but it is the difference in Δ 48 Ca' between two phases that defines whether or not they are at isotopic equilibrium. Cumulate gabbro from the Stillwater complex exemplifies this relationship between orthopyroxene and plagioclase, whereas samples with large differences (such as plagioclase and glass from the experimental tonalite and adjacent whole-rocks 2035A and 2035B, which represent the center and edge, respectively, of a metamorphosed granulite-facies mafic dike hosted in quartzitic country rock) clearly depict the effects of kinetic fractionation and also serve to reinforce the applicability of the theoretically-predicted The other data in Fig. 12 correspond to various granulite-facies rocks and minerals. The largest positive Δ 48 Ca' fractionation is between the center (2035A) and edge (2035B) of a metamorphosed ultramafic dike sample from the Napier complex (Antonelli et al., 2019c), which also follows the trend predicted by the kinetic limit for isotope fractionation (yellow squares labeled 'A' and 'B'). Inter-mineral separates from granulite-facies samples with large Δ 44 Ca differences (including Slave-province xenoliths and Napier complex samples) are also in general agreement with the kinetic limit for mass fractionation. This may be surprising at first, because the kinetic-limit is predicted from diffusion in an ideal gas, however, this simply implies that the mass-law [typically referred to as the exponent 'β' (e.g. [START_REF] Richter | Isotope fractionation by diffusion in molten oxides[END_REF], but not to be confused with reduced partition function ratios (1000lnβ) from ab-initio estimates] is essentially the same for 48 Ca/ 44 Ca and 42 Ca/ 44 Ca. Although only a limited number of samples have been analyzed, the two samples which best agree with equilibrium ab-initio predictions [orthopyroxene and plagioclase from the Stillwater gabbro, along with garnet and feldspar from a lower-crustal garnet amphibolite ('IV-16-19')] have equilibrium Δ 48 Ca' values (Fig. 12). In the absence of experimental data, this observation lends weight to the applicability of currently available DFT estimates for estimating Ca isotopic equilibria between minerals (e.g. Antonelli et al., 2019c;[START_REF] Huang | First-principles calculations of equilibrium Ca isotope fractionation : Implications for oldhamite formation and evolution of lunar magma ocean[END_REF]Wang et al., 2017b).

The triple Ca-isotope method may also allow for the separation of equilibrium and kinetic fractionations in the same sample. As can be seen, for example, in the Stillwater gabbro mineral separates (Fig. 12), despite having the same Δ 48 Ca' values (equilibrium between the minerals), there is a slight offset between these and bulk silicate earth (Δ 48 Ca' = 0). This implies that the parent material may have undergone kinetic fractionations, before subsequent isotopic reequilibration. Given the difficulty of precisely measuring Δ 48 Ca', however, the possibility of distinguishing kinetic fractionations that underwent subsequent equilibrium fractionation (or vice versa) remains to be explored in future work. Although unequivocal confirmation of the kinetic mass-dependence for fractionations between minerals (and adjacent whole-rocks) was accomplished by analyzing samples with large δ 44 Ca differences (up to ~3‰) and by targeting multiple Ca isotopes with the largest mass differences (Antonelli et al., 2019c), an earlier study used unspiked TIMS measurements (corrected with an exponential mass-law) on metasomatized peridotite mineral separates and showed that there were no 40 Ca/ 44 Ca anomalies [START_REF] Zhao | Coupled extremely light Ca and Fe isotopes in peridotites[END_REF], which, although lacking the necessary precision to confirm a specific mass law, strongly suggested a kinetic control on fractionation in these samples.

Kinetic Ca isotope fractionation at high-temperatures

In a general sense, isotopic rate differences for dominantly unidirectional reactions (e.g. rapid crystal precipitation) are responsible for the large disequilibrium effects observed in δ 44 Ca. These kinetic fractionations can occur during transport of Ca isotopes towards a reaction locus, where 40 Ca diffuses faster than 44 Ca, or from isotopic rate differences during bond formation (e.g. at a precipitating crystal surface) or bond breaking (e.g. disruption of the hydration sphere around Ca 2+ ions in water). We refer the reader to DePaolo (2011), [START_REF] Simon | Utilizing Stable Isotopes and Isotopic Anomalies to Study Early Solar System Formation Processes[END_REF], Simon and[START_REF] Simon | Stable calcium isotopic composition of meteorites and rocky planets[END_REF][START_REF] Watkins | Kinetic Fractionation of Non-Traditional Stable Isotopes by Diffusion and Crystal Growth Reactions[END_REF] where these topics are discussed in greater detail.

Of all the kinetic Ca isotope fractionations occurring in high-temperature terrestrial settings, diffusive Ca transport has the largest δ 44 Ca effects, which can be an order of magnitude larger than equilibrium effects at similar temperatures [START_REF] Watkins | Kinetic Fractionation of Non-Traditional Stable Isotopes by Diffusion and Crystal Growth Reactions[END_REF]. Given the potential size of the effects, modeling kinetic Ca isotope fractionation (although slightly more challenging than modeling equilibrium fractionations) is often necessary in order to interpret δ 44 Ca data from natural samples.

In conjunction with kinetic fractionation models, δ 44 Ca data can yield essential information on chemical reaction rates (and durations) and on the nature of Ca transport in various geochemical reservoirs throughout Earth's geologic past. In this section we briefly cover the principles behind kinetic Ca isotope fractionations at high temperatures, and then explore the various models used to gain information from isotopic disequilibria observed in natural samples. These include models constraining: (i) the duration of mantle metasomatism events, (ii) Ca transport mechanisms during high-temperature metamorphism in the lower continental crust, and (iii) crystal growth-rates in volcanic and sub-volcanic systems.

Ca diffusion in experimental silicate liquids

During diffusion in an ideal gas, kinetic isotope effects can be essentially explained by classical mechanics (based on differences in translational energies) and are due to the greater average speeds of lighter isotopes (at a given kinetic energy). Within more complex diffusion media, such as silicate melts, however, cation diffusion occurs by hopping between interstitial sites (potential energy minima within the silicate framework), and can therefore be modeled using transition-state theory [START_REF] Dominguez | The Soret effect and isotopic fractionation in hightemperature silicate melts[END_REF]. The difficulty of estimating the various parameters needed to estimate kinetic isotope effects from transition-state theory, however, has motivated a more empirical approach towards quantifying diffusive isotope fractionation in silicate liquids (F. Huang et al., 2010;[START_REF] Richter | Isotopic fractionation of the major elements of molten basalt by chemical and thermal diffusion[END_REF][START_REF] Richter | Isotope fractionation by chemical diffusion between molten basalt and rhyolite[END_REF][START_REF] Richter | Isotope fractionation by diffusion in molten oxides[END_REF][START_REF] Watkins | Kinetic Fractionation of Non-Traditional Stable Isotopes by Diffusion and Crystal Growth Reactions[END_REF][START_REF] Watkins | Diffusion of multi-isotopic chemical species in molten silicates[END_REF][START_REF] Watkins | Influence of liquid structure on diffusive isotope separation in molten silicates and aqueous solutions[END_REF][START_REF] Watkins | Liquid composition-dependence of calcium isotope fractionation during diffusion in molten silicates[END_REF]. as that shown in Fig. 13, the fractionation factor (often defined as the diffusivity of 44 Ca relative to 40 Ca, D44/D40) depends on a variety of parameters including: (i) the degree of melt polymerization (e.g. the diffusivity of Ca relative to that of the SiO2 framework, as shown in Fig. 14a), where greater isotopic fractionations are correlated with higher SiO2 + Al2O3 in the melt (Fig. 14b), (ii) the direction and strength of chemical gradients for other chemical species, where counter-diffusion (e.g. of Na, as in Fig. 13) generally increases the Ca isotopic fractionation, and

(iii) the direction of thermal gradients imposed on the system, where heavier Ca isotopes are favored at the colder end. A detailed review of the factors influencing isotopic fractionation during diffusion in silicate liquids can be found in [START_REF] Watkins | Kinetic Fractionation of Non-Traditional Stable Isotopes by Diffusion and Crystal Growth Reactions[END_REF], and the reader is referred there for further information.

Ca isotope diffusion in nature: whole-rock and inter-mineral fractionation models

Given that large (>1‰) inter-mineral and whole-rock δ 44 Ca fractionations closely adhere to the kinetic (diffusion-based) mass law in triple isotope space (see Fig. 12), it appears that Ca diffusion is often responsible for isotopic disequilibrium observed in natural samples. However, only a handful of studies have applied diffusion-based Ca isotope fractionation models to hightemperature samples (Antonelli et al., 2019c(Antonelli et al., , 2019b;;[START_REF] Zhao | Coupled extremely light Ca and Fe isotopes in peridotites[END_REF].

The first study used disequilibrium δ 44 Ca measurements (Δ 44 Caopx-cpx down to -0.5‰) in metasomatized mantle xenoliths in order to constrain the product of calcium diffusivity and time (Dt) for mantle metasomatism [START_REF] Zhao | Coupled extremely light Ca and Fe isotopes in peridotites[END_REF]. By assuming various Ca diffusivities and D44/D40 values for volume diffusion (from high-Ca fluids into solid pyroxene grains), and combining them with estimates for grain-sizes, geometry, and original δ 44 Ca, the authors were able to constrain the duration of metasomatism by solving analytical diffusion equations [START_REF] Zhao | Coupled extremely light Ca and Fe isotopes in peridotites[END_REF]. All of the minerals in the calculations were treated as spheres and the composition of the percolating melt was assumed to not change over time, in accordance with the assumption of high flow rates and high melt/peridotite ratios. The 44 Ca/ 40 Ca beta factors used in the successful models varied between 0.05 and 0.10 [which are equivalent to those of an ultramafic and mafic silicate liquid, respectively (see Fig. 14)], however, the models target diffusion into mineral grains and not within the liquid phase, thus the isotopic diffusivity ratios assumed in the models correspond to currently unconstrained fractionation factors for diffusion within solid minerals ('volume diffusion'). Although future studies should focus on understanding Ca isotope fractionations during volume diffusion in different minerals, the study (which also incorporates Fe isotopes) is able to reproduce the measured isotopic fractionations with a large variety of model parameter combinations [including variable clinopyroxene/orthopyroxene size ratios (0.33 to 0.66), temperatures (1000-1200 ○ C), and olivine solid solution compositions (Fo90-Fo60)] and uses these to constrain the duration of the associated metasomatic event to between 10 0 and 10 4 years [START_REF] Zhao | Coupled extremely light Ca and Fe isotopes in peridotites[END_REF].

In a high-grade metamorphic example from the Napier Complex, disequilibrium Δ 44 Ca between adjacent whole-rocks (with known spatial constraints) were used in conjunction with numerical diffusion models. Numerical methods, as opposed to analytical methods, are generally necessary to model diffusion in heterogeneous media (where Ca diffusivity and D44/D40 may vary between layers). In order to constrain effective Ca diffusivities and D44/D40 during granulite-facies metamorphism in the lower crust, the authors used independent estimates for the duration of high-temperature metamorphism in the Napier Complex and found that measured Ca concentrations and δ 44 Ca values were best reproduced by effective Ca diffusivity values ranging from 10 -10 to 10 -7 m 2 /yr (Fig. 15), consistent with previous estimates for Ca diffusion in hightemperature rocks via grain-boundary and volume diffusion (e.g. [START_REF] Dohmen | Diffusion in Polycrystalline Materials: Grain Boundaries, Mathematical Models, and Experimental Data[END_REF][START_REF] Farver | Grain-Boundary Diffusion of Oxygen, Potassium and Calcium in Natural and Hot-Pressed Feldspar Aggregates[END_REF][START_REF] Giletti | Isotopic Equilibrium/Disequilibrium and Diffusion Kinetics in Feldspars[END_REF][START_REF] Keller | Asymmetrically zoned reaction rims: Assessment of grain boundary diffusivities and growth rates related to natural diffusion-controlled mineral reactions[END_REF][START_REF] Vielzeuf | Calcium diffusivity in alumino-silicate garnets: An experimental and ATEM study[END_REF]. The authors also found that isotopic diffusivity ratios (D44/D40) of ~0.99 were necessary in order to reproduce the measured δ 44 Ca of the various units.

Given the existing experimental constraints for D44/D40 in various media, a value of ~0.99 suggests that grain-boundary diffusion likely occurred within silicate melts as opposed to within aqueous fluids [where fractionation factors are expected to be much smaller (e.g. [START_REF] Watkins | Kinetic Fractionation of Non-Traditional Stable Isotopes by Diffusion and Crystal Growth Reactions[END_REF]]. As there is an on-going debate regarding the dominant mechanism through which granulite-facies rocks lose their incompatible elements (e.g. dehydration vs. partial-melting/meltloss), these results, along with those previously described for mantle xenoliths [START_REF] Zhao | Coupled extremely light Ca and Fe isotopes in peridotites[END_REF], serve to highlight the potential for pairing diffusion models with disequilibrium Ca isotope variations in order to constrain both the nature and rates of processes important to both igneous and metamorphic petrology. pronounced Ca concentration and δ 44 Ca differences, with the edge (~1.5 wt% Ca) being +3.3‰ heavier than the center (~3.5 wt% Ca). The initial conditions assume uniform δ 44 Ca (and Ca concentration) across the dike, which slowly increase as Ca diffuses into the country rock. In order to reproduce the measured δ 44 Ca and Ca concentrations at the correct locations, Ca diffusivities must equal ~10 -10 m 2 /yr and ~10 -7 m 2 /yr in the dike and country rock, respectively. The required isotopic diffusivity ratio (D44/D40) is ~0.99, similar to that in dacitic liquids (see Fig. 14b). Additional details and diffusion model results for more complicated scenarios and rock-types can be found in the supplementary information of Antonelli et al. (2019c).

Another emerging application for disequilibrium inter-mineral Δ 44 Ca data is to constrain metamorphic mineral paragenesis. Preliminary results from the previously discussed granulite study (Antonelli et al., 2019c) suggest that Δ 44 Cagrt-mineral fractionations are related to the pressure-temperature-time (PTt) paths for metamorphism, where retrograde garnet formation (e.g. during isobaric cooling from ultrahigh-temperatures) leads to negative disequilibrium Δ 44 Cagrt-mineral values, and prograde garnet formation (clockwise PTt paths) appears to generate positive Δ 44 Cagrt-mineral values closer to predicted isotopic equilibrium (schematically depicted in Fig. 16). Combined with the observed kinetic slopes for negative garnet-plagioclase fractionations in triple isotope space (Fig. 12), this observation suggests that Ca sourced from plagioclase may be subjected to kinetic effects during diffusive transport towards growing garnet grains during isobaric cooling (Antonelli et al., 2019c). The large kinetic effects during retrograde garnet formation, that appear to be essentially absent during prograde and magmatic garnet formation, could be due in part to the removal of volatile elements prior to its formation.

However, very few samples have been measured and further work is required in order to better understand the origins of disequilibrium Ca isotope signatures observed between various metamorphic minerals. paths on the Ca isotopic differences observed between garnet and other minerals, as described in (Antonelli et al., 2019c). Path 1 (dashed green line) corresponds to a 'clockwise' heating path resulting in prograde garnet formation closer to isotopic equilibrium, while path 2 (dashed black line) corresponds to a 'counter-clockwise' path in which garnet is formed during isobaric cooling from very high-temperatures ('retrograde garnet growth') and acquires a negative disequilibrium Δ 44 Cagrt-plag signature (likely due to Ca diffusion, as suggested by Δ 48 Ca' measurements).

By constraining Ca isotope compositions in individual grains (through the development of in-situ δ 44 Ca analyses or more refined sampling techniques) and the average length-scales for Ca diffusion within a given system (e.g. through chemical image analyses), future studies should be able to yield important information on mineral paragenesis and Ca isotope disequilibrium related to metamorphic reactions.

Although our current state of knowledge on this topic is in its infancy, the various possibilities regarding isotopic fractionation during high-temperature metamorphic reactions can still be associated volcanic process. Given that crystal growth rates in magmatic systems vary by at least five orders of magnitude in nature [START_REF] Cooper | Time scales and temperatures of crystal storage in magma reservoirs: implications for magma reservoir dynamics[END_REF], they constitute a major uncertainty in estimating the durations of past volcanic eruption and magmatic recharge events. In order to address this problem, Ca isotope disequilibria have recently been used to independently estimate crystal-growth rates in volcanic and sub-volcanic samples (Antonelli et al., 2019b).

Ca isotope growth rate estimates are based on the model predictions of [START_REF] Watson | Non-equilibrium isotopic and elemental fractionation during diffusioncontrolled crystal growth under static and dynamic conditions[END_REF], where diffusion through a chemically-distinct boundary-layer (BL, formed around rapidly growing phenocrysts) can lead to kinetic isotope effects during crystal growth. If growth rates are rapid relative to Ca diffusivity in the melt (R/D), this is predicted to generate disequilibrium Δ 44 Ca variations in the crystals. As depicted in Fig. 17, crystallization of minerals where Ca is compatible (K > 1, e.g. plagioclase, clinopyroxene) is expected to lead to formation of a Ca-depleted BL (through which 40 Ca diffuses faster than 44 Ca) and negative kinetic Δ 44 Ca, whereas the opposite is expected if Ca is strongly incompatible within the mineral (K < 1, e.g. olivine, orthopyroxene) (Fig. 17).

The predicted kinetic fractionations during crystal growth depend on: (i) crystal growth-rates relative to Ca diffusivity (R/D), (ii) isotopic differences in Ca diffusion coefficients (D44/D40) (iii) mineral-melt Ca distribution coefficients, and (iv) boundary-layer (BL) thicknesses around the growing crystals [START_REF] Watson | Non-equilibrium isotopic and elemental fractionation during diffusioncontrolled crystal growth under static and dynamic conditions[END_REF]. Given the development of a steady-state BL during crystal growth, the kinetic δ 44 Ca fractionation between crystals and melt (Δ 44 Ca) is given by:

= 10 + ,1 - - -. /×12 - (1 -3) (Eq. 4)
where D40 and D44 are the diffusivities of 40 Ca and 44 Ca in the melt, respectively, R is the crystal growth rate, BL is the boundary layer thickness, and K is the Ca distribution coefficient in the mineral vs. the melt (generally K > 1 in plagioclase). Analyzing plagioclase separates from rapid tonalite crystallization experiments (Antonelli et al., 2019b), it was found that the driest samples (1 wt% H2O) have the largest fractionations (Δ 44 Caplag-matrix = -2.2‰) which gradually become smaller with increasing water contents (Δ 44 Caplag-matrix of -1.0‰ and -0.3‰, for the 1.5 wt% and 2 wt% H2O experiments, respectively).

Although the limited data cannot rule out changes in isotopic diffusivity ratios (D44/D40) and BL thicknesses due to increasing H2O (which have not yet been explored), the most simple suggested as maximum values in dynamic systems [START_REF] Lu | Diffusional Gradients at the Crystal / Melt Interface and Their Effect on the Compositions of Melt Inclusions[END_REF]. In (c), D44/D40 = 0.990 and Δ 44 Ca represents kinetic effects based on differences for plagioclase δ 44 Ca (Fisher Lake) or non-mineralogical δ 44 Ca differences between orbicule layers (Emerald Bay). BL sizes are 200 μm to account for larger crystal sizes in the orbs and comb-layers. Rate estimates are directly proportional to BL thicknesses, such that ½ a given BL thickness leads to double the required growth rate. Average values and 2SE for Fisher Lake samples (AM26, AM131) and kinetic layers in OG sample (Emerald Bay) are represented by colored squares (n = 3) with bold error bars in panel (c).

In conjunction with crystal-size distribution (CSD) analyses and measurements of maximum crystal sizes, the authors were able to estimate the durations of crystal growth in various volcanic systems. For the mafic and silicic systems studied, the results yielded similar timescales (ranging from <1 day to ~2 months), suggesting that the previously observed relationship between magmatic timescales and melt compositions [where timescales appear to generally increase from mafic to silicic systems (e.g. [START_REF] Cooper | Time scales and temperatures of crystal storage in magma reservoirs: implications for magma reservoir dynamics[END_REF]] may be due to a priori assumptions regarding crystal growth rates.

Although the initial results are promising, much work remains in order to improve Δ 44 Ca as a proxy for crystal-growth rates in volcanic systems. This includes, for example, careful separation of different phenocryst populations for a given mineral within a sample, where populations of smaller crystals (e.g. plagioclase microlites) are likely to have crystallized the fastest and to contain the largest disequilibrium Δ 44 Caplag-melt signatures. The samples shown in Fig. 18 represent mixtures of different plagioclase crystal populations (up to 3 per sample in the phenocrystic rocks), so the Δ 44 Ca measurements represent minimum values for the rapidlygrown populations within each sample (Antonelli et al., 2019b). Furthermore, most of the parameters in Equation 1are poorly constrained and could benefit from additional crystallization experiments and detailed observations of natural samples. This would then allow future studies to incorporate, for example, (i) the effects of multi-component diffusion into D44/D40 estimates, which could also consider chemical variations in BL compositions, (ii) variations in BL sizes in natural and experimental systems, and (iii) compositional and pressure/temperature dependence of Ca diffusivity in magmas. Nevertheless, preliminary results suggest that Ca isotopes provide an exciting new opportunity for understanding fundamental aspects of crystallization and magmatic recharge across volcanic systems observed on Earth today and in the geologic past.

High fluid/rock and hydrothermal systems

The Ca isotope composition of hydrothermal fluids and minerals has not been as extensively studied as low temperature weathering and carbonate mineral formation processes. Despite the relatively few studies of Ca isotopes in hydrothermal systems, high temperature fluid-mineral reactions are important for the global geochemical cycle because ocean hydrothermal systems are the second largest Ca flux to the oceans [START_REF] Staudigel | Hydrothermal alteration processes in the oceanic crust[END_REF]. Additionally, fossil hydrothermal systems record information about water rock reactions in a diverse set of geologically important environments such as porphyry deposits, continental hydrothermal systems, and ophiolites.

Shallow heat from mid-ocean ridge magmatism drives fluid circulation with the influx of seawater into the ocean crust and the emergence of high temperature (~400 ○ C) fluids at the ridge axis that are chemically distinct from seawater. Chemical reactions in the hydrothermal flow path remove Mg and SO4 and exchange seawater Ca (and Sr) for Ca derived from igneous minerals (e.g. [START_REF] Staudigel | Hydrothermal alteration processes in the oceanic crust[END_REF][START_REF] Von Damm | Chemistry of hydrothermal vent fluids from 9{\textdegree}{\textendash}10{\textdegree}N, East Pacific Rise: {\textquotedblleft}Time zero,{\textquotedblright} the immediate posteruptive period[END_REF]. Because the Ca isotope cycle in the ocean is possibly a robust record of past climate, the effect of hydrothermal reactions on the Ca isotope composition of seawater is an important research question.

The first measurements of mid-ocean ridge hydrothermal fluids were reported by [START_REF] Schmitt | The calcium riverine and hydrothermal isotopic fluxes and the oceanic calcium mass balance[END_REF], while subsequent data were reported by [START_REF] Amini | Calcium isotope (δ44/40Ca) fractionation along hydrothermal pathways, Logatchev field (Mid-Atlantic Ridge, 14 45 N)[END_REF], and most recently by [START_REF] Scheuermann | Calcium isotope systematics at hydrothermal conditions: Mid-ocean ridge vent fluids and experiments in the CaSO4-NaCl-H2O system[END_REF]. Sampled hydrothermal fluids are often mixed with seawater, and thus require some correction to calculate their hydrothermal endmember compositions. Figure 19 shows a compilation of hydrothermal fluid δ 44 Ca values as a function of percent seawater, which assumes that hydrothermal fluids have [Mg] = 0 and seawater has [Mg] = 53 mM (e.g. [START_REF] Lowenstein | The Geologic History of Seawater[END_REF][START_REF] Turchyn | Seawater Chemistry Through Phanerozoic Time[END_REF]. While the data exhibit scatter and are heavily weighted at the near-zero and near-100% seawater endmembers, they suggest that high temperature hydrothermal fluids have Ca isotope compositions similar to MORB (δ 44 CaBSE ≈ 0, which is approximately -1‰ relative to modern seawater). Data are from [START_REF] Amini | Calcium isotope (δ44/40Ca) fractionation along hydrothermal pathways, Logatchev field (Mid-Atlantic Ridge, 14 45 N)[END_REF][START_REF] Scheuermann | Calcium isotope systematics at hydrothermal conditions: Mid-ocean ridge vent fluids and experiments in the CaSO4-NaCl-H2O system[END_REF][START_REF] Schmitt | The calcium riverine and hydrothermal isotopic fluxes and the oceanic calcium mass balance[END_REF].

The hydrothermal fluid data are consistent with near complete Ca isotopic exchange between infiltrating seawater and basaltic minerals and require little or no net Ca isotopic fractionation in high temperature hydrothermal systems. Radiogenic Ca isotope differences reported between seawater and modern hydrothermal vents (Antonelli et al., 2019a[START_REF] Antonelli | Radiogenic 40Ca in Seawater[END_REF]) also suggest that the Ca within hydrothermal fluids is sourced almost entirely from oceanic crust. The Ca isotope system is thus dissimilar to 87 Sr/ 86 Sr, which is incompletely exchanged in modern and some ancient hydrothermal systems [START_REF] Amini | Calcium isotope (δ44/40Ca) fractionation along hydrothermal pathways, Logatchev field (Mid-Atlantic Ridge, 14 45 N)[END_REF][START_REF] Antonelli | Effect of paleoseawater composition on hydrothermal exchange in midocean ridges[END_REF][START_REF] Turchyn | Reconstructing the oxygen isotope composition of late Cambrian and Cretaceous hydrothermal vent fluid[END_REF]. The complete exchange of Ca isotopes and incomplete exchange of Sr isotopes is likely due to the relative distribution between the fluid and rock phases, where [Ca]basalt/[Ca]seawater is an order of magnitude higher than [Sr]basalt/[Sr]seawater [START_REF] Brown | High-temperature kinetic isotope fractionation of calcium in epidosites from modern and ancient seafloor hydrothermal systems[END_REF]. [START_REF] Brown | High-temperature kinetic isotope fractionation of calcium in epidosites from modern and ancient seafloor hydrothermal systems[END_REF] compositions (e.g. [START_REF] Amini | Calcium isotope (δ44/40Ca) fractionation along hydrothermal pathways, Logatchev field (Mid-Atlantic Ridge, 14 45 N)[END_REF][START_REF] Scheuermann | Calcium isotope systematics at hydrothermal conditions: Mid-ocean ridge vent fluids and experiments in the CaSO4-NaCl-H2O system[END_REF] and could lead to distinct hydrothermal δ 44 Ca signatures.

As opposed to oceanic hydrothermal systems, studies of continental hydrothermal systems show relatively large variations in fluid δ 44 Ca up to 1.5‰ [START_REF] Brown | Ca, Sr, O and D isotope approach to defining the chemical evolution of hydrothermal fluids: Example from Long Valley, CA, USA[END_REF][START_REF] Wiegand | Distribution of Sr and Ca Isotopes in Fluids of Lahendong Geothermal Field[END_REF].

The primary observation is that δ 44 Ca of the hydrothermal fluids increases as the fluid Ca concentration decreases, broadly consistent with isotope fractionation during secondary mineral precipitation [START_REF] Brown | Ca, Sr, O and D isotope approach to defining the chemical evolution of hydrothermal fluids: Example from Long Valley, CA, USA[END_REF][START_REF] Wiegand | Distribution of Sr and Ca Isotopes in Fluids of Lahendong Geothermal Field[END_REF]. In the case of Long Valley caldera, the δ 44 Ca shifts are primarily attributed to secondary carbonate mineral precipitation, which is a larger magnitude effect than Ca supply by primary mineral dissolution. The magnitude of δ 44 Ca fractionation (α) for the Long Valley system was inferred using an isotope distillation model, which yielded α=0.99965, similar to observations of calcite-fluid pairs: 0.99945-0.99970 [START_REF] Brown | Ca, Sr, O and D isotope approach to defining the chemical evolution of hydrothermal fluids: Example from Long Valley, CA, USA[END_REF]. While the inferred fractionations are smaller than low temperature processes such as marine carbonate precipitation, the fractionations are greater than would be predicted by temperature-dependent equilibrium models extended from low temperature minerals (Brown et al., 2013).

Hydrothermal minerals offer an additional opportunity to assess the mechanisms and magnitude of Ca isotope fractionation in high temperature fluids. Figure 20 shows a compilation of ∆ 44 Ca for synthetic and natural hydrothermal minerals as a function of fluid temperature. In some cases the fluid and minerals were directly measured [START_REF] Brown | Ca, Sr, O and D isotope approach to defining the chemical evolution of hydrothermal fluids: Example from Long Valley, CA, USA[END_REF][START_REF] Syverson | Experimental partitioning of Ca isotopes and Sr into anhydrite: Consequences for the cycling of Ca and Sr in subseafloor mid-ocean ridge hydrothermal systems[END_REF], for others the fluid composition and temperature have been calculated based on 87 Sr/ 86 Sr and isenthalpic mixing between seawater and vent fluids [START_REF] Amini | Calcium isotope (δ44/40Ca) fractionation along hydrothermal pathways, Logatchev field (Mid-Atlantic Ridge, 14 45 N)[END_REF]Griffith et al., 2008b).

For epidotes, a reactive transport model was used to estimate the fluid composition [START_REF] Brown | High-temperature kinetic isotope fractionation of calcium in epidosites from modern and ancient seafloor hydrothermal systems[END_REF]. experimental anhydrite (yellow), and natural anhydrite (light blue). Data are from [START_REF] Amini | Calcium isotope (δ44/40Ca) fractionation along hydrothermal pathways, Logatchev field (Mid-Atlantic Ridge, 14 45 N)[END_REF][START_REF] Brown | High-temperature kinetic isotope fractionation of calcium in epidosites from modern and ancient seafloor hydrothermal systems[END_REF][START_REF] Brown | Ca, Sr, O and D isotope approach to defining the chemical evolution of hydrothermal fluids: Example from Long Valley, CA, USA[END_REF]Griffith et al., 2008b;[START_REF] Syverson | Experimental partitioning of Ca isotopes and Sr into anhydrite: Consequences for the cycling of Ca and Sr in subseafloor mid-ocean ridge hydrothermal systems[END_REF].

The first and most significant observation is that ∆ 44 Camineral-fluid ≤ 0, with only one sample at unity. Additionally, there is no obvious trend in ∆ 44 Ca with temperature for the dataset as a whole, however, the anhydrites show decreasing fractionation with increasing temperature, while the ∆ 44 Ca of barites are positively correlated with temperature (Fig. 20). [START_REF] Amini | Calcium isotope (δ44/40Ca) fractionation along hydrothermal pathways, Logatchev field (Mid-Atlantic Ridge, 14 45 N)[END_REF] used the temperature dependence of ∆ 44 Ca for anhydrite to predict an average isotopic fractionation of ∆ 44 Cafluid-anhydrite of -0.54‰, but the temperature relationship is less robust if the experimental data of [START_REF] Syverson | Experimental partitioning of Ca isotopes and Sr into anhydrite: Consequences for the cycling of Ca and Sr in subseafloor mid-ocean ridge hydrothermal systems[END_REF] are included (Fig. 20). In contrast, [START_REF] Brown | High-temperature kinetic isotope fractionation of calcium in epidosites from modern and ancient seafloor hydrothermal systems[END_REF][START_REF] Brown | Ca, Sr, O and D isotope approach to defining the chemical evolution of hydrothermal fluids: Example from Long Valley, CA, USA[END_REF] found no ∆ 44 Ca temperature dependence for hydrothermal calcite or epidote. Thus, the extent to which there is a temperature dependence to ∆ 44 Ca for hydrothermal minerals remains an open question. One possible reconciliation of the data is that the relationship between temperature and ∆ 44 Ca is actually related to rapid changes in mineral saturation with fluid temperature changes. In this case, changes in temperature might correlate with mineral saturation (greater oversaturation is generally linked to faster precipitation rates) and by extension ∆ 44 Ca. This suggests that ∆ 44 Ca in hydrothermal minerals is primarily a kinetic effect-similar to the previously discussed examples for igneous and metamorphic mineral assemblages.

Significant ∆ 44 Ca between hydrothermal fluids and secondary minerals offer an opportunity to quantify subsurface processes in mid-ocean ridges, economic geothermal fields, and possibly in mineralizing fluids, however, additional research is required to better understand the underlying fractionation mechanisms and their implications for fluid-rock systems. Future work should include experiments at varying water-rock ratios, fluid chemistry, and temperatures, in order to enable more quantitative interpretations of modern and fossil hydrothermal systems.

Employing radiogenic and stable Ca isotopes as high-temperature source-tracers

The application of Ca isotopes as a petrogenetic tracer is different but complementary to other more commonly used trace element isotope systems (e.g. Sr, Pb, Hf, and Pb). The K-Ca radioactive system (half-life of ~1.25 Ga), however, behaves differently because this parentdaughter system involves elements that make up stoichiometric constituents of major minerals, and therefore the concentrations of K and Ca, and the K/Ca ratio in rocks, are directly related to mineralogy (e.g. [START_REF] Marshall | Calcium isotopes in igneous rocks and the origin of granite[END_REF]. Radiogenic Ca isotope excesses are commonly reported using epsilon notation, after correcting for natural and instrumental mass-fractionation (using an exponential law and 42 Ca/ 44 Ca = 0.31221): 4 = 10,000 6 789 -1: (Eq.

5)

Where ( 40 Ca/ 44 Ca)BSE is generally assumed to be 47.162, though recent work suggests that a value of 47.156 (about 1 epsilon unit lower), may be more accurate (Antonelli et al., 2019a[START_REF] Antonelli | Radiogenic 40Ca in Seawater[END_REF][START_REF] Mills | Calcium and neodymium radiogenic isotopes of igneous rocks: Tracing crustal contributions in felsic magmas related to super-eruptions and continental rifting[END_REF]. The comparatively large mass differences among Ca isotopes cause physiochemical processes to produce relatively large mass-dependent variations in nature, even at relatively high temperature. Although such studies are not routine, as methods become more efficient and measurements become more reproducible, both radiogenic and stable Ca isotope signatures will play important roles in tracking the petrological evolution of planetary to outcropscale processes in igneous and metamorphic rocks.

Carbonate recycling in the mantle

Over 50 years of sample measurements have shown that the mantle remains chemically and isotopically heterogeneous despite the fact that mantle convection has been active for most of Earth's history. Oceanic basalt studies employing radioactive isotope systems (e.g., Sr, Nd, and Pb) have been used to identify compositionally distinct mantle reservoirs that reflect recent mixtures of different source components as well as the time integrated compositional history of ancient mantle reservoirs (e.g. [START_REF] Hawkesworth | Radiogenic Isotopes -Some Geological Applications[END_REF]. Likewise, there are a number of geochemical and isotopic systems that have been used to suggest that recycled marine sediments are important in the petrogenesis of ocean island basalts (e.g. [START_REF] Blichert-Toft | Hf Isotope Evidence for Pelagic Sediments in the Source of Hawaiian Basalts[END_REF][START_REF] Castillo | Marine Carbonates in the Mantle Source of Oceanic Basalts: Pb Isotopic Constraints[END_REF][START_REF] Dasgupta | Deep global cycling of carbon constrained by the solidus of anhydrous, carbonated eclogite under upper mantle conditions[END_REF][START_REF] Frey | Geochemical characteristics of Koolau Volcano: Implications of intershield geochemical differences among Hawaiian volcanoes[END_REF][START_REF] Huang | Ancient carbonate sedimentary signature in the Hawaiian plume: Evidence from Mahukona volcano, Hawaii. Geochemistry[END_REF][START_REF] Huang | Recycled oceanic crust in the Hawaiian Plume: evidence from temporal geochemical variations within the Koolau Shield[END_REF][START_REF] Lassiter | Osmium-isotope variations in Hawaiian lavas: evidence for recycled oceanic lithosphere in the Hawaiian plume[END_REF][START_REF] Plank | The chemical composition of subducting sediment and its consequences for the crust and mantle[END_REF]. The interconnection between recycled materials at subduction zones and more ancient mantle reservoirs remains elusive, however, and debates continue as to whether recycling at subduction zones is related to the isotopic heterogeneity in oceanic basalts.

Ca isotopes potentially provide a powerful tool to test ideas about the various potential source(s) of mantle heterogeneity, in particular for recycling of Ca-rich marine sedimentary materials, because both radiogenic and stable Ca isotope variations yield complementary information.

Significant stable mass-dependent Ca isotope variability is not generally expected to originate at mantle solidus temperatures under which oceanic basalts have been generated (>1200-1500 ○ C, e.g. [START_REF] Putirka | Mantle potential temperatures at Hawaii, Iceland, and the mid-ocean ridge system, as inferred from olivine phenocrysts: Evidence for thermally driven mantle plumes[END_REF][START_REF] Zhang | No measurable calcium isotopic fractionation during crystallization of Kilauea Iki lava lake[END_REF]. However, subduction and recycling of sediments, especially marine carbonates, could introduce distinctive Ca isotopic variations within the mantle because marine carbonates have high Ca concentrations with potentially light δ 44 Ca, up to ~2‰ lower than BSE [START_REF] De La Rocha | Isotopic Evidence for Variations in the Marine Calcium Cycle Over the Cenozoic[END_REF][START_REF] Depaolo | Calcium Isotopic Variations Produced by Biological, Kinetic, Radiogenic and Nucleosynthetic Processes[END_REF][START_REF] Fantle | Variations in the marine Ca cycle over the past 20 million years[END_REF][START_REF] Farkaš | Evolution of the oceanic calcium cycle during the late Mesozoic: Evidence from δ44/40Ca of marine skeletal carbonates[END_REF]Griffith et al., 2008a;[START_REF] Heuser | Calcium isotope (δ 44/40 Ca) variations of Neogene planktonic foraminifera[END_REF][START_REF] Kasemann | Boron and calcium isotope composition in Neoproterozoic carbonate rocks from Namibia: evidence for extreme environmental change[END_REF][START_REF] Watkins | Kinetic Fractionation of Non-Traditional Stable Isotopes by Diffusion and Crystal Growth Reactions[END_REF][START_REF] Zhu | Calcium isotopes in the marine environment and the oceanic calcium cycle[END_REF].

To date, resolvable radiogenic Ca signatures have not been observed in oceanic basalts [START_REF] Huang | Stable calcium isotopic compositions of Hawaiian shield lavas: Evidence for recycling of ancient marine carbonates into the mantle[END_REF][START_REF] Marshall | Calcium isotopes in igneous rocks and the origin of granite[END_REF][START_REF] Simon | Calcium Isotope Composition of Meteorites, Earth, and Mars[END_REF]. This might not be surprising given the work of [START_REF] Caro | 40K-40Ca isotopic constraints on the oceanic calcium cycle[END_REF] who despite finding well-defined excesses of 40 Ca in some river waters draining into the ocean, report that no discernable effects of 40 K decay (within their reported analytical precision of ~0.4 epsilon units, 2σ) were seen in a number of marine carbonate samples ranging in age from Archean to recent. This led the authors to conclude that hydrothermal Ca has dominated the oceanic budget over all of Earth history. This conclusion is at odds, however, with estimates of modern riverine vs. hydrothermal Ca fluxes (e.g. DePaolo, 2004) and with εCa differences observed between modern seawater/carbonates and hydrothermal fluids/mafic igneous rocks (Antonelli et al., 2019a[START_REF] Antonelli | Radiogenic 40Ca in Seawater[END_REF]. However, the importance of both of these studies has been challenged more recently, in part, because marine carbonates deposited over the last ~2.5 Ga have average δ 44 Ca that exhibit little change through time, with values similar to BSE [START_REF] Blättler | Testing Urey's carbonate-silicate cycle using the calcium isotopic composition of sedimentary carbonates[END_REF]. Likewise, it has been recently shown that carbonate metasomatism in the mantle has no discernable δ 44 Ca effects [START_REF] Ionov | Calcium isotopic signatures of carbonatite and silicate metasomatism, melt percolation and crustal recycling in the lithospheric mantle[END_REF]. More work like that of [START_REF] Huang | Stable calcium isotopic compositions of Hawaiian shield lavas: Evidence for recycling of ancient marine carbonates into the mantle[END_REF], therefore, is needed to validate the influence of sediment recycling on the isotopic source compositions of igneous rocks. One possible reconciliation of these models and data could be that the marine carbonate records no longer accurately record their ancient (isotopically light) signatures because they have been reset more recently by exchange with bulk silicate mantle Ca reservoirs.

Crustal Contamination in Magmatic Systems

The Ca isotope ratio determinations of [START_REF] Russell | Ca isotope fractionation on the Earth and other solar system materials[END_REF] provide the basis for all current measurements. Established on sample measurements where radiogenic 40 Ca excesses are negligible, these values are used to represent the normal terrestrial composition at ca. 0.01% resolution. For radiogenic enrichment studies, Ca isotope variations are commonly reported as εCa (Equation 5). Radioactive decay of 40 K within the continental crust can produce a Ca isotopic signature where 40 Ca excesses ≥ 0.5 epsilon units are distinguishable from mantle-like Ca isotope compositions, as shown in [START_REF] Mills | Calcium and neodymium radiogenic isotopes of igneous rocks: Tracing crustal contributions in felsic magmas related to super-eruptions and continental rifting[END_REF]. First employed in modern literature for igneous rocks by Marshall andDePaolo (1989, 1982), the K-Ca radiogenic isotope system has not been applied to many petrogenetic studies primarily because of the difficulty in decoupling the radiogenic enrichments of 40 Ca from stable 40 Ca that makes up ~97% of terrestrial calcium.

Oceanic basalts largely have compositions and isotopic signatures that indicate derivation from partial melting of mantle sources. At levels that were near the analytical capabilities available at the time, however, [START_REF] Marshall | Calcium isotopes in igneous rocks and the origin of granite[END_REF] showed the first evidence for crustal ( 40 Caenriched) calcium in some oceanic island arc magmas. The source components for more evolved magmas erupted through, or emplaced within, continental crust also remain debated. The petrogenic origin(s) of evolved magmas in the continental crust can be complicated because the relative importance of various petrogenetic processes, including magma mixing, fractional crystallization, and assimilation-fractional crystallization (i.e., AFC), controlling magma differentiation can vary [START_REF] Depaolo | Trace element and isotopic effects of combined wallrock assimilation and fractional crystallization[END_REF].

As with studies of crustal recycling discussed above, trace element radiogenic isotope systems have long been used to assess the petrogenesis of felsic magmas. Despite all that has been learned from trace element radiogenic studies, unresolved geologic problems remain. In fact, one of the most important and long-standing challenges is the ability to distinguish the contributions of mantle-derived sources in felsic magmas that are generated from preexisting continental crust [see [START_REF] Mills | Calcium and neodymium radiogenic isotopes of igneous rocks: Tracing crustal contributions in felsic magmas related to super-eruptions and continental rifting[END_REF] and reference therein]. Although old continental crust and asthenospheric mantle sources can be readily distinguished with εNd values, these authors showed that by combining radiogenic Nd and Ca isotope ratios, significant progress can be made. This is because whereas non-radiogenic Nd can be indicative of either old continental crust or lithospheric mantle, Ca isotopes can potentially distinguish between crustal (elevated εCa) and lithospheric mantle (near zero εCa) sources (Fig. 22). Epsilon values for Ca were calculated using the average 40 Ca/ 44 Ca value measured for peridotite and data from [START_REF] Simon | Calcium Isotope Composition of Meteorites, Earth, and Mars[END_REF] for oceanic basalts, which was used to define bulk silicate Earth. The SRM 915a value (shown on left axis) is ~1 ε-unit higher than bulk silicate Earth. Epsilon values for Nd were calculated using 143 Nd/ 144 Nd (CHUR, 0 Ma) = 0.512638 and 147 Sm/ 144 Nd (CHUR, 0 Ma) = 0.1967.

Radiogenic Ca and K-loss in metamorphic systems

The loss of incompatible trace elements (e.g. U, Rb) during melting/dehydration of granulitefacies rocks is well documented (e.g. [START_REF] Rudnick | Large ion lithophile elements in rocks from highpressure granulite facies terrains[END_REF][START_REF] Rudnick | Composition of the Continental Crust[END_REF]. The resultant transfer of heat-producing elements from the lower-crust to the middle-and upper-crust is an essential aspect of many models for the creation of stable continents (through the coupling Using available protolith and metamorphic age estimates for the various localities, the authors placed constraints on the protolith K/Ca values necessary to generate initial εCa recorded by garnet, assuming the oldest protolith ages from each locality in order to arrive at minimum protolith K/Ca estimates (as shown in Fig. 24). This figure also serves to demonstrate the relationship between estimated protolith K/Ca and uncertainties in associated protolith age estimates, where an uncertainty of ±0.3 Ga gives K/Ca estimates consistent within a factor of ~2, even for ancient samples where radiogenic ingrowth of 40 Ca had the biggest effects. Comparing protolith estimates to K/Ca measured in the rocks today, minimum values for K loss (relative to Ca) during lower crustal metamorphism were constrained to ~67% to > 95% (Fig. 25a), confirming that K is effectively lost from the lower-crust during metamorphism. A nonmodal batch melting model [see SI text of Antonelli et al. (2019a) for details] suggests that most samples represent 20-30% granitic melt-loss (Fig. 25b). These estimates are in good agreement with other estimates for melt-loss during granulite-facies metamorphism, (e.g. [START_REF] Yakymchuk | Consequences of open-system melting in tectonics[END_REF], but further constraints on K and Ca partitioning during partial melting, along with sample-specific protolith age estimates, would be greatly useful to further constrain K and melt loss in future studies.

Conclusions and future work

In this review, we have shown that stable Ca isotope fractionations at high-temperatures can equal and often exceed those that occur in low-temperature surface environments. (Antonelli et al., 2019a)], but further exploration is required.

Although there have been no experimental constraints for equilibrium fractionation factors between minerals, ab-initio simulations have provided great insight into equilibrium Ca isotope fractionation at high temperatures. General agreement is reached between the various studies in spite of their different theoretical approaches. Likewise, the existing measurements of equilibrium slopes in triple Ca isotope space ( 48 Ca/ 44 Ca vs. 42 Ca/ 44 Ca) for samples that agree with ab-initio estimates, suggest that they are approximately correct. Thus, there is potential for using inter-mineral Δ 44 Ca measurements for future geothermometry, especially in nonmetasomatized plutonic rocks where minerals tend to have compositions closer to predicted equilibrium. For the moment, however, relatively common observations of inter-mineral disequilibrium preclude the use of Ca isotopes for geothermometry, and additional constraints (e.g. independent temperature estimates combined with ab-initio predictions, and/or Δ 48 Ca' measurements) are necessary in order to further understand the geologic environments in which isotopic equilibrium can be expected.

The large Δ 44 Ca fractionations observed between adjacent whole rocks (which can exceed 3‰) and between coexisting minerals (which can exceed 2‰) have slopes in triple Ca isotope space that agree with the predicted kinetic mass-fractionation law (based on Graham's law), suggesting that they are the result of Ca diffusion. Although modeling disequilibrium isotope fractionation can be more complicated than modeling equilibrium fractionation, Ca isotope diffusion models have been applied to natural samples and yield important information on (i) the timescales of mantle metasomatism, (ii) Ca transport mechanisms during lower crustal metamorphism, (iii) metamorphic mineral paragenesis, (iv) crystal growth rates in volcanic and sub-volcanic rocks, and (v) mineral precipitation rates in hydrothermal systems.

Radiogenic 40 Ca variations (εCa) are especially useful for source-tracing studies, as they are not complicated by the numerous stable isotope effects mentioned above. In rocks with known protolith and metamorphic ages, εCa measurements can be used to estimate initial K/Ca values and constrain K-loss (and potentially K-gains) during metamorphism. Studies have also shown that εCa can be used as a proxy for crustal contamination in mantle-derived continental magmas and (in combination with εNd) to differentiate between lithospheric and asthenospheric mantle sources.

The εCa differences between (modern) carbonates and mantle, however, are not large enough to trace potential contributions from carbonate subduction in igneous samples. Although there is potential for using stable Ca isotopes for this purpose (due to low δ 44 Ca values in rapidlyprecipitated carbonates), this idea has recently been challenged by studies concluding that, on average, carbonates have the same δ 44 Ca as BSE over the last several billion years, and that carbonatite metasomatism has no measurable effects in mantle samples. Thus, further studies are necessary in order to better constrain the situations in which δ 44 Ca can be used as a tracer of carbonate subduction.

In order to further advance our understanding of Ca isotope fractionations during hightemperature processes, future studies of natural samples should pay special attention to: (i) kinetic vs. equilibrium signatures (e.g. by using Δ 48 Ca' measurements or independent temperature constraints combined with ab-initio estimates), (ii) spatial constraints and field relationships suggestive of initial chemical/temperature gradients or open-system processes, and

Fig. 1 .

 1 Fig. 1. Compilation of δ 44 CaBSE in igneous whole rocks (n = 655) grouped by major sample type. All TIMS data have been corrected for radiogenic 40 Ca variations, when applicable, and all data are normalized to Bulk-silicate

Fig

  Fig. 2a-d. (a,b) Bulk chemical and δ 44 Ca data for ultramafic rocks from the Balmuccia (BM) and Baldissero (BD) orogenic massifs, showing predicted effects of partial melting, crystallization, and mantle refertilization after (Chen et al., 2019b). Fertile lherzolite and harzburgite xenoliths are shown for comparison (hollow symbols) (Kang et al., 2017; Zhao et al., 2017). δ 44 Ca data are relative to SRM915a, and the blue horizontal band represents the estimated δ 44 Ca of the Earth's upper mantle (0.94 ± 0.10‰). The long-term external precision from Chen et al. (2019b), ~0.14‰ (2sd) is shown at the bottom left. (c) Plot of whole rock δ 44 Ca versus MgO for igneous rocks with model predictions for partial melting of the mantle at 1 and 2 GPa. Horizontal grey band represents BSE composition estimates with uncertainty, after Zhang et al. (2018). (d) same as (c) but with expanded y-axis and without data, to more clearly show predicted effects of partial melting [model details can be found in Zhang et al. (2018)]. Data

Fig. 3 .

 3 Fig. 3. δ 44 CaBSE for metamorphic whole rocks grouped by principal metamorphic facies and protolith-type (n = 72).

Fig. 4a -

 4a Fig. 4a-c. Equilibrium estimates for reduced partition function ratios (1000lnβ) vs. temperature in various minerals, based on DFT models using local-density approximation (LDA) functionals. (a) Modified from Huang et al. (2019), orthopyroxene and diopside estimates (*) are from Wang et al. (2017b). (b) Orthopyroxene with various solid-

Fig. 5 .

 5 Fig. 5. Equilibrium estimates for reduced partition function ratios (1000lnβ) for 44 Ca/ 40 Ca vs. temperature for minerals based on DFT models using generalized gradient approximation (GGA) PBE functionals (Perdew, Burke, Ernzerhof), modified from(Antonelli et al., 2019c). As opposed to Fig.4, the lines are curved due to use of a linear temperature scale.

Fig.

  Fig. 6a,b. Summary of available equilibrium estimates for 44 Ca/ 40 Ca reduced partition function ratios (1000lnβ) vs. (a) Ca concentrations relative to other major cations, Ca/(Ca+Mg+Fe+Mn+Na), and (b) average Ca-O bond lengths in various minerals. PBE-based estimates (colored symbols) are generally lower than LDA-based estimates. Data are from(Antonelli et al., 2019c;[START_REF] Feng | First-principles investigations of equilibrium calcium isotope fractionation between clinopyroxene and Ca-doped orthopyroxene[END_REF][START_REF] Huang | First-principles calculations of equilibrium Ca isotope fractionation : Implications for oldhamite formation and evolution of lunar magma ocean[END_REF][START_REF] Song | First-principles investigation of the concentration effect on equilibrium fractionation of Ca isotopes in forsterite[END_REF] Wang et al., 2017b).

Fig.

  Fig. 7a,b. Ca isotope compositions in igneous minerals (n = 146), grouped by rock type. (a) measured δ 44 Ca in mineral separates, and (b) measured inter-mineral Δ 44 Ca (n = 69). Data are from(Antonelli et al., 2019b; Chen et al.,

Fig.

  Fig. 8a-c. Mineral-specific δ 44 Ca measurements in igneous rocks (grouped by rock-type) compared with equilibrium ab-inito predictions at various temperatures (in degrees Celsius). (a) δ 44 Caopx vs. δ 44 Cacpx (n = 47), (b) δ 44 Caol vs.δ 44 Cacpx (or δ 44 Caplag) (n = 3), and (c) δ 44 Cacpx vs. δ 44 Caplag (n = 4). Data are from(Antonelli et al., 2019b; Chen et

Fig

  Fig. 9a,b. Ca isotope variations in metamorphic minerals, grouped by metamorphic-facies and protolith composition. (a) δ 44 CaBSE of metamorphic mineral separates (n = 81) and (b) inter-mineral Ca isotope differences (Δ 44 Ca) in metamorphic rocks (n = 66). Triangles correspond to Δ 44 Cagrt-min, diamonds correspond to Δ 44 Camin-fsp, and circles correspond to Δ 44 Camin-cpx. Approximate 2sd uncertainties for δ 44 Ca are ±0.1‰. Opx* corresponds to pure orthopyroxene estimates from unmixing calculations which can have variable uncertainties based on the unmixing parameters [average 2sd of ~0.5‰(Antonelli et al., 2019c)]. Data are from(Antonelli et al., 2019c;

  (a) δ 44 Cagrt vs. δ 44 Cafsp (n = 11), (b) δ 44 Cagrt vs. δ 44 Cacpx (n = 16), (c) δ 44 Cagrt vs. δ 44 Caopx (n = 6), (d) δ 44 Caopx vs. δ 44 Cafsp (n = 12), (e) δ 44 Caopx vs. δ 44 Cacpx (n = 7), (f) δ 44 Cacpx vs. δ 44 Cafsp (n = 5). Asterisk (*) denotes unmixed orthopyroxene δ 44 Ca estimates. DFT predictions are based on PBE functionals

Fig

  Fig. 11a,b. Whole-rock and mineral composition controls on Δ 44 Ca in eclogite, amphibolite, and granulite-facies rocks after Antonelli et al. (2019c). Garnet-mineral and garnet-whole rock Δ 44 Ca versus (a) whole rock Mg# and (b) garnet Mg#. Error bars are generally smaller than the symbols. Colored bands indicate predicted equilibrium at 800-900 °C from ab-initio estimates(Antonelli et al., 2019c): Grt-Opx (purple), Grt-Cpx (green), Grt-Fsp (blue).Orthopyroxene predictions are based on Ca:Mg of 1:63.

Fig. 12 .

 12 Fig. 12. Triple Ca isotope compositions (Δ 48 Ca' vs. δ 42/44 CaBSE) for granulite-facies whole rocks (and minerals) from Diavik, Ivrea, and the Napier Complex, and for an experimental tonalite and cumulate gabbro sample, modified

  figure.

Fig. 13 .

 13 Fig. 13. Example of diffusion experiment results between high-An and low-An plagioclase melts at 1450 ○ C, from[START_REF] Watkins | Influence of liquid structure on diffusive isotope separation in molten silicates and aqueous solutions[END_REF]. The low-Ca side (right) experiences a temporary negative shift in δ 44 Ca as the 40 Ca diffuses into it more quickly, while the high-Ca side (left) experiences a positive shift in δ 44 Ca as the 44 Ca atoms are preferentially left behind. In a closed system, the isotopic deviations level out to zero once 44 Ca has had the time to 'catch up' with 40 Ca, which generally occurs before the Ca concentration gradients reach zero. In this example, the Ca isotopic fractionations are increased due to counter diffusion of Na.

Fig. 14a ,

 14a Fig. 14a,b. Experimental and molecular-dynamic simulation beta-factors versus (a) cation diffusivity relative to the diffusivity of silicon (Di/DSi) and (b) magma SiO2 + Al2O3 (in wt%) after[START_REF] Watkins | Kinetic Fractionation of Non-Traditional Stable Isotopes by Diffusion and Crystal Growth Reactions[END_REF]. 'β' is related to D44/D40 by (m44/m40) β [but must not to be confused with the same variable used for reduced partition function ratios (1000lnβ, as in Figs.4-6) presented earlier in the text], where m44 and m40 are the atomic masses of 44 Ca and 40 Ca, respectively. The efficacy of kinetic isotopic fractionation increases with increasing magma viscosity (SiO2 + Al2O3), with the largest effects in high SiO2 magmas.

Fig. 15 .

 15 Fig. 15. Results of numerical modeling for Ca isotope diffusion between a high-Ca ultramafic dike and low-Ca pelitic country rock, metamorphosed at granulite-facies conditions for 100 Myr from (Antonelli et al., 2019c). The top panel shows changing Ca concentrations and the bottom panel shows the corresponding Δ 44 Ca values over time (where Δ 44 Ca corresponds to δ 44 Cainitial-δ 44 Cafinal). The two analyzed samples (also shown in Fig. 12) come from the center ('2035A') and ~8cm from the edge ('2035B' demarcated by the dashed lines) of a ~1.7m thick dike and have

Fig. 16 .

 16 Fig. 16. Simplified conceptual phase diagram depicting the hypothesized effects of pressure-temperature-time (PTt)

Fig. 17 .

 17 Fig. 17. Schematic diagram showing the effects of Ca diffusion through a chemically-distinct boundary layer (BL) surrounding a growing crystal, based on the model of Watson and Müller (2009), figure from Antonelli et al. (2019b). When Ca is compatible within a mineral, depletion of Ca in the BL leads to an enrichment of 40 Ca at the mineral surface due to diffusion through the BL towards the crystal (top right panel); when Ca is incompatible,build-up of Ca in the BL leads to a relative enrichment of 44 Ca at the crystal surface as Ca diffuses away from the crystal (lower right panel). For further details seeAntonelli et al. (2019b),[START_REF] Watkins | Kinetic Fractionation of Non-Traditional Stable Isotopes by Diffusion and Crystal Growth Reactions[END_REF], and[START_REF] Watson | Non-equilibrium isotopic and elemental fractionation during diffusioncontrolled crystal growth under static and dynamic conditions[END_REF].

Fig. 18a -

 18a Fig. 18a-c. Δ 44 Ca vs. Ca distribution coefficients for analyzed plagioclase separates from (a,b) experimental (n = 3) and natural phenocryst samples (n = 4) along with (c) separates from orbicule and comb-layer samples (Emerald Bay and Fisher Lake, n = 19) with dashed contours representing predictions for different crystal growth rates, after Antonelli et al. (2019b). In (a,b), Δ 44 Ca represents δ 44 Ca differences between plagioclase and glass/matrix, and D44/D40 = 0.980, in accordance with previous work shown in Fig. 14. Boundary-layer (BL) thicknesses are 100 μm,

Fig 19 .

 19 Fig 19. δ 44 Ca of hydrothermal fluids as a function of seawater percentage. Seawater percentage is based on Mg concentrations assuming end-member hydrothermal fluids and seawater have [Mg] of 0 and 53.4 mM, respectively.

Fig. 20 .

 20 Fig. 20. Compilation of Ca isotope fractionation between hydrothermal fluids and hydrothermal minerals with increasing temperatures (n = 20). The different colors correspond to epidote (purple), barite (orange), calcite (grey),

Fig. 22 .

 22 Fig. 22. Plot of age-corrected ε 143 Nd vs. ε 40 Ca showing the potential use of radiogenic Ca for discriminating between different mantle sources. The Data are from Mills et al. (2018) unless otherwise indicated in the legend.

Fig.

  Fig. 23a,b. (a) Whole rock A/CNK, defined as molar Al2O3/(CaO+Na2O+K2O) and (b) εCa data for garnet separates from granulite-facies metamorphic rocks versus garnet grossular (%); after Antonelli et al. (2019a). Approximate compositional fields for various protoliths (based on A/CNK and garnet grossular) are demarcated with dashed lines. Garnet efficiently excludes K from its crystal structure, thus the measured εCa values are the initial values at time of metamorphism. A bulk-silicate Earth 40 Ca/ 44 Ca value of 47.156 was used to calculate εCa.

Fig. 24 .

 24 Fig. 24. Effect of protolith age uncertainty on estimated protolith K/Ca, after Antonelli et al. (2019a). The different lines represent garnet εCa measurements from granulite-facies rocks from the Napier Complex, where the metamorphic age is ~2.5 Ga. Due to the very low compatibility of K in garnet, garnet is assumed to represent wholerock εCa at the time of metamorphism. Although protolith ages are not often constrained with great precision, this figure demonstrates the relatively minor effect that this has on estimated protolith K/Ca.

Fig. 25a ,

 25a Fig. 25a,b. (a) K/Ca modern vs. K/Ca protolith estimates based on garnet εCa and protolith/metamorphic ages, and (b) (K/Ca)mdrn/(K/Ca)protolith vs. modern whole rock K2O (wt%) with melt-loss estimates based on a non-modal batch melting model, after Antonelli et al. (2019a). Arrow shows the potential effects of incomplete melt loss.

  1, we present inter-conversion factors for commonly used Ca isotope standards. To use these factors and report Ca isotope data to different reference materials, it is sufficient to add the listed value from Table 1 to the measured delta value [e.g. δ 44 CaBSE = δ 44 Ca915a + (-0.95)]. Note that for very large values, it is more accurate to recast the delta values into isotopic ratios and then re-calculate new delta values using the 44 Ca/ 40 Ca of desired reference materials, but terrestrial Ca isotope fractionations are not generally large enough to warrant this additional step. To convert δ 44/42 Ca values to δ 44 Ca, one can simply multiply by 2.05 (or divide δ 44 Ca by 2.05 to arrive at δ 44/42 Ca). In this study, we chose to report δ 44 Ca data relative to BSE in order to more clearly represent potential complementarity between various solid-Earth reservoirs and have converted published data using the values listed in Table

Table 1 .

 1 Values for inter-conversion of δ 44 Ca results using common standard materials.

	Measured Ca reference

1.

  reported δ 44 Ca values depending on the analytical technique. TIMS measurements are sensitive to radiogenic 40 Ca variations, whereas multi-collector inductively-coupled plasma mass spectrometry (MC-ICP-MS) analyses (which do not commonly measure 40 Ca due to isobaric interferences from 40 Ar) are not affected by radiogenic Ca variations. The difficulty of achieving a stable signal for 42 Ca using MC-ICP-MS (which requires high-resolution slits in order to separate42 Ca from a large isobaric interference at mass 42), however, stifles the potential dominance of this method over TIMS. The radiogenic 40 Ca enrichments in SRM915a and seawater ( 40 Ca/ 44 Ca of 47.162 vs. ~47.156 for BSE, at a normalizing 42 Ca/ 44 Ca ratio of 0.31221)

are just barely resolvable with current analytical precisions (~0.1‰) and still await further confirmation before renormalization of all TIMS data

  there have been well over 40 studies focused on exploring Ca isotope variations in igneous rocks. A dominant proportion of these studies have measured δ 44 Ca in different types of mantle (ultramafic) whole rocks, while a more limited number of studies explore δ 44 Ca in intermediate to felsic igneous rocks. Ultramafic samples are generally defined by their modal mineralogies, falling into two main categories, pyroxenites (orthopyroxenites, websterites, and clinopyroxenites, with or without olivine) and peridotites (lherzolites, harzburgites, wehrlites, and dunites). In a general sense, starting from fertile mantle and extracting basaltic melt, one is left with residual lherzolite (> 5% Cpx), and with greater melt extraction, residual harzburgite (<

Table 2 .

 2 Compilation of ab-intio predictions and fitting-factors for equilibrium RPFRs (1000lnβ).

								Bond-
					Ca/Ca+Fe+Mg+	1000lnβ		length
	Reference	Type	Mineral		Mn+Na	(1000 K)	Coord.	(Å)	a	b	c
	Huang et al., 2019	LDA	Anorthite	-	1.000	1.30	6.75	2.455	6.21E-05	-5.68E-03	1.307
			Forsterite 1/32	Ca/Mg 1/32	0.031	2.34	6	2.269	1.63E-04	-1.51E-02	2.355
			Forsterite 1/64	Ca/Mg 1/64	0.016	2.36	6	2.267	0.00E+00	0.00E+00	2.362
			Grossular	-	1.000	2.01	8	2.393	1.08E-04	-1.08E-02	2.025
			Oldhamite	-	1.000	0.96	6	2.826	4.89E-06	-1.71E-03	0.957
			Pigeonite 1/8	Ca/Mg 1/8	0.125	1.93	7	2.375	1.15E-04	-1.11E-02	1.942
			Pigeonite 1/16	Ca.Mg 1/16	0.063	1.96	7	2.373	0.00E+00	0.00E+00	1.964
			Pigeonite 1/32	Ca/Mg 1/32	0.031	2.10	7	2.361	0.00E+00	0.00E+00	2.097
	Song et al., 2019	LDA	Forsterite 1/64	Ca/Mg 1/64	0.016	2.37	6	2.267	2.36E-04	-1.57E-02	2.383
			Forsterite 1/32	Ca/Mg 1/32	0.031	2.20	6	2.284	2.09E-04	-1.41E-02	2.216
			Forsterite 1/16	Ca/Mg 1/16	0.063	2.01	6	2.299	1.74E-04	-1.21E-02	2.021
			Forsterite 1/16a	Ca/Mg 1/16	0.063	2.15	6	2.299	3.27E-03	-5.64E-02	2.202
			Forsterite 1/8	Ca/Mg 1/8	0.125	1.88	6	2.327	1.28E-04	-9.83E-03	1.888
			Forsterite 1/8a	Ca/Mg 1/8	0.125	1.87	6	2.327	1.25E-03	-2.46E-02	1.890
			Forsterite 1/4	Ca/Mg 1/4	0.250	1.78	6	2.333	1.27E-04	-9.12E-03	1.789
			Forsterite 1/4a	Ca/Mg 1/4	0.250	1.81	6	2.333	8.88E-04	-1.95E-02	1.831
	Feng et al., 2014	LDA	Opx 1/32	Ca/Mg 1/32	0.031	2.05	6	2.305	1.88E-04	-1.27E-02	2.061
			Opx 1/16	Ca/Mg 1/16	0.063	1.77	6	2.363	1.51E-04	-9.71E-03	1.778
			Opx 1/8	Ca/Mg 1/8	0.125	1.59	6	2.392	1.31E-04	-8.27E-03	1.603
			Opx 1/2	Ca/Mg 1/2	0.500	1.57	6	2.388	1.20E-04	-7.70E-03	1.573
			Cpx 1/2	Ca/Mg 1/2	0.500	1.45	8	2.47	1.18E-04	-6.89E-03	1.458
	Wang et al., 2017b	LDA	Opx 1/32 Fe 3/32	Ca/Mg 1/32 Fe 3/32	0.031	1.91	6	2.326	1.59E-04	-1.11E-02	1.918
			Opx 1/48	Ca/Mg 1/48	0.021	2.09	6	2.3	1.94E-04	-1.32E-02	2.104
			Cpx 3/8	Ca/Mg 3/8	0.375	1.54	8	2.459	1.26E-04	-7.61E-03	1.543
			Cpx 7/16	Ca/Mg 7/16	0.438	1.49	8	2.465	1.22E-04	-7.28E-03	1.498
			Cpx 15/32	Ca/Mg 15/32	0.469	1.50	8	2.469	1.23E-04	-7.28E-03	1.503
			Cpx 17/32	Ca/Mg 17/32	0.531	1.51	8	2.467	1.27E-04	-7.59E-03	1.521
			Cpx 9/16	Ca/Mg 9/16	0.563	1.53	8	2.462	1.28E-04	-7.83E-03	1.535
			Cpx 5/8	Ca/Mg 5/8	0.625	1.56	8	2.453	1.26E-04	-8.01E-03	1.571
			Diopside	-	0.500	1.45	8	2.47	1.18E-04	-6.89E-03	1.458
	Antonelli et al., 2019c	PBE	Diopside	-	0.500	1.32	8	2.529	-	-	-
			Opx 1/16	Ca/Mg = 1/16	0.063	1.72	6	2.3649	Note: 7th Ca-O bond at 2.66Å
			Opx 1/16	Ca/Mg = 1/16, 2c	0.063	1.73	6	2.3649	Note: 7th Ca-O bond at 2.66Å
			Opx 1/32	Ca/Mg = 1/32, 2c	0.031	1.84	6	2.3445	Note: 7th Ca-O bond at 2.67Å
			Opx 1/64	Ca/Mg = 1/64, 2c	0.016	1.85	6	2.343	Note: 7th Ca-O bond at 2.68Å
			Forsterite 1/8	Ca/Mg = 1/8, M2	0.125	1.82	6	2.3314	-	-	-
			Forsterite 1/16	Ca/Mg = 1/16, 2a	0.063	2.10	6	2.2975	-	-	-
			Forsterite 1/32	Ca/Mg=1/32, 2ac	0.031	2.08	6	2.3010	-	-	-
			Forsterite 1/64	Ca/Mg=1/64, 4a2c	0.016	2.11	6	2.2977	-	-	-
			Grossular	-	1.000	1.74	8	2.4288	-	-	-
			Pyrope 1/12	Ca/Mg = 1/12	0.083	2.00	8	2.3797	-	-	-
			Pyrope 1/24	Ca/Mg = 1/24 fcc	0.042	2.03	8	2.3784	-	-	-
			Almandine 1/12	Ca/Fe = 1/12 fm	0.083	1.85	8	2.3876	-	-	-
			Spessartine 1/12	Ca/Mn = 1/12 fm	0.083	1.83	8	2.3999	-	-	-
			Anorthite	-	1.000	1.06	6	2.4608	Note: 7th Ca-O bond at ~2.9-3.0Å
			Labradorite	An50-Ab50 v1	0.500	0.98	6	2.4871	Note: 7th Ca-O bond at 2.97-3.00Å for
										all Ca sites
			Labradorite	An50-Ab50 v2	0.500	1.03	6	2.4740	Note: 7th Ca-O bond at 2.84Å for half
										of Ca sites
			Labradorite	An50-Ab50 v3	0.500	1.01	6	2.4723	Note: 7th Ca-O bond at 2.92Å for half
										of Ca sites
			Apatite	Fluorapatite	1.000	1.21	6	2.4205	Note: Ca1, *7-10th Ca-O neighbors at

  presented simplified reactive transport models for Ca and Sr in mid-ocean ridge hydrothermal systems that suggest 87 Sr/ 86 Sr in hydrothermal fluids can vary as a function of paleoseawater composition [as predicted in Antonelli et al. (2017)], but δ 44 Ca values in hydrothermal fluids are likely to be the same as MORB (or at the most ~0.1‰ displaced toward seawater) in the geologic past. In contrast, it is possible that serpentinite-hosted hydrothermal systems are shifted from local rock

  Stable mass-dependent Ca isotope signatures that correlate with other geochemical parameters (Sr/Nb and 87 Sr/ 86 Sr, shown in Fig.21) have been interpreted to represent recycling of ancient Ca bearing surface materials in Hawaiian lavas (e.g.[START_REF] Huang | Stable calcium isotopic compositions of Hawaiian shield lavas: Evidence for recycling of ancient marine carbonates into the mantle[END_REF]. Although the interpretation of the trends presented by[START_REF] Huang | Stable calcium isotopic compositions of Hawaiian shield lavas: Evidence for recycling of ancient marine carbonates into the mantle[END_REF] is currently under debate due to the low δ 44 Ca and Stable Ca isotope variations versus traditional proxies for carbonate incorporation in Hawaiian shield lavas after[START_REF] Huang | Stable calcium isotopic compositions of Hawaiian shield lavas: Evidence for recycling of ancient marine carbonates into the mantle[END_REF]. (a) δ 44 Ca vs. Sr/Nb, (b) δ 44 Ca vs. 87 Sr/ 86 Sr. In this figure, Ca isotope ratios are

	reported relative to SRM915a (-0.95‰, relative to BSE), and the carbonate end-member component is assumed to
	have 800 ppm [Sr], 0 ppm [Nb], 87 Sr/ 86 Sr = 0.7034, [CaO] = 56 wt%, and δ 44 CaSRM915a = 0.2‰ (-0.75‰ relative to
	BSE).

87 Sr/ 86 Sr values required for the carbonates, it has been reported that that the global ocean (and thus marine carbonates) may have gradually evolved through time towards higher δ 44 Ca values (e.g.

[START_REF] Farkaš | Evolution of the oceanic calcium cycle during the late Mesozoic: Evidence from δ44/40Ca of marine skeletal carbonates[END_REF]

. Fig. 21a,b.

  To date, igneous and metamorphic whole-rock analyses have δ 44 Ca ranges of ~2‰ and ~6‰, respectively, indicative of open-system processes such as fractional crystallization, partial-melt loss, metamorphism/metasomatism, and diffusive Ca transport. Despite this observation, nonmetasomatized peridotites have little δ 44 Ca variability and define a bulk-silicate Earth (BSE) value of +0.95‰ relative to standard SRM915a, while mid-ocean ridge basalts and average continental crust have slightly lower δ 44 Ca (by ~0.1‰ and ~0.2‰, respectively). Basaltic standards analyzed by TIMS generally give values closer to BSE, but this may be due to a +0.1‰ difference in δ 44 Ca compared to the same standards analyzed by ICP-MS (where 40 Ca is not directly measured). This suggests that there may be slight excess 40 Ca in SRM915a (εCa = +1 relative to basaltic igneous rocks), and that BSE has lower 40 Ca/ 44Ca [~47.156 vs. ~47.162

evaluated. When a new phase is created at the expense of an existing phase ('net-transfer reaction'), for example, the Ca isotopic composition of the precursor mineral(s) may be (i) at equilibrium with the product phase (if the forward and backward exchange rates are much larger than the net reaction rate), (ii) directly passed-on to the product phase (which would then require subsequent isotopic re-equilibration with surrounding minerals), or (iii) subjected to kinetic fractionation effects due to isotopic rate differences during bond breaking, diffusive transport (within minerals or along grain-boundaries), and/or bond formation at the new mineral surface (thereby inheriting an isotopic composition different from the precursor). During exchange reactions, which are often used for geothermometry, pre-existing phases exchange components in order to reach chemical equilibrium (e.g. Fe-Mg exchange between ferromagnesian minerals and Ca exchange between clinopyroxene and orthopyroxene). Given that the exchange of components is diffusive in nature, kinetic Ca isotope effects could occur via both volume and grain-boundary diffusion. Upon completion of the chemical exchange, the new mineral may have to isotopically re-equilibrate with surrounding phases in the absence of a chemical potential gradient ('isotopic self-diffusion'), which may be exceedingly slow. Future studies of spatially constrained δ 44 Ca variations across minerals and grain boundary interfaces should allow us to evaluate the possible mechanisms for inter-mineral Ca isotope disequilibrium in metamorphic samples.

Crystal growth rates from Ca isotope disequilibrium in volcanic systems

Crystal growth rates in magmatic systems are especially important for understanding the physical processes leading up to volcanic eruptions. Current methods, such as crystal-size distribution (CSD) analyses or trace-element diffusion profile models [see [START_REF] Cooper | Time scales and temperatures of crystal storage in magma reservoirs: implications for magma reservoir dynamics[END_REF]], however, must generally assume crystal-growth rates a priori in order to arrive at timescale estimates for the explanation is that water leads to increases in Ca diffusivity and decreases in R/D (Antonelli et al., 2019b). Recent work on Li isotope diffusion suggests that water concentrations do not have a significant effect on isotopic diffusivity ratios in silicate liquids [START_REF] Holycross | Diffusive fractionation of Li isotopes in wet, silicic melts[END_REF], but this still remains to be confirmed for Ca isotopes. In natural phenocryst samples, clinopyroxene and other mafic minerals (amphibole, olivine) have slight positive Δ 44 Camineral-matrix (up to +0.3‰) and plagioclase phenocrysts have slightly negative Δ 44 Camineral-matrix (~-0.2‰). δ 44 Ca variations of ~0.5‰ in high-purity plagioclase separates taken from orbicules and comb-layers, however, suggest especially fast growth rates. Assuming that equilibrium Δ 44 Caplag-melt = 0 [in agreement with [START_REF] Zhang | No measurable calcium isotopic fractionation during crystallization of Kilauea Iki lava lake[END_REF]], and using conservative estimates for the other various parameters in Equation 4[see supplementary text of Antonelli et al. (2019b)], the authors were then able to constrain minimum crystal growth rates in the volcanic systems (Fig. 18).

Although the estimated growth rates for orbicules, comb-layers, and several natural phenocryst samples are faster than commonly assumed [typically assumed to be ~0.3 cm/yr in decompressing magmas (e.g. [START_REF] Brugger | Crystal size distribution analysis of plagioclase in experimentally decompressed hydrous rhyodacite magma[END_REF][START_REF] Cooper | Time scales and temperatures of crystal storage in magma reservoirs: implications for magma reservoir dynamics[END_REF][START_REF] Cooper | Rapid remobilization of magmatic crystals kept in cold storage[END_REF][START_REF] Salisbury | Multiple plagioclase crystal populations identified by crystal size distribution and in situ chemical data: Implications for timescales of magma chamber processes associated with the 1915 eruption of Lassen Peak[END_REF]], they agree with previous estimates for orbicule growth [~10 cm/yr, based on textural constraints from superheated experimental magmas [START_REF] Mccarthy | Comb layering monitors decompressing and fractionating hydrous mafic magmas in subvolcanic plumbing systems (Fisher Lake, Sierra Nevada, USA)[END_REF][START_REF] Shea | Kinetics of cooling-and decompression-induced crystallization in hydrous mafic-intermediate magmas[END_REF])], and rates as high as ~30 cm/yr have been described for crystallization of experimental plagioclase spherulites [START_REF] Arzilli | Near-liquidus growth of feldspar spherulites in trachytic melts: 3D morphologies and implications in crystallization mechanisms[END_REF]. of continental crust and lithospheric mantle, 'cratonization'). Given that 40 K was the largest source of radioactive heat on the early Earth, loss of potassium from the lower crust is especially important for cratonization, however, other than by comparison with similar systems (e.g. Rb/Sr, [START_REF] Rudnick | Large ion lithophile elements in rocks from highpressure granulite facies terrains[END_REF] and direct measurements [in rare cases when the unmetamorphosed protoliths are still available (e.g. [START_REF] Ewing | Hafnium isotopes and Zr/Hf of rutile and zircon from lower crustal metapelites (Ivrea-Verbano Zone, Italy): Implications for chemical differentiation of the crust[END_REF][START_REF] Stepanov | Geochemistry of ultrahigh-pressure anatexis: Fractionation of elements in the Kokchetav gneisses during melting at diamond-facies conditions[END_REF]], there have been no direct proxies for K-loss from these systems.

To address this issue, a recent study used radiogenic 40 Ca measurements (εCa) in order to directly constrain K-loss during high-temperature granulite-facies metamorphism of the lower crust (Antonelli et al., 2019a). Analyzing 40 Ca/ 44 Ca in garnet separates from granulite-facies rocks, the authors found large enrichments in εCa (up to +42) correlating inversely with garnet grossular percentage and positively with whole-rock peraluminosity index ('A/CNK', defined as molar Al2O3/CaO + Na2O + K2O, Fig. 23a,b). The large radiogenic 40 Ca enrichments must have occurred through decay of 40 K in protolith rocks, prior to garnet formation, as garnet is very effective at excluding K during crystallization.

(iii) detailed petrographic observations (including textural and paragenetic relationships). A greater number of experimental constraints, including inter-mineral equilibration (to verify fractionation factors), crystallization, melting, hydrothermal mineral precipitation, and diffusioncouple experiments would also be greatly beneficial. Finally, technical advancements such as (i) improved sampling (e.g. micro-drilling across mineral-zones or separation of specific crystal populations) and small sample analysis methods (e.g. TIMS analyses using next generation highresistance amplifiers) and (ii) the development of in-situ Ca isotope analysis techniques (e.g. by SIMS or LA-MC-ICP-MS), would greatly enhance our understanding of the various processes leading to Ca isotope equilibrium and disequilibrium in high-temperature samples. Next generation CC-MC-ICP-MS instruments with low energy collision-cell technology should also allow 40 Ca, 42 Ca, 44 Ca, and 48 Ca to be measured without isobaric interferences from Ar-based polyatomic species [START_REF] Simon | Potential of MC-ICP-CC-MS in cosmochemistry and geochronology J[END_REF][START_REF] Zhao | Calcium isotope determination with the Nu Sapphire Dual-Path MC-ICP-MS[END_REF]), which will be especially useful for determining Ca isotope variations (including εCa, δ 44 Ca, and Δ 48 Ca') in future studies.

Acknowledgments

M.A.A. would like to thank E.M. Nordquist and S.T. Brown for their help with data compilation.

We also thank S. Huang and two anonymous reviewers for their beneficial comments and suggestions. This contribution was partially supported by an ETH Zurich post-doctoral fellowship (19-2 FEL-33) to M.A.A. and by NASA grants (11-COS11-0066, 16-EW16_2-0163, and the NASA Planetary Science Division) to J.I.S.