Major element composition of an Early Enriched Reservoir: constraints from 142Nd/144Nd isotope systematics in the early Earth and high-pressure melting experiments of a primitive peridotite

The Accessible Silicate Earth (ASE) has a higher 142Nd/144Nd ratio than most chondrites. Thus, if the Earth is assumed to have formed from these chondrites, a complement low-142Nd/144Nd reservoir is needed. Such a low-142Nd/144Nd reservoir is believed to have been derived from a melt in the early Earth and is called the Early Enriched Reservoir (EER). Although the major element composition of the EER is crucial for estimating its chemical and physical properties (e.g., density) and is also essential for understanding the origin and fate of the EER, which are both major factors that determine the present composition of the Earth, it has not yet been robustly established. In order to determine the major element composition of the EER, we estimated the age and pressure–temperature conditions to form the EER that would best explain its Nd isotopic characteristics, based on Sm–Nd partitioning and its dependence on pressure, temperature, and melting phase relations. Our estimate indicates that the EER formed within 33.5 Myr of Solar System formation and at near-solidus temperatures and shallow upper-mantle pressures. We then performed high-pressure melting experiments on primitive peridotite to determine the major element composition of the EER at estimated temperature at 7 GPa and calculated the density of the EER. The result of our experiments indicates that the near-solidus melt is iron-rich komatiite. The estimated density of the near-solidus melt is lower than that of the primitive peridotite, suggesting that the EER melt would have ascended in the mantle to form an early crust. Given that high mantle potential temperatures are assumed to have existed in the Hadean, it follows that the EER melt was generated at high pressure and, therefore, its composition would have been picritic to komatiitic. As the formation age of the EER estimated in our study precedes the last giant, lunar-forming impact, the picritic to komatiitic crust (EER) would most likely have been ejected from the Earth by the last giant impact or preceding impacts. Thus, the EER has been lost, leaving the Earth more depleted than its original composition.


Introduction
Recent 142 Nd analyses of terrestrial rocks and chondrites have revealed that terrestrial samples exhibit a nearly uniform 142 Nd/ 144 Nd ratio that is higher than most chondrites (Boyet and Carlson 2005;Jackson and Carlson 2012). Given that 142 Nd is generated through radioactive decay of 146 Sm, the high 142 Nd/ 144 Nd ratio of the terrestrial samples is attributed to either a high 146 Sm/ 144 Nd or initial 142 Nd/ 144 Nd ratio. A high 146 Sm/ 144 Nd ratio would require early silicate differentiation due to the short halflife of 146 Sm (68 Myr; Kinoshita et al. 2012), given the lithophile behavior of Sm and Nd, with Sm being less incompatible than Nd. As such, investigations of 142 Nd anomalies may be key to understanding planetary formation and mantle evolution in the early Earth.
Three major hypotheses have been proposed to explain the higher 142 Nd/ 144 Nd ratios of terrestrial rocks compared with most chondrites. The first hypothesis is that a low-Sm/Nd reservoir was formed in the early Earth, which was then either hidden and isolated from mantle convection or lost from the Earth. If we assume that the Earth formed with initial 142 Nd/ 144 Nd and Sm/ Nd ratios similar to most chondrites, then a low-142 Nd/ 144 Nd reservoir would be necessary to compensate for the high 142 Nd/ 144 Nd ratio of terrestrial samples. The low-142 Nd/ 144 Nd reservoir must have had a low 146 Sm/ 144 Nd ratio at its time of formation and is thought to have been formed as a silicate melt in the early Earth (Boyet and Carlson 2005). In this context, the low-146 Sm/ 144 Nd reservoir is called the Early Enriched Reservoir (EER; Boyet and Carlson 2005), and it appears to be missing from the present-day Earth. In contrast, the residue produced during the EER formation is called the Early Depleted Reservoir (EDR; Boyet and Carlson 2005). Given that the EER is key to understanding silicate differentiation in the early Earth, the composition of the EER and the mechanisms, timing, and place of its formation have been extensively investigated. Boyet and Carlson (2005) and Korenaga (2009a) estimated the trace element composition of the EER based on mass-balance calculations of chondrites, EDR, and continental crust and concluded that the EER would have been generated at upper-mantle conditions. In particular, Boyet and Carlson (2005) argued that the EER would have had a similar composition to lunar KREEP rocks, which are enriched in potassium (K), rare earth elements (REE), and phosphorus (P) and are considered to have formed in the shallow mantle beneath the lunar anorthositic crust. However, the temperature and pressure conditions to form the EER have not been properly constrained. Labrosse et al. (2007) proposed that the existence of a basal magma ocean, which could be the "missing reservoir", can explain the difference in the 142 Nd/ 144 Nd ratios between chondrites and ASE. They estimated the trace element composition of the basal magma ocean using partition coefficients between Mg-perovskite and melt, which were experimentally determined at the uppermost lower-mantle pressures (Corgne et al. 2005). Lee et al. (2007) investigated the trace element systematics in natural Fe-O-S magmas and proposed that a similar Fe-O-S melt, which can account for the difference between the 142 Nd/ 144 Nd ratios of chondrites and the ASE, had formed during the final stages of magma ocean solidification. However, these previous studies mainly focused on the trace element composition of the EER, and the pressure and temperature conditions required to form the EER and its major element composition remain unconstrained.
The second hypothesis is that the Earth was formed from high-Sm/Nd materials, together with the Moon and Mars. Based on the 142 Nd/ 144 Nd and Sm/Nd ratios of lunar and Martian samples, Caro et al. (2008) argued that bulk Moon and Mars have high Sm/Nd ratios almost identical to the ASE and proposed that the Earth, Moon, and Mars were all formed from similarly high-Sm/Nd materials that had experienced partial melting and impact-erosion of crust during collisions with planetesimals. In this model, there is no need to develop models for the formation of the EER or the missing low-Sm/Nd reservoir through solidification of a terrestrial magma ocean (Caro 2011). However, the isotope ratios of lunar and Martian rocks observed on the surface of these bodies need to be corrected for the effect of exposure to cosmic rays, which is not straightforward. As such, it has even been argued that the lunar and Martian rocks have chondritic Sm/Nd ratios (Sprung et al. 2013). At present, the bulk compositions of the Moon and Mars remain imprecisely determined. Moreover, the fractionation between Sm and Nd associated with silicate differentiation during planetesimal collisions is also an unknown factor (Weiss and Elkins-Tanton 2013).
The third hypothesis is that the Earth accreted from material with different nucleosynthetic Sm or Nd than that of chondrites. Previous studies have revealed that ordinary chondrites (OC) and carbonaceous chondrites (CC) have average 142 Nd/ 144 Nd ratios that are 20 and 40 ppm lower than the ASE, respectively, and that enstatite chondrites (EC) have an average 142 Nd/ 144 Nd ratio that is ca. 10 ppm lower than the ASE, with some EC having 142 Nd/ 144 Nd ratios that overlap the terrestrial standard (Boyet and Gannoun 2013). These variations in 142 Nd/ 144 Nd ratios can be generated by p-, s-, and rprocesses and have been interpreted to be the result of imperfect homogenization during the formation of the solar nebula (Qin and Carlson 2016). However, even after corrections for the effects of p-, s-, and r-processes, offsets can still be found among the 142 Nd/ 144 Nd ratios in chondrites and that of the terrestrial standard (Qin and Carlson 2016). For this model involving the heterogeneous isotopic compositions of chondrites, the material that formed the Earth needs to be identified. In order to determine the Earth's accretionary material, previous studies have investigated differences in various stable isotope systems such as O and Si (Drake and Righter 2002;Fitoussi and Bourdon 2012). Assuming that changes in stable isotope ratios are negligible during core-mantle and mantle-crust differentiation, the stable isotope ratios of the ASE should represent those of the source material. Therefore, a chondrite that has stable isotope ratios identical to the ASE might have been the source of accretionary material for the Earth. Unfortunately, no such material has yet been identified.
Each of the above three hypotheses has both strengths and weaknesses, and it is currently unclear as to which is most plausible. In this study, we focus on the missing reservoir hypothesis, which we investigate by combining two novel approaches. The first is to constrain the melt fraction and pressure-temperature conditions required to form the EER from Sm-Nd systematics, experimentally determined phase relations of peridotite melting, and precise mineralmelt partition coefficient data. The second is to determine the major element composition of the EER by performing high-pressure melting experiments on primitive peridotites at the estimated pressure-temperature conditions. The major element composition of the EER allows us to estimate the physical and chemical characteristics of the EER, such as density, viscosity, and contents of heat-producing elements (e.g., K). One of the aims of this study is to determine the density of the EER, since this is essential for determining whether the EER would have been subducted and become hidden in the mantle or would have ascended to the surface and then lost from the Earth. Using these novel approaches, we construct a plausible model of the formation of the EER and explain the difference in 142 Nd/ 144 Nd ratios between the ASE and chondrites, investigate the silicate differentiation of the early Earth, and constrain the composition of the bulk silicate Earth (BSE).

Conditions necessary to form the EER
Here, we estimate the conditions required to form the EER assuming that the difference between the presentday 142 Nd/ 144 Nd ratios of chondrites and the ASE reflects variable Sm/Nd ratios generated during the formation of the EER in the early Earth. If the EER originated from a melt, then the difference in Sm/Nd ratios between chondrites and the ASE should reflect the melting conditions under which the EER formed, such as the degree of melting, mineral assemblage and proportion of the residual solid, and partition coefficients. Given that these parameters are dependent on pressure and temperature, we can then estimate the pressure and temperature conditions under which the EER formed.
We assumed a simple box model involving massbalance calculations where (1) the whole region corresponding to fraction X of the mantle with a chondritic, primitive mantle composition (Chondritic Uniform Reservoir (CHUR)) experienced a single partial melting event with melt fraction F and differentiated into a melt (EER) and residual solid (EDR) in the early Earth (Fig. 1); (2) EER remained isolated until the present; (3) and ASE formed as a mixture of the EDR and an intact region in the mantle. Based on these assumptions, the difference between the Sm/Nd ratios of the EDR and CHUR during formation of the EER can be traced back from the present-day 142 Nd/ 144 Nd ratios of ASE and CHUR ( Fig. 1) where t f and t p are the age of the EER formation and time since the formation of the Solar System, respectively. Then, the difference between the Sm/Nd ratios of EDR and CHUR can be expressed using the relationship among solid phase proportions, melt fraction, and mineral-melt partition coefficients as follows: The right-hand side of Eq.
(2) represents partitioning of Sm and Nd between the source rock and the residual solid (Shaw 1970). D i is the bulk partition coefficient of element i obtained by assuming that the proportion of minerals does not change during partial melting. In reality, the mineral proportions do change as partial melting progresses, and Π i is the partition coefficient of element i obtained from the proportion of minerals entering the melt (Shaw 1970). F is the melt fraction. Given that the left-hand side of Eq. (2) is dependent on Eq. (1), we can obtain the melt fraction as a function of t f . We used the 142 Nd/ 144 Nd ratios of the present-day terrestrial standard and the average value of ordinary chondrites (Boyet and Carlson 2005) in the calculations. The melt fraction was obtained as a function of t f for two cases where the source region corresponds to the upper mantle (X = 0.3) and whole mantle (X = 1), respectively.
To determine Π i and D i , we must first obtain the pressure conditions for the EER formation, as melting phase relations and mineral-melt partition coefficients are dependent on pressure. In this study, we assumed that the EER formed at shallow, upper-mantle pressures because previously estimated trace element abundances of the EER resemble those of a melt generated in the upper mantle (Korenaga 2009a) and, in particular, the upper few hundred kilometers where clinopyroxene is stable (Boyet and Carlson 2005). We then performed our model calculations at three pressure conditions of 1, 3, and 7 GPa. In our model calculations, we need precise partition coefficients and quantitative melting phase relations; i.e., quantitative values for the proportions of solid phases that change as partial melting progresses. While precise partition coefficients have been determined experimentally up to 20 GPa (Nielsen et al. 1992;Salters and Longhi 1999;Frei et al. 2009;Suzuki et al. 2012), data for the quantitative melting phase relations are presently only available at 1, 3, and 7 GPa (Baker and Stolper 1994;Walter 1998), and thus, we have limited our model calculations to these pressures. It is not our intent to preclude the possibility that EER formation occurred at pressures >7 GPa, but this can only be explored when more precise experimental data on quantitative melting phase relations become available for pressures >7 GPa. The partition coefficients for our model calculations are listed in Table 1. The melting phase relations for each pressure were calculated from changes in mass proportions of phases between two temperatures following the method of Walter et al. (1995) (Appendix 1).
The calculated melt fraction is expressed as a function of the timing of the formation of the EER for each pressure condition (Fig. 2). A delay in the formation of the EER would necessitate a larger melt fraction. Therefore, it is essential to constrain the The 142 Nd/ 144 Nd evolution considered in this study. A source region differentiates into the EER and EDR. The EER is hidden within or has been lost from the Earth. The EDR and intact region were mixed to form the ASE. (Bottom) A box model of the EER formation, where X is the fraction of the source region relative to the whole mantle. ASE Accessible Silicate Earth (part of the silicate Earth from which we can obtain materials for analysis), EER Early Enriched Reservoir (low-Sm/Nd reservoir which can compensate for the 142 Nd/ 144 Nd difference between ASE and chondrites), EDR Early Depleted Reservoir (residual solid produced by the EER formation), CHUR Chondritic Uniform Reservoir (putative primitive mantle having a chondritic composition) timing of the EER formation in order to constrain the melt fraction of the EER. We calculated the timing of the EER formation using the method of Boyet and Carlson (2005 Boyet and Carlson 2005;20 Myr in Caro 2011).
We then obtained the maximum melt fraction from our method, for each pressure condition, and for the two different source fractions (Fig. 2). In the case where the source region is limited to the upper-mantle region (X = 0.3), the maximum melt fractions are 2.7 % at 1 GPa, 2.6 % at 3 GPa, and 1.0 % at 7 GPa. In the case where the source is the whole mantle (X = 1), the maximum melt fractions are 0.7 % at 1 GPa, 0.5 % at 3 GPa, and 0.2 % at 7 GPa, respectively. In these results, the maximum melt fractions become smaller at higher pressures. At a higher pressure, an increase in the proportion of garnet increases the D Sm /D Nd ratios of the whole rocks. Therefore, in order to achieve the ( 142 Nd/ 144 Nd) EDR /( 142 Nd/ 144 Nd) CHUR ratio, a smaller (Sm/Nd) EDR /(Sm/Nd) CHUR is needed for higher pressures (larger D Sm /D Nd ). In Eq. (2), (Sm/Nd) EDR /(Sm/Nd) CHUR decreases with decreasing melt fraction. Therefore, the maximum melt fraction becomes smaller with increasing pressure, due to a larger proportion of garnet in the mantle peridotite. Based on the calculations presented in this section, we conclude that the EER should have been generated at very small degrees of partial melting, irrespective of whether the source was the upper or whole mantle. We discuss later the conditions that might enable such small degrees of Hadean melting.

Experimental determination of near-solidus melt compositions
We calculated the melt fractions to form the EER at 1, 3, and 7 GPa because the melting phase relations of a peridotite having primitive-mantle-like compositions have been determined precisely at these pressures by previous experimental studies (Baker and Stolper 1994 for 1 GPa; Walter 1998 for 3 and 7 GPa). Our results show that formation of the EER requires a small degree of melting near the solidus at these pressures. In this section, we determine the major element compositions of near-solidus melts formed at 1, 3, and 7 GPa based on high-pressure melting experiments of primitive peridotite. The near-solidus melt composition of a primitive peridotite at 1 GPa has been determined by Falloon et al. (2008) with the sandwich method (Takahashi and Kushiro 1983) using a fertile peridotite composition (MM3). The near-solidus melt composition at 3 GPa has been determined by Davis et al. (2011) following the Modified Iterative Sandwich Experiment method (MISE; Hirschmann and Dasgupta 2007). Herein, we determined the major element composition of a near-solidus melt of primitive peridotite at 7 GPa with the MISE method.

High-pressure experiment design
In order to determine the near-solidus melt composition at 7 GPa, we employed the MISE method proposed by Hirschmann and Dasgupta (2007). The steps involved in the MISE method are as follows: (1)Equilibrating a putative near-solidus melt with peridotite at solidus temperature (2)Determining a mineral-melt partition coefficient for each element based on the compositions of the resulting quenched melt and residual minerals (3)Calculating a new putative near-solidus melt composition using a set of partition coefficients for each element in each mineral, the modal mineral assemblage of the peridotite, and the composition of the starting peridotite (4)Synthesizing a new putative near-solidus melt Steps (1)-(4) are performed repeatedly until a nearsolidus melt is produced (for more details, see Hirschmann and Dasgupta 2007). The method for assessing whether a near-solidus melt composition has been achieved will be described later.

Experimental procedure Starting materials
We used KLB-1 as the starting material for the peridotite component, as KLB-1 peridotite is considered to represent a fertile peridotite in the present-day mantle (Takahashi 1986), and its major element composition resembles that of the estimated primitive mantle (Ringwood 1966;McDonough and Sun 1995). The first putative near-solidus melt composition was calculated from an experimentally determined relationship between melt compositions and melt fractions of fertile peridotite at 7 GPa in Walter (1998). Powders of KLB-1 and putative near-solidus melt compositions were synthesized from reagent grade oxides and carbonates. The compositions of the synthesized starting powders are shown in Table 2. Powders of SiO 2 , TiO 2 , Al 2 O 3 , Cr 2 O 3 , Fe 2 O 3 , CaCO 3 , Na 2 CO 3 , K 2 CO 3 , and Ca 3 (PO 4 ) 2 were dried at 110°C under atmospheric pressure for 24 h. MgO was dried at 1000°C for at least 12 h, and MnO was kept in a vacuum at room temperature to prevent oxidation to MnO 2 . These powders of oxides and carbonates were weighed with an electronic balance, and the powders, apart from Fe 2 O 3 , were mixed in an agate mortar with ethanol for 1 to 2 h. The mixture was placed in a Pt vessel and decarbonated in air by raising the temperature at a rate of 100°C/h to 1100°C and then held at that temperature for 24 h. After decarbonation, Fe 2 O 3 was added and mixed in with an agate mortar under dry conditions. Pellets of the starting powders were then made and heated at the oxygen fugacity of iron-wüstite in a small Pt basket in a gas-mixing furnace at the Graduate School of Science, Kyoto University. Several pellets were stacked in a Pt basket, and the lowermost pellet in contact with the Pt was discarded to eliminate the effect of Fe-loss to the Pt. The pellets of the KLB-1 composition were heated at 1100°C for 12 h, and pellets of the putative near-solidus melt composition were heated at 1000°C for 12 h. The pellets were then first milled in a tungsten carbide mortar and subsequently ground in an agate mortar for 1 h. The starting powders were stored at 80°C under vacuum to inhibit adsorption of atmospheric water. The H 2 O contents in the starting materials were analyzed using a Karl Fischer Moisture Titrator MKC-610 at the Earthquake Research Institute, University of Tokyo (Table 2). From these analyses, it was confirmed that the starting materials contain negligible amounts of water.

High-pressure experiments
High-pressure experiments were performed at 7 GPa using 1000 and 5000 T Kawai-type multi-anvil apparatus at the Institute for Study of the Earth's Interior, Okayama University. Pressure was generated using eight tungsten carbide cubes with side lengths of 32 mm and truncated edge lengths of 8 mm and semi-sintered MgO (+5 wt% Cr 2 O 3 ) octahedral pressure medium with edge lengths of 14 mm. The relationship between load and sample pressure was calibrated at 1600°C using the following phase transformations: quartz-coesite at 3.7 GPa (Kanzaki 1990); coesite-stishovite at 10.26 GPa (Zhang et al. 1996); Mg 2 SiO 4 (α-β) at 15.2 GPa (Katsura et al. 2004). The starting material was packed in a Re capsule and then sealed in a Pt capsule. The sample capsule was placed in a MgO sleeve to insulate it from the straight LaCrO 3 heater. The junction of the WRe 3 -WRe 25 thermocouple and the sample capsule were positioned symmetrically relative to the center of the heater. The run durations at target temperature were 0.3-4.5 h. After quenching, the capsule was mounted in epoxy resin and polished with SiC and diamond powder. The polished sample was coated with carbon, and its chemical composition analyzed with an electron microprobe. We measured the temperature distribution in the capsule using two-pyroxene thermometry and confirmed that the temperature gradient was negligible (Appendix 2). The difference between temperatures from the thermocouple and two-pyroxene thermometry is within the range of the uncertainty of the two-pyroxene thermometer (~30°C; Gasparik 1989).

Analytical procedures
Textural observations were made using a Hitachi S3500H scanning electron microscope, and quantitative analysis of major elements in minerals and melts was performed using a JEOL JXA-8100 electron microprobe equipped with five wavelength-dispersive spectrometers (WDS) at the Graduate School of Science, Kyoto University. Analytical conditions were based on the method of Kawakami and Hokada (2010). For the WDS quantitative analysis, an accelerating voltage of 15 kV and a ZAF matrix correction routine were used. The beam size was 3 μm for minerals and 10-15 μm for the melt in order to avoid heterogeneity effects. The beam current was 20 nA for the minerals and 10 nA for the melt. The mineral analyses used counting durations for peaks and backgrounds of 30 and 15 s for TiO 2 , K 2 O, and P 2 O 5 and 10 and 5 s for other elements, respectively. The melt analyses used counting durations for peaks and backgrounds of 30 and 15 s for P 2 O 5 and 10 and 5 s for other elements, respectively. Standard deviations were calculated from the analysis of >15 grains of each residual mineral and from >5 melt analyses. The phase proportions were obtained from mass-balance calculations using the composition of the coexisting phases, bulk composition, and least-squares method. Two conditions were utilized in the calculations, which involved including (or not) FeO, and the result with the smaller sum of squared residuals in the calculations was chosen. In the cases where FeO was included, Fe-loss was considered to be negligible.

Determination of solidus temperature and mineral compositions
Experiments to determine the solidus temperature and mineral compositions of KLB-1 at 7 GPa were performed at 1700, 1750, and 1800°C. There was no melt in the quenched sample at 1700°C, and the texture was homogeneous throughout the entire capsule. A large melt pool was present at 1800°C. At 1750°C, a small melt pool formed on the thermocouple side of the capsule (i.e., the hot side of the capsule). In the middle of the capsule, which was positioned symmetrically to the thermocouple junction with respect to the center of the heater, a small amount of melt was observed between mineral boundaries. From these results, we concluded that the solidus temperature is just below 1750°C. At this temperature, olivine, garnet, and clinopyroxene (cpx) were present, but orthopyroxene (opx) was not observed. This observation is consistent with the result of Walter (1998), which reported that opx is unstable near the solidus at 7 GPa at primitive mantle compositions. The chemical compositions of the solidus minerals were determined from the run at 1700°C because even the presence of a small volume of melt at 1750°C would likely affect the composition of the coexisting minerals. The compositions of the solidus minerals are summarized in Table 3. The Mg# (100 × Mg/(Mg + Fe)) of the olivine, cpx, and garnet are 89.5, 89.6, and 86.0, respectively. The calculated mineral proportions are olivine:cpx:garnet = 63.3:22.8:13.9 (in wt%). No Fe-loss was evident in the run at 1700°C.

Approach to the solidus melt composition
The MISE experiments were performed at 7 GPa and 1750°C for both the primitive peridotite and putative near-solidus melt composition. In these experiments, we succeeded in obtaining melt and solidus mineral phases of olivine, cpx, and garnet. The experimental results are summarized in Table 4, and the analyzed compositions of the quenched melts and residual minerals are summarized in Table 5. There was no Fe-loss in the MISE iterations (Table 4). The Fe-Mg exchange partition coefficients between residual olivine and quenched melt were obtained and compared with those calculated from the ratio of non-bridging oxygens to tetrahedral cations (NBO/T) of the melts using the method of Mibe et al. (2006). The Fe-Mg exchange coefficients in the MISE iterations agree within uncertainties with an equilibrium exchange partition coefficient of 0.33. In order to assess whether or not the composition of a quenched melt after an iteration step has sufficiently approached the composition of the true near-solidus melt, Dasgupta and Hirschmann (2007) and Davis et al. (2011) monitored the convergence of the residual mineral compositions to the solidus mineral compositions and the convergence of the mineral-melt partition coefficients and compositions of all phases to a steady state. In addition, Davis et al. (2011) monitored the difference between the analyzed and calculated melt compositions at each iteration step in order to identify equilibrium. However, using the MISE method, we consider that the conditions for generating the near-solidus melt composition can be obtained if there is (1) coincidence of the residual mineral compositions with the solidus mineral compositions and (2) convergence occurs between the analyzed and calculated melt compositions for the next step. Thus, satisfaction of both criteria 1 and 2 at step n is a sufficient condition to obtain the true near-solidus melt composition after the nth iteration. Figure 3 shows the analyzed melt compositions and calculated melt composition for the next step, and Fig. 4 shows the convergence of the chemical compositions of the residual minerals. The difference between the analyzed and calculated melt compositions for the next step diminished at the 5th iteration, and the chemical compositions of residual minerals reached those of the solidus minerals within error at the 5th iteration. From these results, it was concluded that the two criteria were satisfied at the 5th step, so that the calculated melt composition after the 5th iteration is taken to be the near-solidus melt composition.

Major element composition of the near-solidus melt
The major element composition of the near-solidus melt at 7 GPa (this study) is shown together with the nearsolidus compositions experimentally obtained at 3 GPa (Davis et al. 2011) and 1 GPa (Falloon et al. 2008) in Table 6. The near-solidus melt compositions range from basaltic at 1 GPa to komatiitic at 7 GPa. In these solidus melts, both MgO and FeO contents increase with increasing pressure, and the Al 2 O 3 /TiO 2 ratio decreases with increasing pressure. The CaO contents in these near-solidus melts remain relatively constant. The alkali contents in these melts decrease with pressure. Given that the compositions of the near-solidus melts obtained in high-pressure melting experiments range from basaltic to komatiitic, we compared these compositions to those of natural basalts and komatiites (Fig. 5). The near-solidus melt at 3 GPa is picritic and has an intermediate Al 2 O 3 /TiO 2 ratio and FeO content, which falls between those for the near-solidus melts at 1 and 7 GPa.  The FeO content of the near-solidus melt at 3 GPa is about 10 wt%, which is not significantly different from ocean island basalts (OIB) from Isua, Greenland. At 7 GPa, the near-solidus melt has an Al 2 O 3 /TiO 2 ratio of about 2, which is much lower than that of Al-depleted komatiites, and has a higher FeO content than all of the komatiites. The near-solidus melts at 1 and 7 GPa have FeO contents lower than Archean MORB rocks and higher than Archean komatiites. The differences in FeO contents between the near-solidus melts obtained from high-pressure melting experiments and Archean rocks can be explained by their differences in degree of partial melting. Herzberg and O'Hara (2002) constructed forward models to constrain the potential primary melt compositions of mantle peridotites, in particular FeO and MgO contents, by parameterization of experimental data up to 7 GPa. According to their calculations, the FeO content in a melt decreases with decreasing melt fraction below 3 GPa, whereas it increases with decreasing melt fraction above 3 GPa. If it is assumed that the Archean MORB melts were generated at high melt fractions at low pressures below 3 GPa and that the Archean komatiites were generated at high melt fractions in hot plumes at high pressures above 3 GPa, then the differences in FeO contents observed between the experimental near-solidus melt and the Archean rocks are consistent with the results of Herzberg and O'Hara (2002). Figure 6 shows the K 2 O and P 2 O 5 contents of the solidus melt at 7 GPa as compared to those of lunar KREEP rocks (Ryder 1976;Irving 1977;Ryder et al. 1977), normal MORB (N-MORB), and enriched MORB (E-MORB) (Klein 2004 Table 6. The probable pressure of the EER formation in the Hadean Earth and the potential major element composition of the EER are discussed in a later section.

Estimation of the density of the near-solidus melt
In this section, we estimate the density of the nearsolidus melt and primitive peridotite to examine whether the EER would have sunk into the mantle or ascended to the surface. The method for estimating the density of the melt and mantle peridotite is from Matsukage et al. (2005a, b). The density of a melt can be constrained with the molar mass, molar fraction, and partial molar volume of melt component oxides (Bottinga and Weill 1970): where ρ is density, X i is mole fraction, V i is partial molar volume, and M i is the formula weight of component i. The values for V i should be calculated at the target pressure and temperature. Here, we only obtained the V i of SiO 2 , Al 2 O 3 , FeO, MgO, and CaO at the target pressure and temperature, since the other minor elements only have a negligible effect on density. In order to obtain the V i values of the five oxides, we need five or more independent relationships between V i and ρ. Therefore, we used five melts whose compositions range from basaltic to peridotitic and densities have been determined by high-pressure and high-temperature experiments in previous studies (Appendix 3). Given that these densities have been determined at different pressures and temperatures from our target pressure and temperature, we calculated the pressure and temperature dependence After pressure compensation, values of densities at the target pressure were obtained at different temperatures for three of the five melt compositions. We calculated the temperature dependence of density in these melts by averaging the results, based on the assumption that differences in melt composition are not significant. We then calculated the density at the target pressure and temperature for these five melts. Finally, we applied Eq. (4) to these five    Finally, the density of the near-solidus melt with compositions at the target pressure and temperature was calculated using these V i values. The density of the primitive peridotite was determined from the density of the minerals that are stable at the target pressure and temperature. We used thermoelastic parameters for forsterite, fayalite, pyrope, almandine, grossular, orthoenstatite, orthoferrosilite, diopside, and hedenbergite (Matsukage et al. 2005b). The solidus mineral assemblage and composition were taken from Davis et al. (2011) at 3 GPa and from our experimental result at 7 GPa. In the lherzolite melting experiments at 1 GPa in Falloon et al. (2008), spinel and plagioclase were present at sub-solidus conditions but were ignored in this calculation because of their minor abundance and negligible effect on density. Pressure can be calculated as the sum of static P st and thermal pressures ΔP th . The static and thermal pressures in our study were calculated with the third-order Birch-Mürnaghan and Mie-Grüneisen equations of state, respectively (Matsukage et al. 2005b). Figure 7 shows the results of the calculated densities. The densities of the near-solidus melts are lower than the densities of the primitive peridotite. Therefore, an EER formed as a near-solidus melt should have ascended to the surface. The detail of this model is discussed in the next section.

A scenario for the formation of the EER in the Hadean
From the difference in 142 Nd/ 144 Nd ratios between chondrites and the ASE, we conclude that the EER could have formed as a near-solidus melt in the shallow upper mantle within 33.5 Myr of Solar System formation. The extremely small degree of melting in the Hadean mantle would require specific circumstances that allowed the melt to segregate immediately after its generation. One such scenario is shown in Fig. 8 where adiabatically upwelling mantle melts at the pressure and temperature where the mantle geotherm crosses the solidus. If the base of the lithosphere happens to exist just above this depth, then any further mantle upwelling and melting will be prevented, and the near-solidus melt will segregate and ascend into the lithosphere.
Could such a process involving an extremely small degree of partial melting just under a thick lithospheric lid have occurred in the early Hadean? The thickness of the lithosphere largely depends on the style of mantle convection, and so we first examine the style of mantle convection in the early Hadean. Solomatov (1995) showed that thermal convection is exhibited in the stagnant-lid regime when viscosity is dependent on temperature to a sufficiently high degree, and that Earth's mantle and both H 2 O-free and H 2 O-saturated olivine also have highly temperature-dependent viscosities, which inevitably result in the development of stagnant-lid convection. Thus, Earth's mantle is expected to have been in the stagnant-lid convection regime prior to the onset of the plate tectonics. The timing of the onset of plate tectonics is still debated, but recent studies have proposed that convection of the Hadean mantle in the stagnant-lid regime possibly involved intermittent drip-like subduction (Foley et al. 2014;O'Neil and Debaille 2014). A tectonic regime of stagnant-lid convection with intermittent subduction has been supported by results from multiple analytical studies on Jack Hill zircons (Hopkins et al. 2010;Kemp et al. 2010;Tarduno et al. 2015). Hopkins et al. (2010) estimated the temperature and pressure conditions for the formation of Jack Hill zircons using the Ti-in-zircon thermometer and the phengite, Al-in-hornblende, and Tiin-quartz barometers. From the estimated pressure and temperature conditions of zircon formation, they obtained  Tarduno et al. (2015) showed that the paleomagnetic intensity of 3.3-4.2 Ga Jack Hill zircons was 1.0-0.12 times that of the present-day value. The strength of the magnetic field depends on the heat flux across the coremantle boundary, which in turn depends on the subduction of a slab. When subduction is sluggish, heat flux across the core-mantle boundary decreases, resulting in a weakened geodynamo. The low paleomagnetic intensity in the Hadean and the early Archean may be a manifestation of intermittent subduction in the stagnant-lid regime. Based on the model of mantle convection and data from Hadean zircons, it is probable that the mantle was in a stagnant-lid convection mode at the time of formation of the EER i.e., within 30 Myr of Solar System formation.
The next question that follows is whether or not a thick lithosphere could have existed in the stagnant-lid convection regime in the early Hadean. Under stagnantlid convection with temperature-dependent viscosity, the thickness of the lithosphere decreases as the mantle potential temperature (MPT) increases (Korenaga 2009b). However, Korenaga (2009b) showed that when the effects of the partial melting of the mantle, principally dehydration-stiffening and minimal compositional buoyancy of the lithosphere, are taken into consideration, the lithospheric thickness would increase as the MPT increases, with deepening of the initiation of melting. In the dehydration-stiffening model of Korenaga (2009b),   Davis et al. (2011) c Falloon et al. (2008) the base of the lithosphere inevitably coincides with the depth where melting is initiated. Foley et al. (2014) also showed that the viscosity and thickness of the lithosphere increases with increasing MPT due to fast grain growth and healing of lithospheric damage. Although the effects of partial melting of the mantle were not investigated in Foley et al. (2014), this would have enhanced the grain growth in the residual mantle and increased the thickness of the lithosphere up to the depth of the initiation of melting. Thus, a stable, thick lithosphere in the stagnant-lid convection regime can be facilitated by both dehydration-stiffening due to partial melting and by grain growth at high MPT. In both mechanisms, the base of the lithosphere would have been located at the depth where partial melting is initiated. As such, very small degrees of melting would inevitably have occurred in the early Hadean stagnant-lid convection regime. The dehydration-stiffening proposed by Korenaga (2009b) is based on an assumption that the difference in the viscosities between wet and dry mantles is large, by up to three orders of magnitude (Korenaga 2009). This assumption is based on experimental and theoretical studies on the deformation of olivine aggregates (Fo = 90) (Karato et al. 1986;Hirth and Kohlstedt 1996;Mei and Kohlstedt 2000a, b;Jung et al. 2006;Korenaga and Karato 2008), which showed enhanced dislocation and diffusion creep rates in olivine aggregates with the incorporation of water. In the Korenaga (2009) model, experiments with pure forsterite are excluded because the kinetic properties are expected to be significantly different between forsterite (Fo = 100) and mantle olivine (Fo = 90;McDonough and Sun 1995;Mei and Kohlstedt 2000a). However, Fei et al. (2013) showed that the effect of water on mantle viscosity would be smaller than estimated in previous experiments, based on experimentally obtained Si self-diffusion coefficients in single forsterites. Fei et al. (2013) argued that previous experiments using olivine aggregates had been performed in water-oversaturated conditions, which may have been present at grain boundaries and thereby caused grain boundary sliding, resulting in the overestimation of the effect of water. It is still unclear which of these two arguments is valid with regards to the effect of H 2 O on olivine rheology.
Assuming that the dehydration-stiffening model proposed by Korenaga (2009b) is correct, any discrepancy existing between the dehydration-stiffening model and our high-pressure melting experiment would arise from differences in water content. In Korenaga (2009b), the dehydration-stiffening is based on the assumption that  (Falloon et al. 2008;Davis et al. 2011;this study) with the 3.3-3.5 Ga Barberton komatiites (Robin-Popieul et al. 2012), 2.7 Ga Abitibi komatiites (Kerrich and Xie 2002), and 3.8 Ga Isua MORB and OIB (Komiya et al. 2004). MORB mid-ocean ridge basalt, OIB ocean island basalt asthenospheric olivine contains <1000 ppm H/Si. In our experiments, the synthesized starting materials also contained a trace amount of H 2 O (up to 800 ppm). Therefore, no significant discrepancies are expected to exist between the Korenaga model and our high-pressure melting experiment. While such trace amounts of water have negligible influence on the melting phase relations (Green et al. 2014), they can have a strong effect on viscosity (Hirth and Kohlstedt 1996;Mei and Kohlstedt 2000a, b;Korenaga and Karato 2008). Even if the effect of dehydration-stiffening is smaller than that assumed in Korenaga (2009) and fails to produce a sufficiently thick lithosphere, such a thick stagnant-lid could still be generated by fast grain growth and healing of lithospheric damage (Foley et al. 2014). Thus, we conclude that a thick lithosphere may have plausibly existed in the Hadean.
The composition of the near-solidus melt depends on the thickness of the lithosphere and the depth of the initiation of partial melting, which in turn depends on the MPT (Fig. 6). The MPT from the Phanerozoic to the early Archean has been estimated to range from 1400 to 1600°C, based on the MgO content of basalts that were interpreted to have been generated by adiabatic melting of the ambient mantle (Komiya et al. 2004;Herzberg et al. 2010). Korenaga (2011) calculated the thermal budget and MPT in plate tectonic convection from the present day to the Hadean and estimated the MPT at 4 Ga to be 1600°C. The timing of the shift in mantle convection from the stagnant-lid to plate tectonic regime still remains Fig. 6 Comparison of the K 2 O and P 2 O 5 contents of the 7 GPa near-solidus melt with that of lunar KREEP rocks (Ryder 1976;Ryder et al. 1977;Irving 1977) and average normal (N-) and enriched (E-) MORB (Klein 2004). The N-MORB data are for the Kolbeinsey, normal East Pacific, and Mid-Cayman ridges. The E-MORB data are for the Mid-Atlantic Ridge Fig. 7 Densities of the solidus melt and mantle peridotite at 1, 3, and 7 GPa. Diamonds and rectangles show densities of the primitive peridotite and densities of the near-solidus melts, respectively. The solidus melts are less dense than the mantle peridotite. Therefore, the solidus melts should ascend in the mantle unclear, but previous studies have suggested that plate tectonics could have started by 4 Ga (Korenaga 2013). Thus, we adopt the MPT at 4 Ga estimated by Korenaga (2011) and assume a range in the MPT from 1500 to 1700°C for the Hadean mantle. When the MPT is 1500 to 1700°C, melting in the mantle would initiate between 3 and 7 GPa, and the near-solidus melt would also be generated in this pressure range. In this range, the EER would have a picritic to komatiitic composition (Fig. 5).
In order to apply the stagnant-lid convection model to our model, we need to confirm that the source mantle region for the formation of the EER would have experienced adiabatic melting within 33.5 Myr, which is the maximum formation time of the EER after Solar System formation. We calculated the processing time of adiabatic partial melting of a source mantle region under stagnant-lid convection with a MPT of 1500 and 1700°C, using an equation of velocity scaling (Solomatov and Moresi 2000;Korenaga 2009b). The asymptotic form of velocity scaling was expressed in Korenaga (2009b) where v rms is root-mean-square velocity for the entire mantle domain and Ra i is the internal Rayleigh number.
θ is the Frank-Kamenetskii parameter defined by Korenaga (2009b) as: where E is activation energy, R is the universal gas constant, T s is the surface temperature, and ΔT is the temperature constant across the mantle. a v and β v were calculated from Korenaga (2009) and Solomatov and Moresi (2000), respectively, by fitting the above equation for velocity scaling to their numerical data. As a result of our calculations, the obtained processing time with a MPT of 1500 and 1700°C is about ca. 4 and 1 Myr, respectively, when a region corresponding to the upper mantle experiences adiabatic melting. When the whole mantle experiences adiabatic melting, the obtained processing time with a MPT of 1500 and 1700°C is ca. 15 and 5 Myr, respectively. Since these processing times are sufficiently shorter than the maximum formation time of the EER (i.e., 33.5 Myr after Solar System formation), the stagnant-lid convection model can be applied to the formation of the EER for both upper and whole mantle sources at a MPT of 1500 to 1700°C.

Fate of the EER
Based on the results in the previous sections, we conclude that the EER would have been a proto-crust, which  Katsura et al. 2004) and solidus of primitive peridotite (Herzberg et al. 2000). The Hadean mantle potential temperature (MPT) is assumed to have been sufficiently high (1500 to 1700°C) to initiate partial melting at higher pressure. b A schematic diagram of the formation of the EER melt. A small melt fraction needed for EER formation can be generated at the base of thick lithosphere where melting starts. Even at very low degrees of melting, melt can segregate immediately after its generation beneath the base of the thick lithosphere formed from a melt generated at an extremely small melt fraction that then rapidly ascended through the mantle to the surface (Fig. 8b). The proto-crust would subsequently have become hidden somewhere in, or lost from, the Earth because all of the ASE has higher 142 Nd/ 144 Nd ratios than chondrites. The fate of the EER depends on the timing of its formation and the last giant impact. The timing of the EER formation was estimated to be within 33.5 Myr of Solar System formation from 147 Sm-143 Nd and 146 Sm-142 Nd systematics (Boyet and Carlson 2005;Caro 2011; this study). The last giant impact is considered to have generated the Moon, and its timing has been investigated from lunar chronology. The oldest lunar zircon U-Pb age is 150 ± 6 Myr after Solar System formation (Nemchin et al. 2009), and this age has been interpreted to be the age of solidification of the lunar magma ocean. The lunar formation age obtained from the 147 Sm-143 Nd and 146 Sm-142 Nd chronology of the ferroan anorthositic suite falls within the range of 60-125 Myr after Solar System formation (Boyet et al. 2015), and the Rb-Sr, 147 Sm-143 Nd and 146 Sm-142 Nd, and Lu-Hf chronology of the lunar Mg-suite (plagioclase-rich rocks in the lunar highland crust) fall within the range of 150-200 Myr after Solar System formation (Carlson et al. 2014). Jacobson et al. (2014) determined the timing of the last giant impact to be 95 ± 32 Myr after the formation of the Solar System, by calculating the relationship between the timing of the last giant impact and mass of materials added to Earth after core formation, which was estimated from concentrations of highly siderophile elements in the mantle. Based on these results, we consider that the EER formation occurred before the last giant impact. If the EER, or the picritic to Fe-rich komatiitic proto-crust, had been subducted back into the Earth's interior, the ensuing giant impact would have rehomogenized the whole mantle (Canup 2004) and the EER should have become re-mixed into the magma ocean, thus erasing the 142 Nd/ 144 Nd difference between the EER and EDR. Hence, the EER must have been removed from the Earth in order to explain the 142 Nd/ 144 Nd difference between the ASE and chondrites. This could have been achieved by the last and preceding giant impacts of the Earth and impact-erosion of the proto-crust into space, leaving the remaining mantle (ASE) more depleted than the original Earth composition. The removed terrestrial material may have possibly formed the Moon (Canup 2012;Cuk and Stewart 2012) or a part of the asteroid belt (Bottke et al. 2015), although exploration of the asteroid belt is only in its early stages.
However, if the Moon and Earth had accreted from similar materials having higher Sm/Nd ratios than most chondrites and had experienced stripping of the crust during collisions of planetesimals as argued by Caro et al. (2008) and Caro (2011), the need for an EER and a missing reservoir would be negated altogether. In contrast, in our model, the EER is proposed to have formed before the last giant impact, and a need remains to search for the missing reservoir that is the complement to the EER. If the EER has been hidden in the lunar interior, then it follows that the 142 Nd/ 144 Nd ratio of the bulk Moon should be lower than that of the ASE. Given that the 142 Nd/ 144 Nd ratio of the bulk Moon has yet to be precisely determined, there is a need for more 142 Nd/ 144 Nd data on rocks from the interior of the Moon in order to assess the feasibility of our model and alternatives proposed by Caro et al. (2008) and Caro (2011).
We consider that the scenario outlined in this paper is the most plausible at present. However, the timing of the EER formation was estimated based on the average value of 142 Nd/ 144 Nd in chondrites. Due to the significant degree of heterogeneity in 142 Nd/ 144 Nd values of chondrites (Boyet and Gannoun 2013), it is quite possible that the timing of formation of the EER may have been earlier or later than our estimate. We calculated the timing of formation of the EER taking into account the variation of 142 Nd/ 144 Nd values of chondrites (− 0.20 ± 0.14; Boyet and Carlson 2005). The variation in chondritic 142 Nd/ 144 Nd ratios has little effect on the maximum melt fraction, but the timing of formation of the EER is delayed as chondritic 142 Nd/ 144 Nd increases, possibly up to 150 Myr after Solar System formation. Therefore, the formation of the EER could have occurred after the formation of the Moon. If the EER formed after the giant impact, the picritic-komatiitic crust (EER) could have been subducted into the deep mantle and isolated from the convecting mantle until the present. The possible storage area for the subducted EER is the core-mantle boundary layer, but it is still unclear whether the EER could be subducted to the core-mantle boundary and have been stored there without remixing. More precise analyses of 142 Nd/ 144 Nd ratios of chondrites and the ASE, and a more conclusive determination of the age of the last giant impact are needed for improving our understanding of early Earth differentiation processes.

Conclusions
We estimated the major element composition of the EER by combining 142 Nd/ 144 Nd isotopic constraints and results of high-pressure melting experiments on primitive peridotite, which we used to construct a possible scenario for the origin and fate of the EER. The formation of the EER is estimated to have occurred within 33.5 Myr of Solar System formation, through partial melting of the mantle at 3-7 GPa at near-solidus temperatures. The melt fraction was most likely quite low (<2.6 %). The near-solidus melt composition at 3 GPa determined experimentally by Davis et al. (2011) is picritic. The major element composition of the near-solidus melt at 7 GPa determined with the MISE method in this study is more Fe-rich than komatiites and alkali-poor than KREEP rocks. The calculated density of the EER is lower than the density of primitive peridotite. This result supports a scenario in which the EER ascends in the early mantle and forms picritic to komatiitic crust on the Hadean Earth. Since the EER crust is expected to have been generated before the last giant impact, it would have been bombarded by the last giant impact that formed the Moon. Thus, the EER was stripped from Earth and lost into the space, leaving the Earth more depleted than its original composition. However, the EER reservoir has not yet been found, even in the Moon.