J Pancin

S Andriamonje

S Aune

A Giganon

Y Giomataris

J F Lecolley

M Riallot

R Rosa

Piccolo Micromegas: first in-core measurements in a nuclear reactor

Keywords: micromegas, neutron detection, ADS, nuclear reactor PACS: 28.20.-v 29.40.-Cs 28.50.Dr

An Accelerator Driven System (ADS) consists in the coupling of an accelerator with a nuclear reactor. Such systems will need neutron detectors working in a wide energy range and insensitive to X-and γ-rays. Micromegas technology has been proposed to achieve this goal. The ability of Micromegas to detect neutrons over a wide energy range has already been demonstrated and this detector is, under certain conditions, insensitive to γ-rays. A new Micromegas neutron detector called Piccolo Micromegas has been designed to get integrated neutron fluxes on different energy domains. For the first time, Piccolo Micromegas has been placed in the core of a nuclear reactor at Casaccia in Italy. The configuration of the detector will be presented as well as its functioning and the reasons of its insensitivity to γ-rays.

The results of the operation of the detector will also be shown for low reactor power to high reactor power ans some improvements will be suggested.

Introduction

Nuclear energy represents 30% of Europe's electricity, ensuring energy supply security and oil independance combined with low emission of green house gases. However, sustainable development and public concern require a permanent and reliable solution to nuclear radioactive waste. The present leading approch is waste disposal in deep geological formations, which raises problems of site selection, public approval and also long term reliability. Accelerator Driven System (ADS), which consist in the coupling of an accelerator to a target installed in the center of a nuclear reactor, are viewed today as one of the possible ways to reduce the nuclear waste volume and the long term toxicity. It has been experimentaly demonstrated that an ADS has the potential to produce energy [1] but also to destroy nuclear waste [2]. It can also be used for radioactive isotope production, for example for medical applications [3]. ADS has been studied through several experiment projects around the world like the TRADE project [START_REF]TRiga Accelerator Driven Experiment[END_REF]. These hybrid systems are safer than classical nuclear reactors since they work in a sub-critical state. Moreover, they produce less plutonium and nuclear waste when they use a fuel based on thorium oxide.

These new systems need experimental data on their neutron flux in order to confirm or improve simulations. Neutrons are produced by spallation reactions in the target and fission reactions in the fuel. Measuring the neutron energy spectrum would be particularly interesting. Because of the accelerator and the charged particles hitting the target, a lot of γ-rays are produced and it is during this high γ flash that neutrons have to be detected. A neutron detector based on the Micromegas concept [START_REF] Giomataris | [END_REF] has been realised for use with an ADS.

Insensitive to γ-rays in certain conditions, this detector is able to work either in counting or in current mode to detect neutrons in wide neutron energy range for low or high fluxes. Micromegas has already worked as a neutron detector for n TOF (Neutron Time Of Flight) at CERN in 2001 [6,7]. More recently, some studies have been realised in high γ flux environments for the DEMIN project [8]. An other experiment called INPHO has lately demonstrated the ability of Micromegas to detect fast neutrons in a γ flux higher than 10 10 γ/cm 2 /s [9]. It was then natural to try to use Micromegas in the ADS framework. The Piccolo Micromegas detector is presented first with the functioning principle and a technical description. The results obtained from measurements in a nuclear reactor are then reported and some improvements are proposed.

The neutron detector 'Piccolo'

This new detector, named Piccolo for its small size, was designed in 2004 for the TRADE project and it has already been tested with a neutron generator in 2005 [10].

Detector principle

The Piccolo detector is based on Micromegas technology. Micromegas is a gaseous detector composed of two stages separated by a micromesh: a drift or ionization space with a low electric field ( 1 kV/cm) and an amplification space with a high electric field ( 10 kV/cm) [START_REF] Giomataris | [END_REF]. Electrons, produced by the passing of charged particles through the gas, drift toward the micromesh under the action of the low electric field. A funnel is created by the field lines around the micromesh and allow the electrons to pass through it and to be multiplied in the amplification gap. This avalanche creates a current on the anode strips. However, neutrons have to be converted into charged particles in order to be detected by ionization. This can be done by two means, either by elastic collision of the neutrons on gas atoms or by the use of a deposit of 10 B, 6 Li, or 235 U for example, at the entry of the ionization space. Three deposits are used here: 10 B through the 10 B(n,α) reaction (50 µg), 232 Th (140 µg) and 235 U (50 µg) through their fission reaction. The gas used here is a mixture of argon+isobutane(2%) which is a non-flammable gas. A detector part without any deposit is used to detect the recoils of hydrogen coming from isobutane.

The use of different converters allows the detection of neutrons in several energy domains. Boron and uranium permit the measurement of neutrons from the thermal energy to 1 MeV while the thorium fission has a threshold of about 1 MeV. Recoil protons are detected from the threshold energy of the detector to several MeV. Obviously, fission products ionize much more than an alpha particle coming from boron or from hydrogen recoils. Despite the common micromesh, it is possible to get different gains in each detector by applying a different voltage on each pad. Moreover, the detector has a wide dynamic range since it can work in pulse mode at low fluxes and in current mode at higher ones.

Technical description of Piccolo Micromegas

This new detector is placed in a long cylindrical chamber made of stainless steel with 35 mm external diameter, 30 mm internal diameter and long enough to reach the top of the reactor core (see figure 1.a). This small diameter allows the placement of the chamber inside an empty rod of a TRIGA type reactor and to make in-core measurements. The detector is handled at the end of this chamber by a stainless steal rod. In order to use the detector in sealed mode, The chamber is hermetically closed at the top with watertight connections for the signals, the high voltages and the gas. This micromegas detector is in fact composed of 4 micromegas detectors of 5 mm 2 with a common 30 µm thick stainless steel micromesh (see figure 1.b) of 500 LPI. Six wires, going from the detector to the top of the chamber, are necessary to connect the 4 cathode pads, the micromesh and the drift electrode. The common drift electrode is put on a ceramic piece with 4 holes of which one was empty and 3 were holding different neutron converter materials. The drift electrode touches the converter holders to ensure electrical conductivity. The drift gap is 1 mm in the case of the three deposits whereas it is 3 mm for the empty one. The 160 µm amplification gap is ensured mechanically by a stainless steal piece with 4 holes. Contrary to a classical micromegas detector, there is no need for spacers thanks to the small dimensions of the holes diameter. In that case, the micromesh is not warped by the amplification field. Concerning the thermal aspects, a special effort has been made so that the detector supports high temperature with no important gain variation. Moreover, the use of ceramics and stainless steel permits to reduce the activation by the neutron flux.

Electronics and acquisition system

Two working modes have been used, pulse mode or current mode when the counting rate was too high (when the reactor power was high). In current mode, the current is just read out on the high voltage module 1 of the micromesh and the pads. All the other electronic modules consuming current like high voltage filters for the pads or the micromesh preamplifier 2 , are bypassed in order to determine the real detector current.

In pulse mode, the signal of each pad is fed through a fast preamplifier (2 ns of rise time and 50 of gain). These preamplifiers are linked to 1 GHz ADC

6

the pads while at higher power only the current delivered by the high voltage can be registered since the counting rate is too high. This experiment aims at studying the behavior of the detector inside a nuclear reactor. Several problems has been tackled like the linearity of the response versus the reactor power or the aging. The output energy spectra corresponding to the different converters have also been studied, which allows a good understanding of the detector response. one fuel-follower safety rod (SEC) and one regulating rod (REG). Produced thermal power is removed by natural water circulation trough a suitable thermohydraulic loop composed of heat exchangers and cooling towers.

The core

The reactor and the experimental facilities are surrounded by a concrete shield structure. The core and the reflector assemblies are located at the bottom of an aluminium tank (190.5 cm diameter). The overall height of the tank is about 7 m. Therefore, the core is shielded by about 6 m of water. The core, surrounded by the graphite reflector, consists of a lattice of fuel elements, graphite dummy elements, control and regulation rods (see figure 2). There are 127 channels divided in seven concentric rings, named A through G, each containing a number of channels ranging from 1 to 36. One channel houses the start-up Am-Be source, while two fixed channels (at the center and at the periphery) are available for irradiations or experiments. A pneumatic transfer system allows fast transfer from the peripheral irradiation channel and the radiochemistry end station.

The diameter of the core is about 56.5 cm while the height is 72 cm. Neutron reflection is provided by graphite contained in an aluminium container which is surrounded by 5 cm of lead acting as a thermal shield. The core components are contained within top and bottom aluminium grid plates: the top grid has 126 holes for fuel elements and control rods and a central thimble for high flux irradiations. The reactor core is cooled by natural convention of the water in the reactor pool.

The fuel

The fuel elements consist of a stainless steel clad (AISI-304, 0.05 cm thick, 7.5 g/cm 3 density) characterized by an external diameter of 3.73 cm and a total height of 72 cm end cap included. The fuel is a cylinder (38.1 cm high, 3.63 cm diameter, 5.8 g/cm 3 of density) of a ternary uranium-zirconium-hydrogen alloy (H-to-Zr atom ratio is 1.7 to 1; the uranium, enriched to 20% in 235 U, makes up 8.5% of the mixture by weight: the total uranium content of a rod is 190.4 g, of which 37.7 g is fissile) with a metallic zirconium rod inside (38.1 cm high, 0.5 cm diameter, 6.49 g/cm 3 of density). There are two graphite cylinders (8.7 cm high, 3.63 cm diameter, 2.25 g/cm 3 of density) at the top and bottom of the fuel rod. Externally two end-fittings are present in order to allow the remote movements and the correct locking to the grid.

The regulation rod has the same morphological aspect as the fuel rod: the only difference is that instead of the mixture of the ternary uranium-zirconiumhydrogen alloy, there is the absorber (graphite with powdered boron carbide).

The control rods are "fuel followed": the geometry is similar to that of the regulation rod with the bottom graphite cylinder replaced with fuel. The graphite dummies are similar to the regulation rod but without boron inside the central volume.

Data analysis

The study of the linearity of the response as a function of the power is quite straightforward. At high power, the current can be read on the high voltage module. At low power, one has to count the number of reactions detected on each pad on the scaler. The triggering of the scaler is realized with a discriminator module with a threshold tuned to be equivalent to the threshold of the VME acquisition used to register the pulse. However, one has to choose correctly the micromesh and the drift electrode voltage.

Concerning the pulse analysis, each pulse registered by the ADC is processed via a C++ home made program. The pulse amplitude, its time length and its area (proportional to the energy deposit) are then calculated. However one has to be sure to detect neutrons and not γ-rays or simply noise, which can be very high in a nuclear reactor. One example of a pulse registered with Matacq is shown on figure 3.

Working conditions

The noise has been evaluated by determining at the variation of the baseline which has a RMS value of 1 mV. Since the threshold of the VME acquisition is 6 mV, electronic noise can be neglected. Moreover, the mean energy of the noise (that is to say the area between the baseline and the noise oscillations) is around 8 mV.ns. This value is much lower than the one observed on the energy spectra of the nuclei coming from the neutrons conversion.

The γ-ray flux inside the reactor with is about 10 9 γ/cm 2 /s in the center of the reactor at 100 kW. It can be detected by operating Piccolo at high gain (with a micromesh voltage of about -350 V which gives an electric field of 22 kV/cm) but there is no need to apply such a high voltage for charged particles. The detection of H recoils needs high gain. However, the detector sparks because of the recoils of carbon and argon atoms (which ionize much more than hydrogen recoils) if the reactor power is high enough to produce a high neutrons flux and if the potential difference between the pad and the micromesh is over 300 V (in absolute value). Thus, the working voltage was set to -270 V for the 10 B and the recoil pads. For the two fission pads, since the energy deposit of fission fragments is high, a micromesh voltage of -190 V was enough.

To obtain these two different working voltages with the common micromesh, the micromesh voltage was fixed to -100 V while the fission pad voltages was +90 V and the boron and recoil voltage was +170 V. Considering the chosen working voltage values, the gain is lower than the one used during the INPHO experiment by at least a factor of 10, ensuring a good γ rejection [9].

The current versus the micromesh voltage on two pads (uranium and boron) at 10 kW is shown in figure 5. From this curve, it is possible to evaluate the detector gain which is about 10 for the uranium and thorium converters and 30 for the boron and gas converters (H recoil). Therefore, by comparing the Piccolo set up to the n -TOF set up [7], it has been possible to evaluate the detection threshold to 400 keV for the boron and gas converters. The threshold is quite high, which is normal at low gain.

Vmesh [V]

Drift voltage influence

Fixing the micromesh voltage is not sufficient, one has also to choose the drift voltage. Usually in this argon mixture, a field of about 50 V/mm is enough to get optimum results in terms of charge transfer through the micromesh. Figure 6 shows the counting rate dependence of the uranium and of the thorium pad with the drift field for different reactor powers. On the uranium pad, for each power an optimum in the counting rate exists and the corresponding field value increases with the reactor power. It goes from a normal value of 60 V/mm at 10 kW to 250 V/mm at 400 kW. On the contrary, the optimum value for the thorium pad doesn't move with the reactor power.

Since the neutron flux is important in the reactor (about 10 11 n/cm 2 /s at 10 kW), the reaction rates of each pad, and more especially the uranium or the boron pad, are very high. A large number of electrons are created and some space charge effects appear in the detector which degrades the drift field. In that case, it is necessary to increase the drift voltage in order to counterbalance the space charge and sweep the electrons towards the amplification gap. In the case of thorium, the reaction rate is much lower because of the fission threshold and there is no space charge effect. In the rest of this article, all the currents and the counting rates will be given at the optimum value of the drift voltage.

Spectral analysis

The spectral analysis has also permitted to observe the space charge influence. with two different values of the reactor power increasing from left to right. Two peaks appear on these histograms. The detector is not able to distinguish α particles from 7 Li particles and even if these two peaks were representing the two reaction products of the neutron absorption, they shouldn't be so well separated. The lower peak in amplitude is not noise since the noise level is at The figures 8 show the energy spectra from boron and uranium with an optimized drift voltage. In that case, the spectra look homogeneous with no space charge effect (to be compared to the figure 7.c). However, it's not possible to separate the two peaks of the uranium fission.

Aging

The outgassing from its inner material modifies the gas features. The signal amplitude induced by α particles coming from the radioactivity of thorium or uranium has been measured during several days to estimate the gain evolution.

The gain has decreased by a factor of two in two weeks, which can be compensated by decreasing the mesh voltage of -30 V. Moreover, the avalanche occurring in the amplification gap leads to a deterioration of the gas (breaking of the isobutane molecule) and of the micromesh (carbonization) and it is of course worse at high counting rates. Micromegas has already been tested with a ten years equivalent LHC radiation showing no change in the gain [12]. A total decrease of a factor of 2.5 has been observed during the 10 days of test in the TRIGA reactor. It has not been possible to estimate if it was coming from the normal outgassing or from the deterioration of the isobutane, this would need long term dedicated studies.

Concerning the burn up, the most critical deposit is the 10 B with its huge thermal cross section. It could be used only several days at 100 MW for example.

However, during our tests, the power hasn't exceeded 400 kW. On this point of view, the gas recoils are interesting since the number of atoms is higher than for a thin deposit and this converter could last a long time.

Influence of the temperature

In the laboratory, during preliminary tests, the detector has not been heated at higher temperatures than 100 • C, which was the temperature applied on that the thorium pad current remained constant, the uranium pad current increased by 5%, the boron by 20% and the recoil pad by 25%. If this increase were due to activation, it would have increased also on the thorium pad like on the uranium pad of the same absolute value. This increase is proportional to the current and to the gain. Hence, it corresponds to a gain variation probably due to the temperature and the consecutive dilatation. The increase of the boron or the recoil pad gain is compatible with simulations showing that one could get 3% more gain per micron amplification gap decrease and per 100 • C of temperature increase.

Linearity versus reactor power

The figures 10.a and 10.b present the current of three pads corresponding to the uranium, boron and hydrogen recoil converters, as a function of the reactor power in the center and at the edge of the reactor. First of all, all the currents are linear with the power which is a first step towards the feasibility of power measurements with this detector. The thorium converter current is not shown since it was too small to be measured with this set-up due to its low reaction rate. The boron current is the highest because it has a higher gain (with the proton recoil pad) and the higher reaction rate. Moreover, the central measurement (figure 10.a) has been made after 1 hour at 200 kW while the edge measurement has been made directly from 10 to 400 kW at the reactor switching on. The central measurement is then correlated with the activation of the nuclear materials inside the reactor or with the temperature detector increase. It is one of the reasons why the proton recoil current is higher than the uranium current in the center although it was lower at the edge. The proton recoil pad is much more sensitive to temperature since it has a higher gain. The current measurement is also sensitive to γ-rays coming from the fission process inside the reactor. It is not a problem for the uranium and boron converters since their reaction rate is high and their drift gap is very low. Their currents are mainly due to neutron reactions. However for recoils, the drift gap is three times larger and the current due to recoils is probably of the same order than the one induced by γ-rays. earity is not respected anymore for different reasons. Concerning uranium, the reaction rate was too high for the scaler and the linearity is visible only from 10 to 50 kW before the scaler begins to be limited. Concerning the two last pads, H recoil and boron converter, one has to remember that the detection threshold was 400 keV. Only a negligible part of the reaction products (proton recoils, α and 7 Li) deposit enough energy. It is in that case difficult to interpret the results even if the linearity of proton recoils counting is not so bad. However, these counting rates measurements are very interesting since they are totally insensitive to activation and γ rays and could permit to make absolute measurements of the reactor power. On the contrary, it would be very difficult without any reference detector to make a current measurement uncorrelated with γ and β rays. The reaction rates have been evaluated by MCNP simulations using a simulated flux of this TRIGA reactor realized in the LPC/Caen laboratory. The measured reaction rates are much lower than those which have been simulated, still because of the threshold, by a factor of 10 less for thorium to 10 5 for boron.

The current (or counting rate for thorium) ratio between center and periphery is presented on figure 12. Uranium and thorium converters give the same ratio, about 1.7, between center and periphery, which is a realistic value. The result is found by counting for thorium and current for uranium and it is confirmed by the boron converter at high power. It means that the ratio is valid for low energy and for high energy neutrons. However, at low power, the boron current increases since the measurement begins to be sensitive to activation. For the recoils, the current is sensitive to both γ-rays from fission (ratio over 2.5) and from the activation (ratio increases at low power). The γ flux ratio (about 3

following General Atomics data) between center and periphery is higher than the neutron flux ratio, which explains the results obtained with proton recoils. which it is the counting rates ratio).

Future improvements

The first experimental results obtained in the core of the reactor are rather promising. They indicate a good behaviour of the Piccolo Micromegas detector.The counting mode is important for the purpose of background rejection allowing, for instance, γ and β particle discrimination. The detector energy threshold has to be decreased without degrading the γ-ray insensitivity. In order to optimize the performance of the detector, the following improvements are proposed:

-Switching to helium gas would allow to reach a higher gas gain or to improve the sparking limit [13].

-Employing low-noise fast charge preamplifiers would improve the signal tonoise ratio by more than a factor of ten.

-For low-energy neutrons, it would be sufficient to use only an uranium target, a lower counting rate is expected and the larger energy deposition will provide signals of high amplitude.

The various improvements are expected to significantly improve the detector operation and provide comfortable signals at even lower gains; low gain operation is a great step forward to avoid the build-up of space charge and diminish detector aging. Mechanics should also be modified and optimized to avoid as much as possible space charge effects and gain variation with temperature. All these modifications would allow to greatly improve the results.

Conclusion

The performances of a Piccolo-Micromegas prototype have been studied in the core of a nuclear reactor. Up to the highest reactor power (400 kW) mea- 12 Ratio of the central current to the edge current (except for thorium for which it is the counting rates ratio). 19

Fig. 1 .

 1 Fig. 1. (a) Cylindrical chamber handling the detector, (b) the conceptual design of the Piccolo detector.

3. 1

 1 Description of the TRIGA reactor 3.1.1 The TRIGA RC-1 The 1 MW TRIGA reactor of ENEA-Casaccia is a pool thermal reactor with a core contained in an aluminium vessel and placed inside a cylindrical graphite reflector, bounded with lead shielding. The biological shield is provided by concrete having a 2.2 m mean thickness. Demineralized water, filling the vessel, ensures the functions of neutron moderator, coolant and first biological shield. Reactor control is ensured by four rods: two shim rods (SH1 and SH2),

Fig. 2 .

 2 Fig. 2. TRIGA reactor configuration and position of the Piccolo Micromegas detector in the periphery of the core (for the central position, Piccolo was placed in D18).

Fig. 3 .

 3 Fig. 3. Signal registered on the 10 B Matacq channel at P=10 kW.

Figure 4 .

 4 Figure 4.a shows the counting rate of Piccolo inside the reactor switched off

Fig. 4 .

 4 Fig. 4. Piccolo counting rate versus the absolute micromesh voltage at P=0.

Fig. 5 .

 5 Fig. 5. 10 B and 235 U Pads current versus the absolute micromesh voltage at P = 10 kW.

Fig. 6 .

 6 Fig. 6. Counting rate on the uranium part (a) and on the thorium part (b) versus the absolute drift voltage and for reactor powers from 10 to 400 kW.

Figures 7 .

 7 Figures 7.a and 7.b present two amplitude histograms from the boron converter

1Fig. 7 .

 7 Fig. 7. (a) Amplitude histogram from the boron converter at 100 W (b) and 10 kW, (c) energy histogram from the boron converter at 10 kW.

Fig. 8 .

 8 Fig. 8. Energy spectra from boron at 200 kW (a) and uranium at 400 kW (b).

Fig- ure 9 Fig. 9 .

 99 Fig. 9. Uranium converter energy spectra at 10 kW for three consecutive days.

  the shield during pumping (to outgas the detector material) and before gas filling. However, the influence of the temperature on the signal has not yet been measured. The water temperature inside the TRIGA reactor was not exceeding 30 • C at 200 kW while the temperature of the uranium fuel was reaching 140 • C. Although the detector can not be heated up by the converters neutron absorption, it could be heated by the high ambiant radioactivity (300 rad/h/W) leading to an energy deposit from γ-and β-rays in the detector material. It has been estimated that the temperature could have increased up to 200 • C during the test. During one hour at 200 kW, it has been observed

The figure 11 .Fig. 10 .

 1110 Fig. 10. Currents versus reactor power in the center (a) and at the edge (b) of the reactor for the uranium, boron and hydrogen recoil pads.

Fig. 11 .

 11 Fig. 11. Counting rates versus reactor power on 100 s: (a) for the thorium pad at the edge and in the center, (b) for the other converters at the edge (divided by 500 for the uranium pad).

Fig. 12 .

 12 Fig. 12. Ratio of the central current to the edge current (except for thorium for

5 2 9 3 10 4 11 5 10 B 12 6

 5910111012 surements in both continuous (read-out of the cathode current) and counting mode (anode strip read-out) have shown a good linearity with power. The behaviour of the detector inside the reactor under high neutron and γ-ray fluxes has been well understood. Piccolo could perfectly fit the requirements of the future ADS project and perform neutron flux measurements. However, some geometrical modifications are necessary to optimize the detector and especially the electric field configuration in the drift compartment. The choice of a more appropriate gas filling shoud be studied and an adequate low-noise electronics is necessary to decrease the energy threshold. Moreover the long term aging of the detector, which is an important issue in the case of sealed gaseous detectors, would merit a long term dedicated study inside a nuclear reactor. (a) Cylindrical chamber handling the detector, (b) the conceptual design of the Piccolo detector. TRIGA reactor configuration and position of the Piccolo Micromegas detector in the periphery of the core (for the central position, Piccolo was placed in D18). Signal registered on the 10 B Matacq channel at P=10 kW. Piccolo counting rate versus the absolute micromesh voltage at P=0. and 235 U Pads current versus the absolute micromesh voltage at P = 10 kW. Counting rate on the uranium part (a) and on the thorium part (b) versus the absolute drift voltage and for reactor powers from 10 to 400 kW. 13 7 (a) Amplitude histogram from the boron converter at 100 W (b) and 10 kW, (c) energy histogram from the boron converter at reactor power in the center (a) and at the edge (b) of the reactor for the uranium, boron and hydrogen recoil pads. 18 11 Counting rates versus reactor power on 100 s: (a) for the thorium pad at the edge and in the center, (b) for the other converters at the edge (divided by 500 for the uranium pad). 19

Acknowledgments

We would like to thank all the operators of the Casaccia nuclear reactor.


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email: allibert@ipgp.fr

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Quantitative estimates of impact induced crustal erosion during accretion and its influence on the Sm/Nd ratio of the Earth

Keywords:

Dynamical scenarios of terrestrial planets formation involve strong perturbations of the inner part of the solar system by the giant-planets, leading to enhanced impact velocities and subsequent collisional erosion. We quantitatively estimate the effect of collisional erosion on the resulting composition of Earth, and estimate how it may provide information on the dynamical context of its formation. The composition of the Bulk Silicate Earth (BSE, Earth's primitive mantle) for refractory and lithophile elements (RLE) should be strictly chondritic as these elements are not affected by volatile loss nor by core formation. However, an excess in 142 Nd compared to the 144 Nd has been emphasized in terrestrial samples compared to most measurements in chondrites.

crust, which was collisionaly eroded during the late stages of planetary accretion [START_REF] O'neill | Collisional erosion and the non-chondritic composition of the terrestrial planets[END_REF][START_REF] Campbell | Evidence against a chondritic earth[END_REF][START_REF] Boujibar | Cosmochemical fractionation by collisional erosion during the earth/'s accretion[END_REF][START_REF] Bonsor | A collisional origin to earth's non-chondritic composition?[END_REF][START_REF] Carter | Compositional evolution during rocky protoplanet accretion[END_REF][START_REF] Carter | Collisional stripping of planetary crusts[END_REF][START_REF] Shibaike | Excavation and melting of the hadean continental crust by late heavy bombardment[END_REF], leading to the inferred superchondritic Sm/Nd ratio. Total erosion of crust about 1% -1.5 of Earth mass may be required to account for the fractionation of the BSE Sm/Nd ratio [START_REF] O'neill | Collisional erosion and the non-chondritic composition of the terrestrial planets[END_REF]) if Earth's building blocks were made of either ordinary or carbonaceous chondrites. This number is highly dependant on the partial melting rate assumed in the model [START_REF] O'neill | Collisional erosion and the non-chondritic composition of the terrestrial planets[END_REF], on the crustal composition (MORB type here) but also on the dependency with the one-impact erosion assumption. Other estimates can be provided considering other evidences. For example, considering the Mg/Si ratio as well as the hypothesis that the Earth building blocks are made of enstatite chondrites, [START_REF] Boujibar | Cosmochemical fractionation by collisional erosion during the earth/'s accretion[END_REF] estimate that 15wt% of crust may have to be preferentially removed during accretion. However, the Earth is only showing a 10-ppm deficit [START_REF] Burkhardt | A nucleosynthetic origin for the earth's anomalous 142 nd composition[END_REF] in 142 Nd relatively to the 144 Nd for enstatite chondrites instead of 20-ppm when considering the average over all chondrites [START_REF] Boyet | 142nd evidence for early (> 4.53 ga) global differentiation of the silicate earth[END_REF] suggesting a lower loss of crust required to account for this deficit.

Additionally, recent studies have shown that the existence of nucleosynthetic anomalies in pristine planetary precursors such as chondrites and Ca-Al rich refractory inclusions (CAIs) could cause the observed variations in the 142 Nd abundances among planetary materials [START_REF] Burkhardt | A nucleosynthetic origin for the earth's anomalous 142 nd composition[END_REF][START_REF] Bouvier | Primitive solar system materials and earth share a common initial 142 nd abundance[END_REF]. Moreover, few CAIs and enstatite chondrites have also been shown recently to share common Nd isotope signatures with terrestrial rocks [START_REF] Bouvier | Primitive solar system materials and earth share a common initial 142 nd abundance[END_REF][START_REF] Boyet | Enstatite chondrites el3 as building blocks for the earth: The debate over the 146sm-142nd systematics[END_REF]. These observations would imply a chondritic Sm/Nd ratio for the Earth and would preclude dynamical accretion scenarios leading to efficient crustal erosion. However, planetary accretion processes involve energetic impacts with potentially significant erosion of material from embryos [START_REF] Leinhardt | Collisions between gravity-dominated bodies. i. outcome regimes and scaling laws[END_REF][START_REF] Leinhardt | Numerically predicted indirect signatures of terrestrial planet formation[END_REF][START_REF] Bonsor | A collisional origin to earth's non-chondritic composition?[END_REF][START_REF] Marcus | Collisional stripping and disruption of superearths[END_REF]. Thus, it is necessary to quantify the efficiency of collisional erosion of the primordial crust to address the issue of Earth's accretion scenario and the nature of its building blocks. In previous studies, the efficiency of this process has been investigated with either a simple erosion model [START_REF] Shibaike | Excavation and melting of the hadean continental crust by late heavy bombardment[END_REF] focusing only on the period of the terminal lunar cataclysm (TLC), the existence of which is in question [START_REF] Boehnke | Illusory late heavy bombardments[END_REF][START_REF] Zellner | Cataclysm no more: new views on the timing and delivery of lunar impactors[END_REF][START_REF] Morbidelli | The timeline of the lunar bombardment: Revisited[END_REF][START_REF] Hartmann | History of the terminal cataclysm paradigm: Epistemology of a planetary bombardment that never (?) happened[END_REF][START_REF] Mojzsis | Onset of giant planet migration before 4480 million years ago[END_REF] or neglecting the effect of a population of small planetesimals [START_REF] Carter | Collisional stripping of planetary crusts[END_REF]. Notably, [START_REF] Shibaike | Excavation and melting of the hadean continental crust by late heavy bombardment[END_REF] showed that the eventual TLC would not be responsible for removing the entire Hadean crust but most of it. These are estimations based on excavated material, that is defined as the material that is displaced during an impact. This material is not necessarily ejected, so even lower amounts of ejected material can be expected from one bombardment event as suggested by the TLC with a 2×10 23 g mass of impactors. However, this already raised the high influence that impacts from a population of small bodies may have on crustal stripping.

Recently, a first step toward constraining the effect of collisional erosion on the Sm/Nd ratio of Earth was achieved, in a realistic dynamical context, using sophisticated erosion scaling laws (fitted to SPH simulations of impacts between similarly-sized bodies) coupled to N-body numerical simulations of Earth's accretion [START_REF] Carter | Collisional stripping of planetary crusts[END_REF]. However, computational resources limited the resolution (i.e. number of particles forming planetary bodies) in such simulations: collisions in which the impactor was less than 1% of the target mass were neglected [START_REF] Carter | Collisional stripping of planetary crusts[END_REF].

As a consequence, SPH numerical simulations do not address the physics of smaller cratering impacts [START_REF] Holsapple | A crater and its ejecta: An interpretation of deep impact[END_REF][START_REF] Housen | Ejecta from impact craters[END_REF][START_REF] Svetsov | Cratering erosion of planetary embryos[END_REF], thus hampering quantitative estimates of total eroded material. Planetesimals likely contribute to the majority of impacts, and so collisional erosion may accordingly be driven by small impactors. The effect of these small impacts on the budget of RLE has been neglected in previous works [START_REF] Carter | Collisional stripping of planetary crusts[END_REF]. In addition, [START_REF] Carter | Collisional stripping of planetary crusts[END_REF] only explored the collisional stripping of the early Earth proto-crust during the first stages of planetary growth (between 1 Myr and 23 Myr) [START_REF] Carter | Collisional stripping of planetary crusts[END_REF] while the presence of numerous planetesimals in the early solar system is predicted up to at least 200 Myr. Only during this period of time and only considering relatively high mass ratio impacts, they already show that a substantial amount of crustal stripping can be reached (6-9% of the planetary mass). That raises the question on the effects that smaller impacts may have in addition to giant impacts.

Here we evaluate the hypothesis that crustal erosion from planetesimal impacts can eventually fractionate Earth's Sm/Nd ratio. The evolution of the Sm/Nd ratio is monitored during the accretion of Earth (from 1-3 Myr to 200 Myr) using an approach combining analytical modeling of the cratering process and N-body planet formation simulations.

In section 2 we describe the numerical model as well as the two dynamical scenarios explored.

In section 3 we present our results for both Grand Tack and classical model. Then we summarize our results and discuss them in the context of the collisional erosion hypothesis and nucleosynthetic anomalies hypothesis.

Method

Code description

We use N-body numerical simluations from [START_REF] Raymond | Building the terrestrial planets: constrained accretion in the inner solar system[END_REF]; Jacobson and Morbidelli (2014) to provide the collisional history of terrestrial embryos (whose sizes are defined for each scenario later on) in the proto-planetary disc. These simulations last for about 200 Myr, and have initial conditions representing the state of the solar disk 3-4 Myr after CAIs. Erosion from small impactors (these do not include the impacts fragments that may contribute to the population of small planetesimals) is tracked through the entire accretion process until proto-Earth embryos reach Earth-like mass. We explore two scenarios for the giant planet's orbital evolution : (1) the Grand Tack (Jacobson and Morbidelli, 2014) and (2) a classical scenario where the orbits of the giant planets are excited by mutual perturbations [START_REF] Raymond | Building the terrestrial planets: constrained accretion in the inner solar system[END_REF].

The impacts are divided in two groups : giant embryo-embryo impacts and planetesimal-embryo impacts. The respective ranges of mass for embryos and planetesimals are dependent on the Nbody simulation and are described later on. The planetesimal-embryo impacts are treated with an analytical cratering model based on scaling laws for rocky targets [START_REF] Svetsov | Cratering erosion of planetary embryos[END_REF][START_REF] Holsapple | A crater and its ejecta: An interpretation of deep impact[END_REF]. In our model, giant impacts have no effect on the BSE Sm/Nd ratio as they are assumed to fully melt and mix the crusts and mantles of the colliding embryos. This assumption is a simplification that is made because the melting processes resulting from giant impacts are still poorly constrained. Since evidences have been enlighted by previous studies arguing for only incomplete mantle melting even for very energetic impacts (e.g. [START_REF] Nakajima | Melting and mixing states of the earth's mantle after the moon-forming impact[END_REF][START_REF] Carter | The energy budgets of giant impacts[END_REF], a first step toward estimating the influence of this hypothesis is made by proposing the three distinct sets of assumptions for the chemical mass balances. The results are presented in the supplementary material. After quantifying the mass transfer following each impact, a geochemical model is used to follow the evolution of the amounts of Sm and Nd in the crust and mantle of the different embryos. As numerous works suggest an early differentiation of planetary bodies [START_REF] Lugmair | Early solar system timescales according to 53mn-53cr systematics[END_REF][START_REF] Srinivasan | 26al in eucrite piplia kalan: Plausible heat source and formation chronology[END_REF][START_REF] Bizzarro | Rapid timescales for accretion and melting of differentiated planetesimals inferred from 26al-26mg chronometry[END_REF][START_REF] Amelin | The u-pb systematics of angrite sahara 99555[END_REF][START_REF] Kruijer | Protracted core formation and rapid accretion of protoplanets[END_REF], we assume that all embryos in the simulations are differentiated.

The processes of crust-mantle reequilibration after an impact are poorly known. Consequently, we assume that giant impacts induce a full crust-mantle reequilibration, while planetesimals impacts only affect the composition and mass of the crust and leave the mantle unchanged. Since the N-body numerical simulations used here assume perfect merging, there is no way of tracking the ejected material. In consequence possible capture of ejected material by other embryos, following an impact, has been neglected. Accordingly, the fractionation computed in this work might be overestimated. We introduce the parameter describing the chemical fractionation of the Sm/Nd ratio in the mantle of growing Earth like embryos (i.e. BSP for Bulk Silicate Planet) with respect to the initial chondritic composition: The details of the model (i.e. quantification of ejected and accreted masses, chemical mass balances and post-processing N-body simulations of accretion) are presented hereafter.

= Sm N d t BSP -Sm N d ini BSP Sm N d ini BSP

Cratering model

The first step of the work consists in the calculation of the eroded and accreted masses during a single impact event. As we consider embryos as differentiated bodies, the eroded mass is assumed to have a crustal composition when the total ejected mass after considering all impacts from a given distribution is lower than the crustal mass. This assumption is made to take into account the fact that overlapping between the craters may occur since it physically implies that the model assumes that all planetesimals from a given distribution are coming from all direction. For greater ejected masses, the eroded material is assumed to be made of both crust and mantle (the remnant mass after the entire crust is considered is then ejected mantle). We assume that planetesimals have a bulk chondritic composition which imply that accreted mass has a chondritic composition.

The ejected and accreted masses are deduced from previous studies analytical laws constructed as a function of the impact velocity at encounter, the impactor density and the target density and masses of both the target and impactor [START_REF] Holsapple | A crater and its ejecta: An interpretation of deep impact[END_REF][START_REF] Shuvalov | Atmospheric erosion induced by oblique impacts[END_REF][START_REF] Svetsov | Cratering erosion of planetary embryos[END_REF].

These parameterizations assume that the impactor mass is much smaller than the target mass (i.e. M i /M e < 0.01) so they cannot be used for giant impacts. The treatment of the giant impacts in the proposed model is dependant on the mixing scenario choosen. This point is described in the section 2.3. p-c represents the fraction of proto-crust with respect to the entire embryo's mass. We adapt this melting equation to our specific problem and define the term f 1 p-c as the melt fraction normalized to the primitive mantle mass only, except for the first stage (before any impact) made to estimate the initial composition of the embryo's crust. In the second stage, a f 2 p-c fraction of the proto-crust is eroded (note that f 2 p-c < f 1 p-c ). The concentrations of an element M in the crust and mantle after the first stage can be described following:

Mass balance model

C p-c M C 0 M = 1 D M + f 1 p-c (1 -D M ) , (1)
With D M the partition coefficient of M between melt and solid, C 0 M the chondritic composition for M , and C p-c M its concentration in the proto-crust. It is possible to estimate the concentration in the BSP (Bulk Silicate Planet, i.e. crust and mantle of the growing planetary embryo) after the second stage as (assuming no mantle erosion):

C BSP -eroded M C 0 M = f 1 p-c (1 -D M ) + D M -f 2 p-c (D M + f 1 p-c (1 -D M ))(1 -f 2 p-c ) , (2)
Where C BSP -eroded M is the concentration of M in the BSP. f 1 p-c is initially fixed to a value of 0.026 following O' Neill and Palme (2008). It is then assumed that the partial melting rate remains constant over the accretion history (inducing variations in the crust thickness because of the embryo mass increase) but f 1 p-c takes a value of 0.037. Three different scenarios are considered for crustal formation after an impact, with notably two of them being end-members (we call them scenario 1 & 2. ).

Further results (section 3) are presented for the most realistic case only (called scenario 3), however they are all discussed in appendices.

Scenario 3: "remixing only for giant impacts" model

This scenario is an intermediate case in between (1) and (2) that respectively refer to two extreme end members. For the scenario 1, all impacts are assumed to melt the entire proto-Earth mantle and to fully re-equilibrate the chemical composition in the BSP. Scenario 2 assumes that none of the impacts produce any mantle remixing. Accordingly, we propose an intermediate case that is much more realistic: scenario (3). Planetesimal-embryo impacts are treated following scenario (2), see eq. 12 while Embryo-Embryo impacts are treated following scenario (1)(i.e. full melting of the BSP with formation of a new crust). Accordingly, the composition of the mantle changes each time a giant impact occurs. All giant impacts are considered as perfect mergers.

For simplifications, we assume that this results in a mass of chondritic composition equal to the impactor mass is accreted to the target.

Planetesimals size-frequency distribution

N-body simulations need high computational requirements, so they can not include a realistic amount of planetesimals for the calculations. This is why we have chosen here to distribute the mass of a single planetesimal impactor into a size-frequency distribution of smaller impactors (i.e. each impact from a planetesimal in the data file is assumed to be a serie of impacts with impactors distributed on a size-frequency distribution (SFD)). That serie of impactors is assumed to fall onto the target at the same time. Considering the mass transfer, this assumption implies that the erosion caused by all impactors in the SFD of a given superparticle is applied first. Then accreted mass is added.

The differential size distribution of impactors follows a power law :

dN sf d (> r) dr = -Kr -α , (3)
with α and K standing for positive constants. N sf d (> r) is the cumulative size distribution (the number of bodies with a radius larger than r). We get updon integration :

N sf d (> r) = K r 1-α max -r 1-α 1 -α (4)
and the total mass of the distribution can be described as

M T = rmax rmin - 4 3 πr 3 ρ dN sf d dr dr, (5)
with ρ standing for the mean density of the planetesimals. So replacing

dN sf d dr
in the later expression we get,

M T = rmax rmin 4 3 πr 3 ρKr -α dr. (6)
It is then possible to obtain K:

K = 3M T 4πρ × 4 -α r 4-α max -r 4-α min , (7)
where M T (the total mass of the distribution) equals the mass of the impactor given in the Nbody simulation. r min and r max have to be fixed. They are arbitrarily chosen as 8m and 800km

respectively. The number of bodies in each mass bin is evaluated using the cumulative mass distribution, rounded to an integer number using a random number generator. Accordingly, 800km

is taken as the upper bound of this distribution to be slighlty lower than the typical size of a planetesimal in the simulations [START_REF] Raymond | Building the terrestrial planets: constrained accretion in the inner solar system[END_REF]Jacobson and Morbidelli, 2014). It may be noted that a 800km-sized body is likely differentiated. However, whether the planetesimals are differentiated or not makes no difference in our model since we only investigate here the evolution of an elemental ratio of two RLEs. However the change in the minimum radius of a body within the SFD is not critical on the model outcomes (less than 1% change in the fractionation when choosing minimum radius values of few kilometers). The physical characteristics of the planetesimals are as follows :

• The mean density of the target, ρ m , is assumed to be 5200 kg/m 3 .

• The crust density, ρ t , is assumed to be 2900 kg/m 3 .

• The density of impactors (chondritic material), ρ imp , is assumed to be 2600 kg/m 3 .

• The impact velocity is assumed to be:

V imp = V 2 inf + v 2
esc with V inf the velocity of the impactor at infinity (beyond the gravitational attraction of embryos), which is here assumed to be the same for all impactors in the same impact event. This V inf is deduced from the data of the N-body simulation.

• The radii of target and impactors are calculated using their mean densities and respective masses.

In this study, the slope of the SFD (α) is set at 3.5 which corresponds to the actual value of the distribution of bodies in the asteroid belt. However, the influence of the choice for α needs to be examined, this is the subject of the section 4.2 in the appendices.

Dynamical Scenarios

The impact model presented in the previous section is applied to two dynamical accretion scenarios of the terrestrial planets accretion: (1) Grand Tack or (2) classical. Within each dynamical model, a variety of parameters are explored such as (i) variable eccentricities and semi-major axis of Jupiter and Saturn (for classical model), (ii) different surface densities of the disc (classical) and (iii) different initial embryo masses (Grand Tack). These different dynamical scenarios are summarized in Table 1 (Classical) andTable 2 (Grand The classical model (e.g. [START_REF] Wetherill | Accumulation of the terrestrial planets[END_REF][START_REF] Wetherill | Occurrence of giant impacts during the growth of the terrestrial planets[END_REF][START_REF] Wetherill | The formation and habitability of extra-solar planets[END_REF][START_REF] Kokubo | Formation of protoplanets from planetesimals in the solar nebula[END_REF][START_REF] Chambers | Planets in the asteroid belt[END_REF][START_REF] Raymond | Building the terrestrial planets: constrained accretion in the inner solar system[END_REF] is the reference model concerning terrestrial planets formation, however it is well known now that it fails in reproducing several key features of our present day solar system. Other models have been further proposed to better reproduce the structure of the Solar System at its present state. Notably, the Grand Tack model is very popular as it among the first models proposed that can explain the small masses of Mars and the asteroid belt. It also reproduces the excited orbits of the bodies in the main asteroid belt, as well as the mixing between inner and outer solar system objects.

Classical scenario

The main assumption considered in the Classical model is that the formation of the terrestrial planets can be treated independently compared to the giant gaseous planets [START_REF] Wetherill | Formation of the earth[END_REF][START_REF] Wetherill | An alternative model for the formation of the asteroids[END_REF][START_REF] Chambers | Planets in the asteroid belt[END_REF][START_REF] Raymond | High-resolution simulations of the final assembly of earth-like planets i. terrestrial accretion and dynamics[END_REF][START_REF] Raymond | Building the terrestrial planets: constrained accretion in the inner solar system[END_REF]Raymond et al., , 2014)). The initial conditions thus assume that Jupiter and Saturn are already formed and placed on fixed orbits. Then, it assumes that the terrestrial planets formed essentially from material originated close from their current position within the disk. Such a growth dynamics induces similar sizes for bodies on adjacent orbits (Kokubo and Ida, 2002). Classical models fail to reproduce the size of Mars.

In our study all simulations start with a disk (from 0.5AU to 4.5AU) of embryos and planetesimals with Jupiter and Saturn already formed. They contain 85-90 embryos and 1000-2000 planetesimals [START_REF] Raymond | Building the terrestrial planets: constrained accretion in the inner solar system[END_REF]. The total mass of the system is equally distributed between planetesimals and embryos. However, eight different sets of initial conditions are explored [START_REF] Raymond | Building the terrestrial planets: constrained accretion in the inner solar system[END_REF]. They refer to different orbits of Jupiter and Saturn and different disk surface densities (See Table 1 for details). Depending on the initial conditions, the initial mass of the embryos in the disk is comprised between 0.005 and 0.1 terrestrial masses.

Grand Tack scenario

The grand tack scenario was designed to explain the small size of Mars as this was one of the major issues with the classical accretion model [START_REF] Walsh | A low mass for mars from jupiter's early gas-driven migration[END_REF]Raymond and Morbidelli, 2014).

The inward migration of Jupiter, followed by Saturn before they both migrate outward -when Jupiter gets caught in the 2:3 resonance with Saturn [START_REF] Masset | Reversing type ii migration: resonance trapping of a lighter giant protoplanet[END_REF][START_REF] Morbidelli | The dynamics of jupiter and saturn in the gaseous protoplanetary disk[END_REF][START_REF] Pierens | Two phase, inward-then-outward migration of jupiter and saturn in the gaseous solar nebula[END_REF])-has been proposed as a mechanism to deplete and excite the Mars region leading to a small size of Mars and a distribution of orbital elements in the asteroid belt compatible with current observations. In the simulations we use here, the system contains a population of embryos and two different populations of planetesimals (i.e. internal planetesimalswithin 3AU -and external smaller planetesimals beyond 3 AU) (Jacobson and Morbidelli, 2014).

Each simulation begins with approximately 100 embryos and 2000 planetesimals. The N-body numerical simulations data used here consider the system is equally distributed between the different planetesimals and embryos masses (Jacobson and Morbidelli, 2014). However, the initial typical mass of an embryo before the onset of Jupiter's inward migration is a free parameter. Three different values are tested here: 0.025M ⊕ , 0.05M ⊕ or 0.08M ⊕ (referenced respectively as 0.025, 0.05 and 0.08 in Table 2). The corresponding average values and their 1σ uncertainty are detailed in Table 2.

Results and discussion

Fig. 1 shows how evolves during the accretion of eight representative planets extracted from different N-body simulations: (i) four embryos evolved within a Grand Tack scenario with different initial masses and (ii) four other embryos evolved within a classical scenario for different Jupiter and Saturn orbits. Only a small fraction of the impacts have a significant influence on the fractionation. This effect is a consequence of the impact configuration, especially the impact velocity: the higher the velocity, the more crust is typically ejected. However, successive impacts by planetesimals produce a fractionation that cannot be neglected. The less massive the embryo the easier it is for impactors to erode the surface, leading to a higher fractionation in Sm and Nd in the BSP. The impact history varies considerably from planet to planet and from simulation to simulation. In the case of the Grand Tack scenario, most of the fractionation is produced during the first Myrs, when the system is dynamically excited by the migration of the giant planets. In the classical scenario, the fractionation occurs over the entire accretion history and the final BSP concentration becomes stable later than for the Grand Tack scenario.

On average neither the classical accretion model ( = 0.029 ± 0.015) nor the Grand Tack ( = 0.024±0.013) can account for the eventual superchondritic BSE Sm/Nd ratio ( = 0.06), especially not within the first 30 Myrs of accretion. The final values are presented in the fig. 2 as a function of the final masses of embryos at the end of the 71 N-body numerical simulations. Most of the surviving embryo's masses range between 0.03M ⊕ and 1.4M ⊕ . Smaller final embryos present a much larger range of values, especially in the case of the Grand Tack scenario (fig. 2). This is due to the fact that collisional erosion and consequent chemical fractionation is more efficient for smaller embryos. For the Grand Tack scenario especially, the system is very excited during the first Myrs because of the early migration of Saturn and Jupiter. At this time, the embryos can easily be eroded because of their lower masses and escape velocities. However, for bodies considered as relevant Earth analogs (i.e. with a final mass between 1/2M ⊕ and 1.2 × M ⊕ and final semi-major axis between 0.5 AU and 2 AU), the range of variation of values is narrower than for smaller bodies and almost never reaches the Earth target value of 0.06.

Note however that we have used a partial melting rate of 2.6% as proposed by O' Neill and Palme (2008) to produce the newly formed crust. Such a low partial melting rate is maybe appropriate for the Earth (O'Neill and Palme, 2008) whereas it differs significantly from estimates of differentiated bodies such as Vesta (possibly up to 20%), Ceres or the Moon (between 5% to 10%) [START_REF] Yamaguchi | Metamorphic history of the eucritic crust of 4 vesta[END_REF][START_REF] Ruzicka | Vesta as the howardite, eucrite and diogenite parent body: Implications for the size of a core and for large-scale differentiation[END_REF]. However, increasing the partial melting rate in our model The markers refer to the type of accretionary scenario (circles for the classical scenario and square for grand tack). The colors refer to the final semi-major axis of the embryos. The bottom panel is a zoom of the top panel corresponding to the good Earth analogs. The green line is the average value of for the grand tack simulations, the associated light green area is the 1σ uncertainty. The pink line and light pink area refer respectively to the average and 1 σ uncertainty for the classical accretionary scenario.

would lead only to a decrease in the fractionation of the Sm/Nd ratio (i.e;. lower epsilon value) and could accordingly never account for the observed Sm/Nd value of the BSE. Thus, collisional erosion seems unlikely to explain the possible superchondritic Sm/Nd ratio of the BSE except in marginal cases discussed hereafter. Morevover, the present model assumes striclty chondritic composition for the accreted material which may be invalid especially in the case of embryo-embryo impacts. A higher fractionation could be expected without this assumption because the impacting embryo might have been fractionated in their own earlier accretion history. Considering the present results, estimating the accurate influence of this parameter cannot be done. This will be the object of further investigation in future works. Additionally, O'Neill and Palme (2008) predicted a need of 54% of the crust mass to be ejected in order to account for a 6% fractionation into the Sm/Nd ratio.

This corresponds to about 1.5% of the Earth's mass that should be ejected as crustal material. This is way lower than our estimates of ejected masses than can go up to 17% in most erosive cases, even if the fractionation in Sm/Nd predictions remain under the 6%. For Grand Tack simulations, and if we consider only Earth analogs, the average fraction of ejected mass relative to the final mass of the embryo is 7.5%±6.9%. It should be noted here that O'Neill and Palme ( 2008) assume a single impact on a potential Earth analog that would have its current mass. It implies that the effect of accreted material through time (possibly reducing the enrichment of the crust and BSE in Nd relative to Sm during accretion) is fully neglected. Additionally, melting and re-equilibration processes during accretion are not mention in this model neither [START_REF] O'neill | Collisional erosion and the non-chondritic composition of the terrestrial planets[END_REF]. These different results cannot be directly compared to each other.

Using our results, we also evaluate the hypothesis of enstatite chondrites as the dominant accreting material for Earth as they share strictly similar isotopic compositions with Earth [START_REF] Javoy | The integral enstatite chondrite model of the earth[END_REF][START_REF] Javoy | The chemical composition of the earth: Enstatite chondrite models[END_REF]. In this case, collisional erosion has been proposed to account for the elevated Mg/Si ratio of the BSE compared to enstatite chondrites [START_REF] Boujibar | Cosmochemical fractionation by collisional erosion during the earth/'s accretion[END_REF] if 15% of the early Earth mass is eroded [START_REF] Boujibar | Cosmochemical fractionation by collisional erosion during the earth/'s accretion[END_REF]. Only 9% of the total planetary mass was predicted to be eroded from recent numerical simulations [START_REF] Carter | Collisional stripping of planetary crusts[END_REF] neglecting the impacts of small planetesimals. Our results show that an erosion of 17% of the final embryo's mass can be reached.

Furthermore, this 15% estimation [START_REF] Boujibar | Cosmochemical fractionation by collisional erosion during the earth/'s accretion[END_REF] is based on considering a single impact stripping away that amount of a crust already highly enriched in Si compared to Mg (because of the large difference in the degree of compatibility of Si and Mg at the high pressure and temperature conditions considered for differentiation [START_REF] Boujibar | Cosmochemical fractionation by collisional erosion during the earth/'s accretion[END_REF]). Additionally, the accretion onto its surface of chondritic material is not considered by [START_REF] Boujibar | Cosmochemical fractionation by collisional erosion during the earth/'s accretion[END_REF] and should lead to a lower fractionation (as illustrated in the present work with the Sm/Nd ratio). Thus, preferential erosion of the crust induced by low energy impacts is unlikely to fully account for the observed Mg/Si ratio of the BSE if Earth was mainly accreted from enstatite chondrites. However, an aternative scenario has been proposed by [START_REF] Dauphas | Planetary and meteoritic mg/si and δ30si variations inherited from solar nebula chemistry[END_REF] that may explain the Earth Mg/Si non chondritic ratio by nebular fractionation effects, requiring in that case no or low collisional erosion to happen during accretion.

For Grand Tack simulations (but not classical model simulations), we find a positive correlation between the chemical fractionation in Sm and Nd of the BSP and the timing of the last giant impact (fig. 3). However there is no correlation between the number of giant impacts and the final fractionation. Thus, the observed correlation between the timing of the Moon-forming event and the fractionation is the consequence of the fact that late last giant impacts are due to a lack of dynamical friction in the disk (Jacobson and Morbidelli, 2014), which consequently creates more erosive collisions. An other key factor for this correlation to the timing of the last giant impact is the crustal growing rate as it grows mostly through the input of chondritic material. Indeed, such amount of chondritic material input further leads to a decrease of the incompatible elements concentration within the crust. This means that for every giant impact a newly more incompatiblerich crust is formed. This makes easier the chemical fractionation for the next impacts responsible for a preferential loss of crust relative to the mantle. Considering chemical measurements on terrestrial and lunar samples (for Hf/W, Mg-suite crustal rocks, highly siderophile elements, I-Xe) [START_REF] Chyba | Terrestrial mantle siderophiles and the lunar impact record[END_REF][START_REF] Walker | Highly siderophile elements in the earth, moon and mars: update and implications for planetary accretion and differentiation[END_REF][START_REF] Touboul | Late formation and prolonged differentiation of the moon inferred from w isotopes in lunar metals[END_REF][START_REF] Touboul | Tungsten isotopes in ferroan anorthosites: Implications for the age of the moon and lifetime of its magma ocean[END_REF][START_REF] Touboul | 182w evidence for long-term preservation of early mantle differentiation products[END_REF][START_REF] Touboul | Tungsten isotopic evidence for disproportional late accretion to the earth and moon[END_REF][START_REF] Pepin | Xenon isotope systematics, giant impacts, and mantle degassing on the early earth[END_REF], lunar rocks dating [START_REF] Norman | Chronology, geochemistry, and petrology of a ferroan noritic anorthosite clast from descartes breccia 67215: Clues to the age, origin, structure, and impact history of the lunar crust[END_REF][START_REF] Nyquist | Feldspathic clasts in yamato-86032: Remnants of the lunar crust with implications for its formation and impact history[END_REF][START_REF] Borg | Chronological evidence that the moon is either young or did not have a global magma ocean[END_REF][START_REF] Boyet | Sm-nd systematics of lunar ferroan anorthositic suite rocks: constraints on lunar crust formation[END_REF][START_REF] Carlson | Rb-sr, sm-nd and lu-hf isotope systematics of the lunar mg-suite: the age of the lunar crust and its relation to the time of moon formation[END_REF] and numerical simulations (Jacobson et al., 2014;Jacobson and Morbidelli, 2014), estimates of the age of the moon support a late forming giant impact ranging in between 50

Myr and 150 Myr (after the CAIs formation), with a preferred value around 100 Myr [START_REF] Carlson | Rb-sr, sm-nd and lu-hf isotope systematics of the lunar mg-suite: the age of the lunar crust and its relation to the time of moon formation[END_REF][START_REF] Touboul | 182w evidence for long-term preservation of early mantle differentiation products[END_REF][START_REF] Pepin | Xenon isotope systematics, giant impacts, and mantle degassing on the early earth[END_REF]Jacobson and Morbidelli, 2014). In addition, two recent studies have argued for a younger estimate of the Moon time formation that would be 50 Myr after CAIs [START_REF] Barboni | Early formation of the moon 4.51 billion years ago[END_REF][START_REF] Thiemens | Early moon formation inferred from hafnium-tungsten systematics[END_REF]. These are based on methods (from Lu-Hf system or Hf-W) that have the advantage of presenting relatively low uncertainties. In addition, U-Pb estimates for the age of the terrestrial silicate differentiation is 4480±20 Myr (e.g. [START_REF] Manhes | Lead-lead systematics, the "age of the earth" and the chemical evolution of our planet in a new representation space[END_REF][START_REF] Albarede | Unscrambling the lead model ages[END_REF][START_REF] Allègre | The major differentiation of the earth at 4.45 ga[END_REF], suggesting that the Moonforming impact is unlikely to have occured after 80Myrs. The age of the Moon is still debated, however most estimates point out to a no younger than 50 Myr Moon-formation. In our Grand Tack simulations the superchondritic Sm/Nd ratio of the BSE can be produced only in the case of a last giant impact occurring after 60 Myrs in agreement with most of these recent estimates.

While only a modest fraction (2%) of our sample of Grand Tack simulations have Earth analogs that suffer a last giant impact after 100 Myr, those accretion histories are favored for another reason. Planetesimals that collide with Earth after the last giant impact deliver highly-siderophile elements (HSEs) to Earth's crust and mantle; matching the very low abundance of HSEs in the BSE [START_REF] Walker | Highly siderophile elements in the earth, moon and mars: update and implications for planetary accretion and differentiation[END_REF] requires a late last giant impact (Jacobson et al., 2014). However, after 103 Myr half of the 146 Sm should have decayed into 142 Nd already. This implies that further fractionation between elemental Sm and Nd should have no (or low) effect on the 142 Nd/ 144 Nd.

In that case, Sm/Nd ratio evolution cannot be a proxy for the 142 Nd/ 144 Nd after a few tens of Myr and should only be interpreted as an elemental fractionation. Additionally, if an excess in Sm/Nd ratio is produced after 30 Myr, it would cause an anomaly in 143 Nd/ 144 Nd larger than observed [START_REF] Boyet | 142nd evidence for early (> 4.53 ga) global differentiation of the silicate earth[END_REF]. The addition of the evidences for nucleosynthetic anomalies as responsibles for the excess in 142 Nd/ 144 Nd and the observables on 143 Nd/ 144 Nd ratio lead us to favor a low erosive process during terrestrial planets accretion (i.e. no late last giant impact).

However, it is important to mention here the fact that no erosion is assumed in our model during giant embryo-embryo collisions. This has been done by [START_REF] Carter | Collisional stripping of planetary crusts[END_REF] for the 20 first Myr of evolution of the proto-planetary disk. They find fairly similar results than we do, implying that if the results of both studies are applicable together, then the combined effect of larger impacts and cratering impacts could be sufficient to explain a high Sm/Nd ratio acquired during the 30 first Myr. This adds an other argument to the fact that a low erosive environment should be favored for the accretion process not to produce higher anomalies than any observed. Additionally, as mentioned before, the partial melting rate assumed in our present model is low. A simple way to decrease the fractionation in Sm and Nd even for highly erosive cases could be to significantly increase (by several percents at least) the partial melting rate when forming new crusts. The effects of the partial melting rate are presented in the supplementary materials. Nucleosynthetic anomalies has been emphasized as an explanation for the excess in 142 Nd of the Earth [START_REF] Burkhardt | A nucleosynthetic origin for the earth's anomalous 142 nd composition[END_REF][START_REF] Bouvier | Primitive solar system materials and earth share a common initial 142 nd abundance[END_REF]. That case would require at most 2% of chemical fractionation of the Sm/Nd ratio induced by collisional erosion in order to maintain the initially implanted Sm/Nd ratio in planetary bodies by nucleosynthetic anomalies [START_REF] Burkhardt | A nucleosynthetic origin for the earth's anomalous 142 nd composition[END_REF][START_REF] Bouvier | Primitive solar system materials and earth share a common initial 142 nd abundance[END_REF]. Therefore, the dynamical and geochemical scenarios leading to such a low final fractionation need to be explored and might provide another set of constraints on the initial conditions of the solar system.

Impacts with a high velocity (higher than 20 km/s at encounter) are highly erosive [START_REF] Svetsov | Cratering erosion of planetary embryos[END_REF][START_REF] Shuvalov | Atmospheric erosion induced by oblique impacts[END_REF]. However, they represent only a small fraction of the total. Overall, the impact velocity averaged over 200 Myrs and over the whole disk is higher for non Grand Tack scenarios and for a steeper disc surface densities. Additionally, since the embryos grow faster in the frame of the Grand Tack than in the classical scenario, is it made earlier harder to eject material. As a consequence of these two aspects, the Grand Tack scenarios generally induce a slightly lower fractionation among the different embryos, but the timing of the onset of the Grand Tack might also influence the fractionation yields. Erosion may be more efficient when the system is excited early as embryos have smaller sizes than in a later case. Therefore, future works could constrain the timing of the onset of the Grand Tack (if it existed) in order to preserve the original Sm/Nd fractionation implanted by nucleosynthetic anomalies or, in opposition, to induce efficient stripping of the crust.

Modeling the effects of collisional erosion for other RLE will eventually provide further insights on whether a given terrestrial accretion history can be compatible with observed nucleosynthetic anomalies in chondritic components.

Outcomes dependency on the dynamical scenario

We emphasized that in the case of the remixing scenario reliable on, the grand tack tends to produce a slightly lower final fractionation for relevant Earth analogs compared to the classical scenario (see tables 1 & 2). However, in the case of smaller embryos, the range of final values is larger in the grand tack dynamical scenario than in the classical dynamical scenario. Collisional erosion and consequent chemical fractionation is more efficient for smaller embryos. Within a Grand Tack dynamical scenario, the embryos reach a Earth-like mass faster (within 20-25Myr) and become accretive faster, limiting sooner the erosive power of collisions. The erosive power of collisions is improved at earlier times in the grand tack dynamical scenario since the system is excited by the migration of the giant planets Jupiter and Saturn. This combined effect is responsible for the differences between the dispersion observed in fig 2 for Classical Grand Tack scenarios. However, besides the higher dispersion for the Grand Tack scenario, the final average fractionation is lower for the Grand Tack than for the classical scenario. This is a surprising result that can be explained by the fact that in the Grand Tack scenario the embryos grow much faster.

After 20 Myr the embryos ending as good Earth analogs have already reached most of their total masses. From that point, it becomes complicated to produce enough erosion to induce significant fractionation (because of (1) the composition of the crust that is most of the time close to chondritic because of accretion and of (2) the escape velocity that tends to limitate the erosive power of a given impact).

Conclusion

We have quantified the efficiency of collisional erosion during Earth accretion for two dynamical scenarios ( classical scenario and Grand Tack scenario) and computed the resulting BSE Sm/Nd ratio using a combination of analytical modeling of cratering processes [START_REF] Housen | Ejecta from impact craters[END_REF][START_REF] Svetsov | Cratering erosion of planetary embryos[END_REF], simple mass balance estimates (O'Neill and Palme, 2008) and N-body numerical simulations [START_REF] Raymond | Building the terrestrial planets: constrained accretion in the inner solar system[END_REF]Jacobson and Morbidelli, 2014). It is shown (Fig. 1) that only a small amount of impacts (compared to the total) is responsible for a significant fractionation of the BSP Sm/Nd ratio, notably for the classical scenario. During the Grand Tack scenario, most of the fractionation is acquired during the first Myrs (owing to the low escape velocities of small embryos and the rapid decrease of the planetesimal population) of the system evolution while the classical scenario presents a more progressive fractionation trend, spreading all over the accretion history. We find, that a maximum of 17% of the proto-planet's mass erosion can be reached for the most erosive cases. In average, over all surviving embryos, only 5% of the proto-planet's mass is ejected as crust. Even though this amount of ejected crust is large, we show that neither the classical model, nor the Grand Tack model can always account, for the 142 Nd possible excess in terrestrial rocks. However, we find a strong correlation between the fractionation in Sm and Nd with the timing of the Moon-forming event in the case of the Grand Tack scenario. Especially, a large fractionation (about 5%) is reachable for a Moon-forming impact (last-giant impact) occurring after 50 Myr after CAIs, that could account for the proposed off-set in 142 Nd. New evidences of nucleosynthetic heterogeneities in primitive planetary components obviates the need for a superchondritic Sm/Nd ratio of the BSE and impose limited loss of crust preferential erosion during accretion. Such observations seem to challenge the most eroding scenario explored in this of the mean values in the BSP after the growth of the embryos considered as Earth analogs (with a final mass comprised between 1/2M ⊕ and 1.2M ⊕ and a semi major axis comprised between 0.5 AU and 2 AU) in a classical accretion scenario. The standard deviation associated to each value is also available in the table . (1) refers to the mixing scenario 1, for wich a full reequilibration between crust and mantle is assumed after an impact.

(2) denotes the opposite scenario for which there is no reequilibration at all, the giant impact are thus ignored and for the small impacts, it is assumed that everything happens in the crust, remaining the mantle intact. Finally, (3) refers to the intermediate scenario. Giant impacts are treated in agreement with scenario (1) and small impacts are treated the same way than in scenario (2). The values are given for each kind of simulations, with different initial conditions. A quick description of each type of N-body simulation is given. a j and as refer respectively to the Jupiter and Saturn semi-major axis. e j and e S denote the eccentricities of Jupiter and Saturn respectively. χ is used to varie the effect of the disc surface density. The disc surface density can be expressed as:

(r) = 1 r 1AU -χ . The r -1 simulations formed slightly fewer planets, contained less total mass in planets and had longer formation timescales for Earth for the final systems than the r -3/2 simulationsRaymond et al. (2009). "MMSN" refers to the "Minimum Mass Solar Nebula" modelWeidenschilling (1977).

work corresponding to a Grand Tack combined with a late moon forming impact. However, it is important to keep in mind the assumptions of the model presented here. Notably, a low fractionation could be achieved even in an erosive dynamical scenario if the partial melting rate for forming the crust is much larger or under alternative crust compositions as suggested by [START_REF] Carter | Collisional stripping of planetary crusts[END_REF]. Further works should investigate systematically the effect of collisional erosion on the budget of other refractory and lithophile elements and provide an acceptable set of conditions for the dynamical accretion of the Earth that could account for the observed compositions of these elements in the BSE. (Jacobson and Morbidelli, 2014). The standard deviation associated to each value is also available in the table . (1) refers to the mixing scenario 1, for wich a full reequilibration between crust and mantle is assumed after an impact. (2) denotes the opposite scenario for which there is no reequilibration at all, the giant impact are thus ignored and for the small impacts, it is assumed that everything happens in the crust, remaining the mantle intact. Finally, (3) refers to the intermediate scenario. Giant impacts are treated in agreement with scenario (1) and small impacts are treated the same way than in scenario (2). The values are given for the three different sets of initial conditions concerning the initial mass of an embryo (corresponding to the mass that basically has an embryo when the inward migration of Jupiter begins). M ⊕ refers to the actual Earth's mass. 4.3. Correlation with the last giant impact is not due to the number of giant impacts Figure 6 shows 1) on the left panel the number of impacts suffered by the embryos surviving to Grand Tack simulations as a function of the timing for the last giant impact; and 2) on the right panel, the epsilon value for epsilon as a function of the number of giant impacts for Grand-Tack surviving embryos. We see no correlation between any of these values. The only thing that can be pointed out by this figure is the dispersion after a few Myr that becomes extremely large. The denotes the fact that the correlation observed for the Grand Tack scenario betweem the fractionation of the Earth analogs and the time they registered their last involvement into a giant impact is not linked to the number of giant impacts they went through. The correlation is a correlation to the timing of the last chemical re-equilibration and not on how often a re-equilibration has occured.

Influence of the choice for the cratering model

The scaling laws for rocky targets [START_REF] Holsapple | A crater and its ejecta: An interpretation of deep impact[END_REF] has been chosen when integrated over all impact angles [START_REF] Svetsov | Cratering erosion of planetary embryos[END_REF] for the description of the eroded mass. However, [START_REF] Svetsov | Cratering erosion of planetary embryos[END_REF] explores and improves an other scaling law that has been suggested [START_REF] Shuvalov | Atmospheric erosion induced by oblique impacts[END_REF] based on simulations of impacts in the presence of planetary atmospheres. This very last description is tested in this study, to explore the influence that could have the choice of the cratering model used. This formulation of eroded mass is described before discussing the associated results.

The escaped mass is estimated as [START_REF] Svetsov | Cratering erosion of planetary embryos[END_REF]:

M escaped = m imp C(V imp ) V 2 imp -C 1 (V imp )v 2 esc v 2 esc V imp 20v esc a , (8)
with m imp and V imp being respectively the impactor mass and the impact velocity. Here, v esc is assumed to be the two-body escape speed:

v esc = 2G(m tar + m imp ) R tar + R imp , (9)
with m tar the target mass, R tar and R imp are respectively the target and the impactor radius. The different dimensionless coefficients needed for the calculation of M escaped , are defined as follows:

C 1 (V imp ) = 0.7 + (V imp /20km/s) 2 , (10)
a = 0.15 -0.0003(V imp -15km/s) 2 , (11)
And, concerning C (wich is also a dimensionless number):

• if V imp ≥ 20 km/s, then C(V imp ) = 0.02, • if V imp = 10 km/s, then C(V imp ) = 0.035, • if V imp ≤ 5 km/s, then C(V imp ) = 0.07,
• if 5 km/s < V imp < 10 km/s, then C(V imp ) = 0.07 + [(0.035 -0.07)/5](V imp -5),

• if 10 km/s < V imp < 20 km/s, then C(V imp ) = 0.02 + [(0.02 -0.035)/10(V imp -10).

The two different scaling laws are tested over all the collisional histories of the different embryos issued from the numerical simulations. Depending on the range of impact velocity, either one model overestimates the escaped mass compared to the other, either it underestimates it. However, the difference between the two different models is not critical. For both models the final values are similar within a few percents and show a below 0.05.

Three sets of assumptions for mass balance modeling

We present here the description of the two additional mixing scenarios for chemical mass balance resulting from impacts. These are end-members meant to provide lower-bound and maximumbound fractionation.

Scenario 1: "the full remixing" model

The BSP is molten and the newly formed crust reequilibrates with the mantle after each impact.

Giant impacts are considered as perfect mergers, and a full reequilibration between crust and mantle is still assumed. Expressions 1 and 2 are used for estimating the resulting BSP concentration in a given element M compared to its previous value.

Scenario 2: "no remixing" model

Only small impacts are taken into account for estimating the masses of eroded crust and accreted chondritic material. All accreted material is assumed to fall onto the surface so that only the crustal composition is modified while mantle composition (and its mass) remains constant. No new crust is formed. Accordingly, the new composition of the crust in M at a given time i is expressed following:

C i M crust = C i-1 M crust m i-1 crust + C chondritic M m i accreted -C i-1 M crust m i ejected m i-1 crust + m i accreted -m i ejected , (12)
where m crust is the crustal mass, C chondritic M is the chondritic concentration for the element M , m accreted is the accreted mass (from impactor) and m ejected is the escaped mass.

In this scenario, the giant impacts are not treated at all in terms of mass balance. It means that when a giant impact is recorded into N-body collisions file, our model is ignoring it, it behaves exaclty as if proto-Earth remained perfectly intact. No new crust is formed from partial melting.

It is far from realistic scenario, but is meant to represent a end-members.

The scenario 1 (full remixing) and 2 (no remixing) represent unrealistic end members. As a consequence, previously, only the most realistic case, the scenario 3 (remixing only for giant impacts) has been presented. The results concerning the other two cases are presented here in details (7 and 8). The values of range from 0.029 ± 0.003 to 0.05 ± 0.02 for the classical scenario in the case of the mixing scenario (1) (fig. 7) which is clearly the most favorable to fractionate the Sm/Nd ratio with respect to chondritic composition. In this case, a new crust is formed after each impact and Sm and Nd are re-distributed according to their partition coefficients between the solid mantle and the liquid crust. This leads to an increase of the Nd/Sm ratio of the newly formed crust at each step of the simulation as Nd is more incompatible than Sm. However, this scenario is not appropriate as low energy events are not likely to produce large amounts of melt and can be consider as an upper bound for the final modeled Sm/Nd ratio of the bulk silicate Earth. In the case of the mixing scenario (2) (fig. 8, takes values between 0.0037 ± 0.0004 and 0.009 ± 0.006 for a classical accretion scenario. In this scenario the crust is refertilized in chondritic Sm/Nd after each impact and absence of melting leads to the lowest chemical fractionation of the Sm/Nd value.

The crust mass increases and becomes more and more chondritic during the course of accretion.

This scenario can be considered as the opposite as scenario 1, and provides a lower bound for values. The global trand of those observation is found also concerning the grand tack simulations.

However, the final fractionation values are much more spread out. Especially, concerning the most favorable case to fractionation: full remixing; the values of the fractionation are very high in average (0.12 ± 0.03). Such fractionation is unexpected as we do not sample any extraterrestrial body or meteorite that would present such values of Sm/Nd ratio. However, since the grand tack has a powerful effect of fractionation during the first Myrs of accretion, is the crust is enriched in Nd at each step because of a new crust-forming event du to complete melting of the silicate Earth, it is clear that the total fractionation will take unrealistically large values. As a conclusion, those two end members gives us clues about the very lower and upper bound of the fractionation, but they still quite unrealistic and they are not strong enough to allow a better approach and constrain the Sm/Nd ratio evolution over accretion histories. That is the reason why the intermediate case

(3) -remixing only for giant impacts -has been chosen alone to follow this fractionation. It provides a good first approximation of the fractionation as a response to collisional erosion.

  /Nd ratio of the BSP at any time t of the simulation and Sm N d ini BSP the Sm/Nd ratio of the BSP at time zero of the simulation (i.e. chondritic value).

O

  'Neill and Palme (2008) proposed a mass balance for a two-stage model of crustal erosion: with (1) crust-forming process and (2) collisional erosion. The first stage assumes the formation of the embryo's crust with fractionnation between compatible and incompatible elements. The newly formed proto-crust is enriched in incompatible elements relatively to the residual mantle. In the calculation proposed by O'Neill and Palme (2008), f 1

  Tack) with their associated results for our model in terms of fractionation in Sm and Nd.

Figure 1 :Figure 2 :

 12 Figure1: Evolution of the fractionation parameter as a function of time for 8 relevant Earth analogs accretion histories: four evolving within a classical accretion scenario with different orbits of Jupiter and Saturn and four evolving within a Grand Tack scenario. They respectively end up with masses of 0.6M ⊕ , 0.69M ⊕ , 1M ⊕ , 1.01M ⊕ , 1.2M ⊕ , 1.2M ⊕ , 1.01M ⊕ , 0.86M ⊕ and semi major axis of 1.16 AU, 0.6 AU, 0.76 AU, 0.62 AU, 0.72 AU, 0.9 AU, 0.95 AU and 0.71 AU, from the most fractionated to the less fractionated. A zero fractionation value refers to a chondritic composition. Time zero is the beginning of the simulation:3 Myr after CAIs. In the case of the Grand Tack scenario, time zero corresponds to the very beginning of the Jupiter's migration. Stars refer to the last giant impact for Grand Tack and classical scenarios.

Figure 3 :

 3 Figure 3: Sm-Nd fractionation with respect to chondritic Sm/Nd values of the different Earth analogs at the end of Grand Tack N-body simulations reported as a function of the timing of the last giant impact suffered. The markers sizes refer to the final mass of an embryo. The purple area refers to the time laps of the moon forming impact.The red curve is our quadratic fit for the relation between the fractionation value and the timing of the last giant impact, the light red envelop defines the 1σ error associated.

Figure 4 :

 4 Figure 4: Partial melting rate influence on the final embryos fractionnation evidenced by representation of mean values for the different dynamical scenarios in a classical accretion disk. Solid lines represent the model presented in the main text while dashed lines represent the results for other α tested here. The colors and markers refer to the mixing scenarios.

Figure 5 :

 5 Figure 5: α (SFD slope) influence on the final embryos fractionnation evidenced by representation of mean values for the different dynamical scenarios. Solid lines represent the model presented in the main text while dashed lines represent the results for other α tested here. The colors and markers refer to the mixing scenarios.

Figure 6 :

 6 Figure 6: (1) Left panel. Number of impacts suffered by the embryos surviving to Grand Tack simulations as a function of the timing for the last giant impact. (2) Right panel. value for epsilon as a function of the number of giant impacts for Grand-Tack surviving embryos. In both figures the sizes of the circles stand the final masses of the surviving embryo.

Figure 7 :

 7 Figure 7: (Sm-Nd fractionation) with respect to chondritic Sm/Nd values of the different surviving embryos at the end of classical and Grand Tack N-body simulations reported as a function of their final masses. The remixing model adopted here is the most favorable to fractionation (scenario 1: full remixing). The markers refer to the type of accretionary scenario (circles for the classical scenario and square for grand tack). The colors refer to the final semi-major axis of the embryos. The red box shows the Earth analogs.

Figure 8 :

 8 Figure 8: (Sm-Nd fractionation) with respect to chondritic Sm/Nd values of the different surviving embryos at the end of classical and Grand Tack N-body simulations reported as a function of their final masses. The remixing model adopted here is the less favorable to fractionation (scenario 2: no remixing). The markers refer to the type of accretionary scenario (circles for the classical scenario and square for grand tack). The colors refer to the final semi-major axis of the embryos. The red box shows the Earth analogs.

Table 1 :

 1 Table

			cjs1	cjs15	cjsecc15	jsres	jsresecc	eejs15	ejs1	ejs15
	(1)	¯ mean fraction-ation	0.029	0.04	0.04	0.041	0.05	0.05	0.04	0.05
		Standard deviation 0.003	0.01	0.01	0.009	0.02	0.015	0.016	0.02
	(2)	¯ mean fraction-ation	0.007	0.0049	0.004	0.0037	0.004	0.005	0.009	0.009
		Standard deviation 0.002	0.0008	0.001	0.0003	0.0015	0.002	0.006	0.006
	(3)	¯ mean fraction-ation	0.020	0.025	0.021	0.025	0.030	0.028	0.03	0.03
		Standard deviation 0.002	0.004	0.006	0.005	0.008	0.008	0.01	0.01
	Grand tack ? no	no	no	no	no	no	no	no
	Description	Circular Jupiter and Saturn. Mutual in-clination = 0.5 • . a j =5.45AU and a s =8.18AU. χ = 1	Same as but χ = 3/2 cjs1 with (MMSN model)	cjs eccentric with orbits. Jupiter and Saturn placed at their cjs semimajor axis 5.45AU and 8.18AU with e j =0.02 and e s =0.03 and a mutual in-clination of 0.5 •	Jupiter and Saturn in resonance, placed in their mu-tual 3:2 mean mo-tion res-onance. a j =5.43AU, a s =7.30AU and e j =0.005, e s =0.01 with a mu-tual incli-nation of 0.2 •	Jupiter and Saturn in resonance on eccen-tric orbits. Same as jsres but with e s = e j =0.03	Extra centric Jupiter and Saturn. ec-Placed at their actual semi major axis but with high eccentric-ity orbits. a j =5.25AU, a s =9.54AU, 1.5 • clination is mutual in-e s =0.1. e j =0.1 and	Eccentric Jupiter and e s =0.06 e j =0.05, Satur. Ini-tial orbits around ac-tuals. Same semi ma-jor axis and mutual inclina-in eejs15. tion than	Same as but χ = 3/2 ejs1 with instead of 1.

Table 2 :

 2 Table of the mean values in the BSP after the growth of the embryos considered as good Earth analogs in a grand tack accretion scenario

	0.025	0.05	0.08

Acknowledgement J.S. acknowledges support from the French National Research Agency (ANR project VolTerre, grant no. ANR-14-CE33-0017-01) And Institut Universitaire de France. Parts of this work were supported by the UnivEarthS Labex programme at Sorbonne Paris Cité (ANR-10-LABX-0023 and ANR-11-IDEX-0005-02).

Appendice

Influence of the partial melting rate

When forming a new crust, the mantle partial melting rate is determinant for the repartition of the chemical elements between crust and mantle. As it is evidenced in the eq.1, the higher the partial melting rate f 1 p-c , the lower the concentration in Nd within the crust. As a consequence, the preferential loss of Nd compared to Sm is lower for a higher partial melting rate. A given amount of eroded crust leads to a lower fractionation in the Sm/Nd ratio for a higher f 1 p-c . The results for three different partial melting rates are presented in fig. 4. This parameter clearly has a non neglectable influence on the final results; however, since when it increases the fractionation decreases. Accordingly, f 1 p-c has been chosen low (2.6%) to estimate an upper bound of possible impact-induced fractionation.

Influence of the Size-frequency distribution

The final results concerning the evolution of the fractionation value ( ) over time have been computed for 3 other α: 2.5, 3 and 4.1. As the concern is only the influence of the slope over the fractionation, we present it only for the different classical simulations, that in average, produce a higher fractionation value than the Grand Tack scenario. They are presented in fig. 5. The solid colored lines represent the results presented above (with α = 3.5), the dashed lines represents the results of the three different mixing scenarios for α = 2.5 and the last represents the results for α = 4.1. The higher α (i.e. more mass in the smaller bodies), the lower . It may be expected that earlier in the planetary formation history, the slope would have been slightly higher [START_REF] Tanaka | Steady-state size distribution for the self-similar collision cascade[END_REF]. As a consequence, the fractionation estimated in this study could be overestimated.

However, the difference between the different models from different α are very low, so the choice of this parameter is not critical.