Paul Noël

Maxen Cosset-Cheneau

Victor Haspot

Vincent Maurel

Christian Lombard

Manuel Bibes

Agnès Barthelemy

Laurent Vila

Jean-Philippe Attané

Negligible thermal contributions to the spin pumping signal in ferromagnetic metal-Platinum bilayers

Spin pumping by ferromagnetic resonance is one of the most common technique to determine spin hall angles, Edelstein lengths or spin diffusion lengths of a large variety of materials. In recent years, rising concerns have appeared regarding the interpretation of these experiments, underlining that the signal could arise purely from thermoelectric effects, rather than from coherent spin pumping. Here, we propose a method to evaluate the presence or absence of thermal effects in spin pumping signals, by combining bolometry and spin pumping by ferromagnetic resonance measurements, and comparing their timescale. Using a cavity to perform the experiments on Pt\Permalloy and La0.7Sr0.3MnO3\Pt samples, we conclude on the absence of any measurable thermoelectric contribution such as the spin Seebeck and anomalous Nernst effects at resonance.

I Introduction

Spinorbitronics is based on the interconversion of charge currents into spin currents, by the spin Hall Effect in bulk materials or by the Edelstein effect at surfaces and interfaces. Determining the spin to charge current conversion efficiency, i.e., the spin Hall Angle or the inverse Edelstein length, and the spin diffusion length, is a key point to understand experimental results and develop applications. Materials with large spin Hall angles are indeed required for a variety of foreseen applications such as 3-terminal SOT MRAM [START_REF] Prenat | Spintronics-Based Computing[END_REF], Magnetoelectric Spinorbit Logic [START_REF] Manipatruni | [END_REF] or Terahertz emitters [3]. Since the early 2000's several techniques have emerged to evaluate this conversion efficiency, using lateral spin valves [4 ] , Spin Seebeck measurements [5 ] , spin pumping FMR (SP-FMR) [6 ] , magneto-optical Kerr effect measurements [7], etc. The spin pumping FMR technique has been widely used to evaluate the spin Hall angles and inverse Edelstein lengths of a large numbers of materials, including heavy metals [6,8,9], semiconductors [10], Rashba interfaces [11,12] and topological insulators [13,14,15,16]. The reason of such a wide use relies on the compatibility of SP-FMR with any multilayer stack containing a ferromagnet, and the absence of any complex and costly nanofabrication process. Moreover ferromagnetic resonance was an already widely used spectroscopy technique to evaluate the dynamical and non-dynamical properties of ferromagnetic thin films Even for a widely studied material as Platinum the estimated values of spin diffusion length and spin Hall angle determined by different techniques spread over more than one order of magnitude,with spin diffusion length ranging from 1.2 nm [17] to 11 nm [18] and spin Hall angle from 1.2% [19] to 38.7% [20]. This large discrepancy can be partially explained by differences in Pt resistivity or accounted for by interface-related phenomena such as the spin memory-loss [9,21 , 22 , 23 , 24 ] , but it remains mostly unexplained [25]. In this particular context, concerns regarding the reliability of the SP-FMR technique have been pointed out. A thermal gradient could indeed arise at the ferromagnetic resonance, due to the energy absorption in the ferromagnetic layer [26 , 27 , 28 , 29 ] and give rise to several thermoelectric and spin-caloritronics contributions  in particular the anomalous Nernst effect (ANE) and the spin Seebeck effect (SSE)  that would add up to the spin pumping inverse spin Hall effect (ISHE) signal. In this picture, the signal would thus be due to a combination of the ISHE signal, the spin rectification effects (SRE) and thermal effects [30].

While the separation of ISHE from SRE has already been vastly discussed and can be achieved from the angular dependence in different measurement geometries [30,31,32,33], disentangling the ISHE signal from thermal effects remains an open question. While the ordinary Seebeck effect (OSE) and ordinary Nernst effect (ONE) contributions may be extracted from angular dependences, it is not the case of the ANE and SSE contributions, which possess an angular dependence similar to that of the ISHE signal. Recent analysis of spin-pumping FMR results have even been based on the hypothesis that the observed signals are dominated by the SSE [27]. In that case the nature of the spin pumping signal would be incoherent -thermal-and the extensively used coherent spin pumping model would strongly misestimate the injected spin current [34]. It is therefore possible that due to a large contribution of incoherent spin pumping, the estimation of the Spin Hall Angle by spin pumping FMR in Pt is inaccurate. More generally, a contribution of ordinary thermoelectric effects such as the ordinary Nernst effect is also a possible explanation to the discrepancy between the lack of spin charge conversion in Silver Bismuth bilayer measured by means of Longitudinal Spin Seebeck Effect (LSSE) [35], and the large one measured by means of spin pumping [11,12]. Therefore, estimating possible thermoelectric or spincaloritronics contributions is of high importance to obtain an accurate evaluation of the spin charge conversion efficiency.

II Experimental procedure

Figure 1 a) depicts the dynamical spin injection process as described by Tserkovnyak et al. [36], which is the model used to analyze the signal observed in spin pumping FMR measurements. As suggested by Yamanoi et al. [27], the additional dissipation at the FMR could lead to the appearance of a voltage along the x direction. The absorption at resonance would lead to a temperature increase of the ferromagnet, and thus to a thermal gradient perpendicular to the layers. This thermal gradient would lead to the injection of a pure spin current along Z towards the non-magnetic material, converted by ISHE into an electric field along X through a process known as the longitudinal Spin Seebeck effect as seen in figure 1.b. Owing to the existence of a thermal gradient, the anomalous Nernst effect in the FM layer could also appear, creating an electric field along X as depicted in figure 1c. We propose to test these hypothesis in two multilayers. The first one is a Pt\Permalloy bilayer, archetypal of spin pumping ISHE experiments [6,8,17,19], with a large ANE coefficient in Permalloy (Py) [37], and the second one is a La0.7Sr0.3MnO3\Pt bilayer, in which a large SSE contribution is expected [38]. The characteristic timescale of the FMR spin injection mechanism is the FMR precession period, which is of the order of the nanosecond. But the temperature increase timescale, the time needed to reach a thermal equilibrium, is of several seconds [39 , 40 ] : thermal and non-thermal effects have different dynamics. We thus propose a technique that can be adapted to any SP-FMR experiment to disentangle the two mechanisms, by measuring the time dependence of the spin pumping signal and of the temperature increase.

III Results and Discussion

We performed SP-FMR measurements on a SiO2\\Pt( 10)\Py( 20) multilayer (the numbers in parenthesis represent the thickness in nanometers), on a 2.40.4 mm² structure positioned in a Bruker MS5 loop gap cavity. We performed a FMR measurement at different sweeping rates, at a power of 100 mW. The scheme of the measurement is shown in figure 2 a), and consists in the measurement of the voltage at the ferromagnetic resonance. The DC magnetic field can be applied in the plane of the sample, and perpendicularly to the electrical contacts. This configuration is the parallel configuration. The sample is then turned by 180°, in a position corresponding to the antiparallel configuration. As seen in figure 2 b) and c) in both the parallel and antiparallel configurations, the signal is symmetric and independent of the sweeping time. In both cases we subtracted the offset signal and divided by the square of the radiofrequency magnetic field hrf to have comparable results. The amplitude of the radiofrequency field was determined by measuring the Q factor with the sample placed inside the cavity using the following equation: hrf 2 =4PQ/500, with P the microwave power in Watt [33]. 10)\Py( 20) sample (top view). b) Signal obtained in the in-plane parallel configuration for various sweeping rates, normalized by the square of the rf magnetic field ℎ 𝑟𝑓 2 for a power of 100mW. The Q factor was 690 and the symmetric voltage of 14.7 µV before normalization. c) Similar measurement in the inplane antiparallel configuration. The Q factor was 537 and the symmetric voltage of -11.8µV before normalization d) Out-of-plane angular dependence of the spin signal fitted using the ISHE angular dependence provided in ref [33] e) Power dependence of the symmetric part of the signal as a function of the microwave power in the parallel configuration, the Q factor was 510.

Regarding the possible contribution of the spin rectifications effects in Py, the out-of-plane angular dependence has also been performed (cf. figure 2 d). The obtained symmetric signal can be fitted with the ISHE angular dependence model described in reference [33] as sin(M) with M being the magnetization angle with respect to the out of plane direction. This also excludes any contribution of the ordinary Seebeck effect, which would be field independent, and of the Nernst effect, which would depend on the applied field perpendicular to the thermal gradient. The signal is also linear with the power (cf. figure 2 e) indicating a negligible change of magnetization when increasing Power.

The signal possesses the ISHE angular dependence, and there is no trace of thermal drift, which implies that if there is a thermal component to the signal, a steady state with the thermal gradient has to be reached in a characteristic time well below one second. Let us now evaluate this characteristic time. A temperature increase can occur at the ferromagnetic resonance, due to the increased microwave absorption at resonance [26,39,40,41]. To evaluate the time dependence of this effect, we adopt the measurement scheme shown in figure 3 a), where the field is applied out of plane to avoid ISHE or SRE voltage contribution. We used a fixed DC current of 1 mA, fixed power of 100 mW and measured the change of resistance at resonance, known as the bolometric effect. As can be seen on figure 3 b) we observe an increase of resistance at resonance; more importantly, this increase is highly dependent on the field sweeping rate. The resistance increases from 7.2±1 mΩ for a sweeping rate of 12 mT/s, to 28.5±1 mΩ for a sweeping rate of 0.18 mT/s. The result is in stark contrast with fig. 2 where the signal is independent on the sweeping rate. The temperature increase characteristic time is thus of several seconds, as the time spent near resonance at a sweeping rate of 0.18 mT/s is of 20s for a linewidth of 3.5 mT. This timescale is similar to what has been observed in previous Electrically Detected FMR experiments [39,40]. A similar result was obtained with in plane field.

We estimate the corresponding temperature increase using the temperature dependence of the resistance of the sample measured inside the cavity (cf. figure 3c). The linear behavior of the resistance leads to a temperature dependence of 96±2 mΩ/K in a range from 100K to Room temperature. The temperature increase as a function of the sweeping rate is shown in figure 3d. The maximum temperature increase is of 297±10 mK for the slowest sweeping rate, and of 75±10 mK for the fastest. The temperature increase is thus found to be strongly dependent on the sweeping time, the temperature stabilization is not reached even after several seconds near resonance. Therefore, any effect originating from a temperature change should vary with the sweeping time. The spin signals measured in the configuration of fig. 2 being totally independent of the sweeping time, we can conclude that in SiO2\\Pt( 10)\Py( 20) the longitudinal spin Seebeck effect and anomalous Nernst effect are negligible, and that the observed signals are due to coherent spin pumping. This comparative measurement can also be performed using a coplanar waveguide setup where bolometric effects with a time constant of several seconds were previously observed [40]. Some hundreds of mK might appear to be a small temperature increase when compared to the several Kelvins usually used in LSSE measurements. Nonetheless due to the large thickness of the substrate compared to the Ferromagnetic thin films, the temperature difference between the bottom and the top of the ferromagnetic layer is usually below 1mK in LSSE measurements [ 42 ]. For the ordinary Nernst contribution a temperature difference of only a few mK could also give rise to signals of some µV [43].

In order to verify this lack of thermal contribution in Pt we also performed a combined bolometric and SP-FMR measurements on a LSAT\\La0.7Sr0.3MnO3(13.8)\Pt(8.2) sample, measured along the [100] direction. La0.7Sr0.3MnO3 (LSMO) possesses a high resistivity compared to Permalloy and Platinum moreover the LSMO\Pt structure is expected to possess a smaller ANE coefficient but a larger SSE contribution than Py\Pt as demonstrated in Longitudinal Spin Seebeck experiments [38]. Therefore the possible contribution of SSE in this multilayer is expected to be enhanced compared to Pt\Py. In figures 4b and 4c similarly to the case of Pt\Py we can see that the thermal equilibrium is still not reached even for the slowest sweeping rate, the total temperature increase is of comparable amplitude and up to 199±3mK. As can be seen in figure 4e and4f, the obtained spin signal is independent on the sweeping rate. Here again, this shows that in this system the ANE and SSE contributions are negligible compared to the spin pumping ISHE signal. As already observed in LSMO layer, in the out of plane configuration the resonance peak is slightly asymmetric, because of magnetic homogeneities [44], this leads to an asymmetry of the bolometric response as observed in figure 4 b). This is not the case in Permalloy, because the small grain size leads to a strong exchange narrowing [45] and thus to a small inhomogeneity. This inhomogeneity is not observed in plane and ISHE signal in figure 4 e) and f) are consequently symmetric.

The NM and FM stacking order is inverted in the Pt\Py sample, which is why the spin signal is of opposite sign when compared to LSMO\Pt and to previous results on Co\Pt [9], as expected for ISHE symmetries [46]. The normalized ISHE signal is the ISHE voltage divided by the square of the rf field, the width and the total resistance of the device. The obtained values are of 0.78 mV.G -2 Ω -1 m -1 in LSMO\Pt and 1.11 mV.G -2 Ω -1 m -1 in Pt\Py, similar to the value of 0.85 mV.G -2 Ω -1 m -1 to 1.13 mV.G -2 Ω -1 m -1 that was previously reported in SiO2\\Co\Pt of similar thicknesses at X-band [9] indicating a similar injected spin currents in these three structures. Another experiment can be done to demonstrate the absence of thermal contribution to the spin pumping signal. The sample was placed in the parallel configuration and the external field was swept as fast as possible from 20 mT below the resonance to the resonance field Hres at a fixed rf power of 100 mW. The sweeping rate in this experiments was limited to 1 mT/s to avoid a large overshoot of the field when stopping at the resonance field and thus allowing a fast stabilization of the field comparable to our time resolution.

In a first step, a 5 mA current is applied in the sample, so that the signal variations correspond mostly to resistance variations. The voltage resulting due to Ohm's Law is of 5 µV/mΩ using a current of 5mA while the total spin pumping signal is of around 10 µV at a power of 100 mW. The results, shown in figure 5, exhibits a resistance increase when reaching the resonance field. The time constant of the temperature increase is of around 10 seconds. In a second step, the same experiment is performed in the open circuit conditions commonly used for spin pumping experiments. In that case, the maximal signal is obtained immediately after reaching the resonance field. This implies that the signal measured in open circuit conditions is not linked to the slow temperature increase at resonance but to the fast dynamical spin injection mechanism. In this study we focused on the possible thermal effects contributions at resonance. As previously shown there is also in our measurements an offset signal (existing at and out of resonance), that is usually subtracted, and which can also be of thermal origin [29]. The change of temperature due to microwave absorption out of resonance is orders of magnitude larger than that due to the resonance [40], and it can lead to larger thermal gradients. In order to study that point we used a technique very similar to that shown in figure 5, but out of resonance.We thus compared the evolutions of the resistance of the sample and of the offset voltage when increasing the power entering the cavity. We observed a temperature change more than one order of magnitude larger than the change observed at resonance at a similar power, and a similar evolution for both the resistance and offset voltage in both samples. These observations confirm that the offset signal has a thermal origin, and that actually the temperature increase of the sample is mostly due to microwave absorption, independently of the resonance. Note that this signal contribution, which exists both at and out of resonance, does not contribute to the spin signal amplitude.

IV Conclusion

We observed in SP-FMR experiments in cavity that the temperature increase at resonance is limited to a few hundreds of mK, even at a large rf power of 100 mW, and is further reduced to dozens of mK for faster field sweeping around the resonance. Moreover, regarding the angular dependencies and the absence of link between the detected signal and the temperature increase at resonance, we can conclude that the SSE and ANE are absent in the signals at resonance for both SiO2\\Pt\Py and LSAT\\LSMO\Pt multilayers, and that only dynamical spin injection is involved. The experiment presented here can be generalized to any system measured using spin pumping, and in particular to Rashba interfaces and topological insulators. Indeed, these two groups of materials gather a large number of materials of interest for spinorbitronics but possess a very high thermoelectric figure of merit [47 ] . For example in Bi, Bi2Se3, or Bi2Te3 it could give rise to non-negligible thermal signals, unrelated to the spin-charge interconversion. It might also be a useful way to further study some recently evidenced thermal effects such as the valley Nernst effect [ 48 ] or to acquire a better understanding of the nature of the spin pumping signal in antiferromagnets [49].

Figure 1 :

 1 Figure 1: Schematic representation of the possible spin injection mechanisms at the FMR and of thermal gradient related effects. a) Dynamical spin injection. Because of the magnetization precession, the spin current 𝐽 𝑆 𝑆𝑃 is injected from the FM layer towards the NM layer. An electromotive force 𝐸 𝐼𝑆𝐻𝐸 then arises along X, due to ISHE, which can be detected as a voltage in open circuit. b) Spin injection due to the thermal gradient. At the ferromagnetic resonance the temperature of the FM layer increases creating a thermal gradient 𝛻𝑇 along the z direction and thus a thermal spin current injection 𝐽 𝑆 𝛻𝑇 . This spin current is then converted into an electromotive force 𝐸 𝐼𝑆𝐻𝐸 . c) Thermal gradient within the ferromagnet could give rise to an anomalous Nernst related electromotive force 𝐸 𝐴𝑁𝐸 .

Figure 2 :

 2 Figure 2: a) Schematic representation of the measurement device used to detect spin pumping ISHE at resonance in the Pt(10)\Py(20) sample (top view). b) Signal obtained in the in-plane parallel configuration for various sweeping rates, normalized by the square of the rf magnetic field ℎ 𝑟𝑓 2 for a power of 100mW. The Q

Figure 3 :

 3 Figure 3: a) Schematic representation of the measurement device used to detect a resistance change at resonance in the Pt(10)\Py(20) sample (top view). b) Change of the two probes resistance around the resonance field, for various field sweeping rates (with base resistance subtracted). c) Resistance of the sample as a function of the temperature, measured in a 2 probes configurationon the same sample. The slope value is 96±2mΩ/K d) Temperature change as a function of the sweeping rate, estimated from the increase of resistance at resonance.

Figure 4 :

 4 Figure 4: a) Schematic representation of the measurement device used to detect resistance change in the LSAT \\La0.3Sr0.7MnO3(13.8)\Pt(8.2) sample (top view). b) Change of resistance for various sweeping rate. Inset shows the FMR response out of plane in LSMO is asymmetric, with a narrow and a wide peak c) Temperature change as a function of sweeping rate. Inset shows the resistance as a function of temperature, slope is of 303±6 mΩ/K. d) Schematic representation of the measurement device used to detect spin pumping (top view). e) Signal obtained in the in-plane parallel configuration for various sweeping rates, normalized by the squared rf magnetic field ℎ 𝑟𝑓 2 . The Q factor was 510 and the symmetric voltage of -18.8 µV before normalization. f) Similar measurement in the in-plane antiparallel configuration. The Q factor was 269 and the symmetric voltage of 11.8 µV before normalization.

Figure 5 :

 5 Figure 5: Change in the measured output voltage as a function of time during the sweeping of the field from out of resonance to resonance in the SiO2\\Pt(10)\Py(20) sample for parallel to the plane configuration with an input current of 5 mA (in black) and in open circuit conditions (in red).

Acknowledgments:

We acknowledge the financial support by ANR French National Research Agency Toprise (ANR-16-CE24-0017), ANR French National Research Agency OISO (ANR-17-CE24-0026) the Laboratoire d'excellence LANEF (ANR-10-LABX-51-01) and European Commission via the TOCHA project H2020-FETPROACT-01-2018 under Grant Agreement 824140. We are grateful to the EPR facilities available at the national TGE RPE facilities (IR 3443). We thank Jean-François Jacquot and Serge Gambarelli for their help and advices on the FMR measurement setup.

Data availability:

The data that support the findings of this study are available from the corresponding author upon reasonable request.


Xin Jin

Alexandre Boulle

Alain Chartier

Jean-Paul Crocombette

Aurélien Debelle
email: aurelien.debelle@ijclab.in2p3.fr

Analysis of strain and disordering kinetics based on combined RBS-channeling and X-ray diraction atomic-scale modelling

Keywords: Irradiation eect, RBS/C, X-ray diraction (XRD), Molecular dynamics simulations, Defects

Ion beams delivered by particle accelerators are routinely used to emulate harsh, radiative environments and they also constitute the foundations of the modern microelectronics industry. To characterize irradiated materials, numerous experimental and computational techniques can be implemented, but it is extremely dicult to eectively intertwine them, and to compare the associated data. In the present work, we present an integrated, experimental and computational approach that uses a same set of molecular dynamics simulations to generate signals of Rutherford backscattering spectrometry in channelling condition and X-ray diraction, with UO 2 as a test-case material. From these signals, parameters to monitor the damage level are computed, compared and confronted with experimental data. Although the evolution of the strain and disordering kinetics obtained by simulations differ on an absolute scale from those obtained experimentally (a discrepancy inherent to the method used to generate the atomic-scale data), a very good relative agreement is obtained, which demonstrates the validity of the approach, hence providing a new tool for the ne study of irradiation eects in materials.

Introduction

In our modern society, there is an increasing number of activities involving the use of energetic ions delivered by particle accelerators, which encompass (but are not limited to) health (with hadrontherapy) [START_REF] Amaldi | Present challenges in hadrontherapy techniques[END_REF], high-energy and nuclear physics [START_REF]Combined Measurement of the Higgs Boson Mass in pp Collisions at √ s =7 and 8 TeV with the ATLAS and CMS Experiments[END_REF][START_REF] Durante | Applied nuclear physics at the new high-energy particle accelerator facilities[END_REF], astronomy [START_REF] Bringa | Energetic Processing of Interstellar Silicate Grains by Cosmic Rays[END_REF], and materials science. In this latter domain, several industries and research elds have beneted greatly from the ion irradiation process which can result in both the controlled introduction of foreign atoms in the lattice materials and the tailored modications of the atomic structure. Ion doping of semiconductors, for instance, represents one vital step for modern integrated-circuit manufacturing [START_REF] Chason | Ion beams in silicon processing and characterization[END_REF][START_REF] Rubin | Ion implantation in silicon technology[END_REF], and it is considered as a promising route for developing materials with applications in novel optoelectronic devices [START_REF] Nogales | Cathodoluminescence of rare earth implanted Ga 2 O 3 and GeO 2 nanostructures[END_REF] as well as in spintronics [START_REF] Theodoropoulou | Unconventional Carrier-Mediated Ferromagnetism above Room Temperature in Ion-Implanted (Ga, Mn)P:C[END_REF] and nanophotonics [START_REF] Ocelic | Subwavelength-scale tailoring of surface phonon polaritons by focused ion-beam implantation[END_REF]. In a broader framework, ion irradiation is explored to tailor various properties of nanomaterials including mechanical, electrical and magnetic properties [1012]. Ion irradiation can also be used to shape materials down to the nanoscale [1315]. In the eld of nuclear energy, ion irradiation is used worldwide as a proxy to neutron irradiation for testing and qualifying nuclear materials [1618].

Ion irradiation clearly represents a powerful tool in materials science because it allows driving the materials in out-of-equilibrium conditions. This process has a large number of advantages that span from the possibility to perfectly control the nature and the quantity (also called uence, i.e. (ion)/cm 2 ) of the ions, the ux and energy of the beam, to applicability to nearly all types of materials. However, it has its drawbacks, the main one being the quasi-systematic disordering of the perfect stacking of the lattice atoms because of the permanent displacement of these latter during the energy deposition process. Indeed, if the energy transferred to target atoms is larger than a threshold displacement energy, the primary knock-on atom is ejected, sometimes with a velocity sucient to induce a subsequent collective displacements of target atoms; this phenomenon is referred to as a collision cascade [START_REF] Robinson | Basic physics of radiation damage production[END_REF]. After this collisional stage, an energy dissipation phase occurs, followed by a diusional phase, and defect recombination may occur. At the end of this multi-step process, the target atoms may remain displaced, inducing a residual disorder. Increasing the irradiation uence usually leads to defect clustering and formation of extended defects or even amorphous clusters [START_REF]Ion Beam Modication of Solids: Ion-Solid Interaction and Radiation Damage[END_REF]. As a consequence, the irradiation process can result in macroscopic changes of the material properties that include mechanical [START_REF] Victoria | Microstructure and associated tensile properties of irradiated fcc and bcc metals[END_REF], thermal [START_REF] Ronchi | Eect of burn-up on the thermal conductivity of uranium dioxide up to 100.000 MWdt -1[END_REF], and chemical properties (e.g., dissolution rate [START_REF] Weber | Radiation-induced amorphization in Ca 2 Nd 8 (SiO 4 ) 6 O 2[END_REF]), as well as frequent signicant volume change [START_REF] Hastings | Irradiation-induced volume changes in commercial UO 2 fuel: Comparison with model prediction[END_REF][START_REF] Jiang | Direct determination of volume changes in ion-beam-irradiated SiC[END_REF]. Therefore, it is of crucial importance to have a comprehensive understanding of the basic mechanisms underlying defect formation, accumulation and reorganization during ion irradiation. For this purpose, advanced tools for material characterization and modelling are required.

Investigating irradiation eects in materials has fueled a huge amount of experimental and computational studies over the last decades. Experimentally, transmission electron microscopy (TEM) is an invaluable tool for direct defect imaging. Techniques such as Raman spectroscopy, positron annihilation spectroscopy, Rutherford backscattering spectrometry in channeling mode (RBS/C), and X-ray diraction (XRD) are widely used to evaluate the disorder level in irradiated materials (see ref. [START_REF] Zhang | Advanced techniques for characterization of ion beam modied materials[END_REF] and references therein). Most of the time, they rely on a phenomenological approach where disorder is quantied through a simple parameter such as the weakening of the Raman lines, the fraction of positron annihilations with core or shell electrons, the backscattering yield in RBS/C, or the intensity attenuation in XRD. Simulation methods for modelling irradiation eects encompass a vast range of time and length scales, from the femtoseconds to centuries, from the atomic level to macroscopic dimensions (see refs. [START_REF] Becquart | Modeling microstructure and irradiation eects[END_REF][START_REF] Nordlund | Historical review of computer simulation of radiation eects in materials[END_REF] for reviews). Most widely used methods are molecular dynamics (MD) for the study of primary defects, kinetic Monte-Carlo (KMC) that addresses the defect migration, dislocation dynamics to handle complex extended defects, and nite element modelling (FEM) for predicting the elastoplastic response of large systems. Note that these techniques can be parameterized using quantum-mechanical calculations. Ideally, experiments and simulations are intertwined, which allows validating computational techniques and interpreting experimental data. Another step towards an integrated experimental and computational approach lies in the generation of experimental-like signals using computational data as input data. This approach provides an ecient way to establish direct connections between actual defective structures (hardly accessible by experiments) and phenomenological parameters (derived from experiments).

In the present paper, we propose such an integrated approach. MD defective cells, used to mimic irradiated materials, serve as input data to dedicated codes to produce RBS/C and XRD signals. Uranium dioxide, U O 2 , has been chosen as a test-case material, as it is the most widely used nuclear fuel worldwide and has thus received a considerable attention during the past 50 decades. A recent review of irradiation eects in U O 2 can be found in, e.g., ref. [29]. The well documented behavior of this material under irradiation, as well as the substantial quantity of corresponding data (including our own data), were two major reasons to select this material to implement our integrated approach. We show here that RBS/C and XRD signals can be generated from a same set of MD data and used to compare the computed results with experimental characterizations.

2 Simulation and analysis methods

MD simulations and defect analysis methods

Radiation damage was reproduced with the Frenkel pair (FP) accumulation method which has the advantage of bypassing the collision cascade stage and directly simulates the evolution of defects by continuously introducing FP or by reaching a nal concentration of FP at once, resulting in considerably shorter computing times [START_REF] Crocombette | Atomistic simulation of amorphization thermokinetics in lanthanum pyrozirconate[END_REF][START_REF] Aidhy | Kinetically evolving irradiation-induced point defect clusters in UO 2 by molecular dynamics simulation[END_REF]. It has proven ecient to study the complete disordering processes in Ni and Ni-based alloys [START_REF] Aidhy | Point defect evolution in Ni, NiFe and NiCr alloys from atomistic simulations and irradiation experiments[END_REF], in UO 2 [START_REF] Chartier | Early stages of irradiation induced dislocations in urania[END_REF], but also the amorphization processes in lanthanum pyrozirconate [START_REF] Crocombette | Atomistic simulation of amorphization thermokinetics in lanthanum pyrozirconate[END_REF] and SiC [START_REF] Debelle | Interplay between atomic disorder, lattice swelling, and defect energy in ion-irradiation-induced amorphization of SiC[END_REF], and the complex defects in irradiated Fe [START_REF] Chartier | Rearrangement of interstitial defects in alpha-Fe under extreme condition[END_REF].

The cells used in the MD simulations have a cubic shape. The side length of a pristine one is 21.6 nm. Each cell contains 768000 atoms. The z direction of the MD cell is along the [001] direction of a UO 2 unit-cell. The nominal disorder of the cell is dened as the displacements (of uranium atom) per uranium atom (dpU, similar to the displacements per target atoms (dpa) but referring only to uranium atoms) which ranges here from 0.01 to 7.85. More details about the MD simulations can be found in ref. [START_REF] Chartier | Early stages of irradiation induced dislocations in urania[END_REF]. The conversion of dpa to dpU can be calculated as follows:

dpU = dpa N t N U d U d t (1)
in which N t is the total target atom number, N U is the total Uranium atom number, d t is the displacements of total atoms at a certain position and d U is the displacements of Uranium atoms at the same position. According to SRIM Full-Cascade calculations [START_REF] Ziegler | SRIM[END_REF] We must highlight that there is a true dierence between, on the one hand, the dpU level obtained in the current MD cells produced by the FP accumulation method (MD-dpU) and, on the other hand, the dpU level derived from the SRIM Full-Cascade calculations (SRIM-dpU) which we performed in the framework of the irradiation experiments. It is known that SRIM simulations (using the Kinchin-Pease model or the Full-Cascade option, see ref. [START_REF] Stoller | On the use of SRIM for computing radiation damage exposure[END_REF]) overestimate the actual number of stable defects. Indeed, they rather provide the total number of displaced atoms without considering any physical mechanism for a potential recombination, as it occurs, based on MD simulations, during the thermal spike phase of collision cascades [START_REF] Van Brutzel | Molecular dynamics studies of displacement cascades in the uranium dioxide matrix[END_REF]. In addition, all the FPs introduced in the MD cells, in the current work, do contribute to defect evolutions, whereas even part of the stable defects surviving at the end of the collision cascade may further recombine or annihilate due to thermal migration. Hence, we cannot compare directly MD-dpU and SRIM-dpU levels, and we can expect onsets of defect changes to take place at MD-dpU lower than SRIM-dpU. These remarks will be used in the following to explain the dierence between computed and experimental strain and disordering kinetics.

The OVITO code [START_REF] Stukowski | Visualization and analysis of atomistic simulation data with OVITO-the Open Visualization Tool[END_REF] was used to visualize and analyze the MD cells. Concentrations of point defects, namely interstitials (I) and vacancies (V), were derived from a Voronoi analysis method [START_REF] Stukowski | Structure identication methods for atomistic simulations of crystalline materials[END_REF]. Dislocation types and densities were determined by the dislocation analysis (DXA) method [START_REF] Stukowski | Extracting dislocations and non-dislocation crystal defects from atomistic simulation data[END_REF][START_REF] Stukowski | Automated identication and indexing of dislocations in crystal interfaces[END_REF]. Fig. 1(a) presents the visualization of dislocations in damaged MD cells, in which 1/3<111> Frank dislocations (blue) are clearly in the form of loops, whereas it is dicult to distinguish 1/2<110> perfect loops and lines (green).

2.2

RBS/C simulations and analysis RBS/C simulations were performed with a recently developed Monte-Carlo code called RBSADEC which stands for Rutherford Backscattering Simulation in Arbitrary Defective Crystals [START_REF] Zhang | Simulation of Rutherford backscattering spectrometry from arbitrary atom structures[END_REF]. This code was subsequently improved by us in order to extend its applicability to more simulation conditions [START_REF] Jin | New developments in the simulation of Rutherford backscattering spectrometry in channeling mode using arbitrary atom structures[END_REF]. The code is based on the Binary Collision Approximation (BCA) and can simulate RBS/C spectra from targets containing arbitrary atomic structures. The trajectories of probing ions are determined by a search-and-collide process taking into account actual target crystalline structures [START_REF] Robinson | Computer simulation of atomic-displacement cascades in solids in the binary-collision approximation[END_REF], while an algorithm describing the interaction between ions and amorphous targets is used to calculate the trajectories of backscattered ions in order to accelerate the simulations.

For the current simulations, the probing ion was dened as a 3.085 MeV He ion (corresponding to most available experiments). The detector energy resolution was xed at 15 keV. The one-dimensional root mean square (rms) thermal vibration magnitudes for U and O atoms were set to 6.5 pm and 9 pm respectively [START_REF] Garrido | Channeling investigation of the crystalline structure of U4 O9-y[END_REF]. For each simulation, periodic boundary conditions were applied to the x and y directions of the MD cells, while the projectile entered the target along the z direction. Thus, the simulated target can be regarded as a thin layer with a 22 nm thickness. Three sets of simulations were performed. For each set, before applying the periodic boundary conditions, the cells were rotated in order to have the z direction along, alternatively, the [001], [010] and [100] directions. The purpose was to detect a potential anisotropy of material response to defect formation. In addition, a simulation in a random conguration was also performed for each cell in order to have the maximum backscattering yield as a normalization criterion. Considering that the total number of the cells was high (897 cells), a TRIM-based amorphous slowdown approach was used, which allowed a quicker calculation in RBSADEC, see ref.

[? ] for more details. In this case, it is reasonable to assume that, essentially, direct scattering events contribute to the yield, and the dechanneling component can be neglected. This assumption facilitates the determination of the (RBS/C) disorder fraction, f d , in each of the MD cells, which can be calculated as follows:

f d = y a -y p y r -y p (2)
in which y r represents the total yield of a random spectrum, y a represents the total yield of an aligned spectrum for a defective cell, and y p represents the total yield of an aligned spectrum generated from a pristine sample.

XRD simulations and analysis

Similar to the RBS/C simulations, the MD cells were used to generate XRD signals and the three [001], [010] and [100] directions have been considered. For each orientation, two sets of calculations have been performed with the x direction of the MD cell parallel to two dierent directions of the unit-cell, and the corresponding strain and disorder were averaged. For instance, for a [001] orientation of U O 2 , the [100] and [010] directions are successively set parallel to the x direction. For each data set, the three-dimensional (3D) amplitude distribution, A, was calculated in the vicinity of the 004 Bragg reection using the following equation [START_REF] Warren | X-Ray Diraction[END_REF]:

A = j f j (S)exp(2πiS • r j ) ( 3
)
where S is the scattering vector, f j is the atomic scattering factor of atom j, r j is the position of atom j. Since in the disordered cell the atoms are not regularly distributed on a 3D lattice, fast Fourier transform algorithms can not be used. Instead we used a direct summation approach accelerated by the use of a general purpose graphical processing unit (GPU).

The 3D amplitude distribution is converted to a 2D intensity distribution (i.e., a Reciprocal Space Map, RSM) with [START_REF] Channagiri | Diuse X-ray scattering from ion-irradiated materials: A parallel-computing approach[END_REF]:

I(S x , S z ) = dS y • A(S)A * (S) (4)
where the S i represents the component of S along the direction indicated by the subscript i.

Complete details concerning this approach will be given in a separate article. For the sake of the present article, it is sucient to mention that this approach allows to derive the lattice strain and the disorder parameter (the Debye-Waller (DW) factor) for all orientations of the MD cells. Examples of 004 RSMs of UO 2 MD cells for increasing damage levels are given in Fig. 1(c). These RSMs display the 2D intensity distribution in a (H, L) reference frame where H and L are the coordinates of the scattering vector [START_REF] Boulle | Strain and damage build-up in irradiated crystals: Coupling X-ray diraction with numerical simulations[END_REF]:

S(H, K, L) = Ha * + Kb * + Lc * (5)
where a * , b * and c * are the reciprocal space basis vectors.

The inspection of Fig. 1(c) reveals several features. The RSM from the pristine cell correspond to the theoretical signals of a perfect crystal, i.e., a 2D Laue function. For increasing uence, we observe both a shift of reciprocal lattice point towards lower L values, and a spread of the intensity out of the specular row (characterized by H = 0). The former is indicative of the development of tensile strain, e, within the MD cell, which can be directly quantied through [START_REF] Boulle | Strain and damage build-up in irradiated crystals: Coupling X-ray diraction with numerical simulations[END_REF]:

e = L 0 -L L 0 (6)
where L 0 is the L value of the Bragg peak in a pristine sample (here L 0 = 4). The weakening of the specular intensity is quantied by the DW factor and can be estimated from the RSMs via:

DW = I Hmax I Hmax 0 (7)
in which I Hmax is the intensity integrated along the row H = H max where the spot of the maximum intensity is located, and the subscripts 0 refers to the pristine material for which H max = 0. The disorder fraction obtained from the XRD simulations is dened as 1 -DW .

3 Results

Defect evolution sequence

Let us rst consider the defect evolution obtained by a direct analysis of the MD cells in which radiation damage was mimicked by the Frenkel pair (FP) accumulation method. Fig. 2(a) gives the point defect concentration and Fig. 2(b) gives the dislocation densities expressed in m -2 (which corresponds to the total dislocation length of a given type, divided by the cell volume) as a function of irradiation dose -dpU.

The defect evolution follows a sequence that has been previously identied, and, overall, conrmed by TEM analysis [START_REF] Chartier | Early stages of irradiation induced dislocations in urania[END_REF]. The sequence can be decomposed as follows. Below 0.09 dpU, point defects are predominant, and the dislocation density is very low. From 0.09 dpU, 1/3<111> Frank dislocation loops form and constitute the main defect type up to 0.25 dpU. Formation of these dislocation loops is accompanied with a decrease in the interstitial concentration, which strongly suggests that the loops are of interstitial type. At around 0.12 dpU, 1/6<112> Shockley partial dislocations are detected, but they rapidly react with Frank loops to form 1/2<110> perfect dislocations [START_REF] Hull | Introduction to dislocations[END_REF]. These latter initially form as loops, but gradually coalesce and transform into lines. Note that, unlike Frank loops which are easily distinguishable among other defects, perfect loops and lines can be hardly separated. Above 1 dpU, 1/2<110> perfect dislocation lines are predominant over the other types of dislocations. At the same time, the point defect concentration reaches a steady state, with vacancies found in a much larger concentration than interstitials. These results indicate that at this stage, interstitial point defects are eciently trapped at dislocation lines. This general sequence of defect evolution is in accordance with recent combined experimental and computational results [START_REF] Chartier | Early stages of irradiation induced dislocations in urania[END_REF]5254]. A question that now arises is how this sequence of transformations aect the XRD and RBS/C signals, and what can we learn from the latter?

Disordering kinetics

In order to compare the results of the simulations of the two techniques, we rst plotted disordering kinetics using the disorder parameters obtained by XRD and RBS/C. Fig. 3(a) shows the disorder obtained from XRD, while Fig. 3(b) shows that derived from RBS/C; results along the [001], [010] and [100] directions of the U O 2 unit-cell are presented. In inset of the gures, the abscissa is in linear scale. The two disordering kinetics exhibit a very similar shape, which means that the two techniques could be used interchangeably to monitor the disordering process in irradiated materials. From a practical standpoint, this is an important result because, although RBS/C is to date more popular to evaluate the disorder in irradiated materials, XRD is more easily accessible and this could facilitate the characterization.

Specicities of the techniques in terms of defect sensitivity are given in the following. The absolute values of disorder obtained by XRD and RBS/C dier; this is expected as both techniques rely on very dierent physical phenomena, namely the interference of scattered Xrays on the one hand and the backscattering of energetic ions on the other hand. Moreover, the disorder measured by XRD is dependent on the considered HKL reection since the Debye-Waller factor, which is used to quantify the disorder [START_REF] Boulle | Statistical Nature of Atomic Disorder in Irradiated Crystals[END_REF], is a function of the scattering vector S(H, K, L), while the backscattering yield changes with the probing ion energy.

Nonetheless, the relative evolution of both disorders remains perfectly comparable. Both kinetics show common features that can be related to the stages of the sequence of defect evolution above-described (a quantitative support for this correlation is given in the Discussion part). First, the disorder grows at a relatively low rate with increasing dpU when, principally, point defects are present. Then, when dislocation loops form, the disordering rate considerably increases and the disorder reaches a maximum value at around 0.4 dpU. Finally, after this maximum, the disorder starts to decrease (at least with the current evalua-tion tools), which can be related to the presence of, essentially, dislocation lines surrounding defect-depleted regions.

It should be noted that at high dpU, for both techniques, the disorder in the dierent directions start to diverge and exhibit seemingly random uctuations. Presumably, these features could be due to the so-called polygonization process, in which dislocations pile-up and lead to the formation of sub-crystallites with slightly dierent orientations [START_REF] Matzke | Polygonization of single crystals of the uorite-type oxide UO 2 due to high dose ion implantation[END_REF][START_REF] Nogita | Radiation-induced microstructural change in high burnup UO 2 fuel pellets[END_REF]. Such resulting boundaries can act as ecient defect sinks. An in-depth analysis of the MD cells (not shown here) indeed indicates that, depending on the orientation of the cell, 2 to 3 rotated sub-crystallites co-exist within the MD cells, i.e., at high uences the structure of the cells diers depending on the direction. It can be expected that in larger cells, such divergence would statistically disappear. A detailed investigation of that point will be given in a forthcoming paper. Fig. 3(c) presents the disordering kinetics obtained from RBS/C experiments on U O 2 irradiated with 500 keV La ions at room temperature (see the supplemental material in ref. [START_REF] Debelle | How relative defect migration energies drive contrasting temperature-dependent microstructural evolution in irradiated ceramics[END_REF]). The displacements per target atoms (dpa) obtained from SRIM Full Damage cascade (FD) calculations [START_REF] Ziegler | SRIM[END_REF] were converted into dpU. Qualitatively, the curve from the RBS/C simulations and the experimental one exhibit very similar trends, and even the apparent decrease in disorder at high dpU is reproduced.

Quantitatively, there are some dierences we address hereafter. Compared with the simulation results, the maximum disorder in the experiment is almost 4 times lower. As explained in ref. [START_REF] Chartier | Early stages of irradiation induced dislocations in urania[END_REF], this is because the dislocation density in the MD simulations is much higher than that in the experiments due to a very high damage rate, K 0 , in the simulations. According to SRIM calculations, the experimental dose rate is around 3 × 10 -3 dpU/s at the damage peak, while it is 1.5625 × 10 9 dpU/s for the MD simulations. Assuming that the dislocation density is proportional to K 1/6 0 [START_REF] Rest | A model for the inuence of microstructure, precipitate pinning and ssion gas behavior on irradiation-induced recrystallization of nuclear fuels[END_REF], the dislocation density in the MD cells is thus around 100 times higher than that in the real samples.

Another dierence is that it takes around 20 times more dpU to reach the maximum level in the experiments. The rst reason for this is, obviously, the higher defect densities resulting from the higher damage rate in the simulations. Thus, defect transformations can occur earlier in the MD cells. The second reason is due to the very denition of SRIM-dpU and MD-dpU as mentioned in Sec.2.1. The former corresponds to displaced atoms calculated in SRIM, whereas the latter is related to the actual number of defects introduced in the MD cells, which fully contribute to the evolution of defects. It is known that the SRIM results tend to overestimate the actual defect numbers, i.e., the remaining displacements after potential recombination [START_REF] Van Brutzel | Displacement cascade initiated with the realistic energy of the recoil nucleus in UO2 matrix by molecular dynamics simulation[END_REF][START_REF] Crocombette | Molecular dynamics simulations of high energy cascade in ordered alloys: Defect production and subcascade division[END_REF]. Nonetheless, and above all, it is important to emphasize that the computed and experimental disordering kinetics agree qualitatively, which represents a notable result because it demonstrates the relevance of our procedure and in the same time, it provides a support to previous interpretations of experimental data that were based on phenomenological descriptions. Furthermore, considering the good agreement between the computed RBS/C and XRD data, we can conclude that the whole integrated approach is valid. A very similar disordering kinetics has been observed in other materials including cubic yttria-stabilized zirconia (YSZ) [START_REF] Moll | Multistep damage evolution process in cubic zirconia irradiated with MeV ions[END_REF][START_REF] Debelle | Comprehensive study of the eect of the irradiation temperature on the behavior of cubic zirconia[END_REF], MgO [START_REF] Moll | Damage processes in MgO irradiated with medium-energy heavy ions[END_REF], Ni [START_REF] Veli³a | Multi-axial and multi-energy channeling study of disorder evolution in ion-irradiated nickel[END_REF] and Ni-based equiatomic binary alloys [START_REF] Jin | Ion irradiation induced defect evolution in Ni and Ni-based FCC equiatomic binary alloys[END_REF]. One common feature of these materials is that either the entire crystal or the cationic sub-lattice has a face-centered-cubic (FCC) structure, indicating the possibility of having similar defect evolutions (as already partly shown). Therefore, the present qualitative description of the defect transformation sequence should hold for these materials as well, and this conclusion can be based on experimental data only (there is no need to perform MD calculations).

Elastic strain kinetics

In addition to disorder, XRD permits the determination of the elastic strain, which can be considered as another quantity to monitor the radiation damage. Fig. 4(a) shows the irradiation-induced elastic strain kinetics derived from the XRD simulations using MD cells as input data. As for RBS/C, the [001], [010] and [100] directions were studied. Similar to the disorder, the strain kinetics displays distinct features that can be ascribed to a given type of defects (a quantitative support for this correlation is given in the Discussion part as well). When point defects are dominant, the elastic strain buildup is fast. When dislocation loops dominate, the strain keeps on increasing (with a slightly lower rate), to reach a maximum of 1.9 % at around 0.28 dpU. From this point, the strain is partially relieved, which can essentially be related to perfect dislocations being the dominant defects. A slightly anisotropic evolution is observed in the three crystallographic directions, similarly to what is observed in the disordering kinetics. It is also worth noting that the maximum of the elastic strain is reached at a lower dpU value as compared to that of the disorder (0.28 instead of 0.4 dpU); we will comment this dierence hereafter. Fig. 4(b) and Fig. 4(c) show the elastic strain kinetics from UO 2 irradiated with 20 keV He ions and 500 keV Ce ions, respectively. As for the disorder addressed earlier (with La ions, which are equivalent to Ce), the computed strain kinetics exhibit a qualitative agreement with the experiments. Due to the very high dislocation density in the MD cells, the maximum strain computed from the simulations is around 3 to 4 times higher than that in the experiments. Regarding the delay in reaching the maximum level, since Ce and La are similar in terms of mass and solubility in UO 2 , a same explanation as for the disordering kinetics holds for the strain. In contrast, the strain kinetics from 20 keV He irradiations is shifted towards higher dpU values by only a factor of 6 (instead of 20 for the disorder). As shown in ref. [START_REF] Crocombette | Molecular dynamics simulations of high energy cascade in ordered alloys: Defect production and subcascade division[END_REF], the discrepancy between BCA models and MD calculations intensies with increasing the primary knock-on atoms (PKA) energy, with much less surviving defects in MD (than for BCA) at high energy. In the current case, the median PKA energy of 500 keV Ce ions is obviously larger than that of 20 keV He ions. Therefore, it is expected that the dierence between SRIM-dpU and MD-dpU is smaller for He than for Ce irradiations, explaining why the delay is reduced for the He irradiation experiment.

Discussion

From the previous section, it can be concluded that strain and disorder are remarkably correlated with the defect densities derived from the quantitative defect analysis of the MD cells. On the one hand, strain develops as soon as point defects are formed, and keeps increasing when interstitial defects agglomerate into Frank dislocation loops. When the concentration of Frank loops decreases, the strain decreases as well. On the other hand, disorder is more related to the presence of dislocations than point defects. To go beyond these simple qualitative observations, we propose hereafter to decompose the computed strain and disordering kinetics into components associated with the concentration of each type of defects. We averaged the results obtained along the three main directions to smooth out the observed uctuations.

Let us start with the strain. The elastic strain kinetics is tted assuming a linear relationship between the defect concentration or density and the induced strain. Note however that this assumption is valid only if interactions between elastic strain elds are limited, which might not be entirely justied for high defect concentrations, but which yet proves to be reasonable considering the results presented below. For point defects, the elastic strain is proportional to their so-called relaxation volume [START_REF] Debelle | Lattice strain in irradiated materials unveils a prevalent defect evolution mechanism[END_REF][START_REF] Dederichs | Lattice theory of point defects[END_REF][START_REF] Bachiller-Perea | Study of the initial stages of defect generation in ion-irradiated MgO at elevated temperatures using highresolution X-ray diraction[END_REF]. For dislocation loops, not only is it aected by the loop density, but it is also dependent on the size of the loops [START_REF] Dederichs | The theory of diuse X-ray scattering and its application to the study of point defects and their clusters[END_REF][START_REF] Ullah | Evolution of irradiation-induced strain in an equiatomic NiFe alloy[END_REF]. However, the characteristic features of the dislocation loops are not completely known in the current work in terms of size, geometry, etc., because they are often signicantly entangled. Moreover, as stated earlier, there is no simple way to detect the loop character of a dislocation in an automated way. We therefore rely on the Burgers vector and visualization of dislocations to determine the loop character: 1/3<111> are (Frank) loops, but for perfect dislocations 1/2<110>, as stated above, we are unable to distinguish between loops and lines. Henceforth, for the sake of simplicity, we assumed that the elastic strain induced by dislocations is simply proportional to their density. Finally, the total elastic strain, ε def , reads:

ε def = 1 3 [C V V rel V Ω + C I V rel I Ω + (S F ρ F + S p ρ p )] (8)
where C V and C I are concentrations of vacancies and interstitials, respectively, V rel V and V rel I are the relaxation volumes expressed in atomic volume unit Ω, ρ F is the density of the Frank loops and ρ p is the density of perfect dislocations; the terms S F and S p , which have the dimension of a surface, are the proportionality factors between concentration of Frank loops and perfect dislocations, and the corresponding strain, respectively. It should be noticed that the sign of the relaxation volume of U vacancy-type defects is still the object of debate in the literature [START_REF] Goyal | The conundrum of relaxation volumes in rst-principles calculations of charged defects in UO 2[END_REF][START_REF] Freyss | Point defects in uranium dioxide: Ab initio pseudopotential approach in the generalized gradient approximation[END_REF]. In the present study, we performed a pressure relaxation (MD) simulation to have direct access to this value. We obtained an average 0.6 Ω value for V rel V , where the average assumes an equiprobable distribution of single U vacancies, U divacancies and Schottky defects (2U+O). We therefore performed ttings imposing a positive sign for V rel V . The tting of the strain kinetics with Eq.8, shown in Fig. 5, was performed using a non-linear least squares optimization algorithm by minimizing the root-mean-square deviations (RMSD). Overall, the elastic strain kinetics is well tted using Eq.8, demonstrating that the elastic strain determined from the MD simulations can be straightforwardly connected to the defect densities. Table I reports the values of the tting parameters of Eq.8. It can be noticed that the value of V rel V was kept free during the tting in order to allow for deviations from the simple equiprobable defect averaging. S F is found to be 75 Å 2 while S p is 10 Å 2 . These values demonstrate that the contribution of Frank loops to the overall strain level is around 8 times larger than that of perfect loops.

This nding was expected since Frank loops, in contrast to perfect loops, have a nonvanishing average Burgers vector of 1/3<111>. This dierence implies that, while Frank loops essentially generate homogeneous (elastic) strain, perfect loops principally produce local (i.e., heterogeneous) strain. The same reasoning holds for perfect lines, where a random spatial distribution of dislocations or their arrangement into dipoles yield a vanishing average Burgers vector. Therefore, because of their limited contribution to the overall strain level, the inability to distinguish perfect dislocation loops from perfect dislocation lines does not have any measurable consequence. We show below that the situation is far dierent for the disorder.

Let us now consider the disorder, both derived from XRD and RBS/C. In a rst step, similarly to what was done for the elastic strain, we assumed a linear relationship between defect densities and the induced disorder. Contrarily to the strain, vacancies do not contribute signicantly to the disorder; their role was therefore neglected in the following. Fig. 6 presents the corresponding tting of the RBS/C disordering kinetics averaged over the three directions. It can be readily observed that this model provides a good description of the disordering process up to 0.4 dpU, which corresponds to the uence at which the density of perfect dislocations reaches a maximum. Above 0.4 dpU, the model fails to describe the disorder, the most likely reason being that perfect loops and lines contribute signicantly, but dierently, to disorder, as opposed to their contribution to the strain. Being able to distinguish between perfect dislocation loops and lines is therefore crucial to get an accurate, quantitative description of the disordering process.

One solution to cope with this problem lies in the introduction of an ad hoc perfect line density, ρ line p , as a parameter that can be tted and from which the perfect loop density, ρ loop p , can be derived, following the relationship: as ρ loop p = ρ p -ρ line p . Based on the t presented in Fig. 6, we can safely assume that ρ line p is not signicant until around 0.4 dpU, i.e. where the t quality deteriorates, and ρ line p reaches a saturation level in the nal line stage. This qualitative description of ρ line p indicates that the shape of the corresponding disorder component should feature a high incubation time most likely followed by a sigmoidal growth spanning from 0.4 to 1.5 dpU range, and a height determined by the nal dislocation density, i.e. 0.26 × 10 17 m -2 . The total disorder, f d , can nally be represented by:

f d = r I C I + r F ρ F + r loop p ρ loop p + r line p ρ line p (9)
where r I is the disorder constant for the interstitials, and r F , r loop p and r line p are the disorder constants for the Frank loops, perfect loops and perfect lines, respectively, with a m 2 unit. Fig. 7 shows the ts of the disordering kinetics obtained from (a) RBS/C and (b) XRD simulations using Eq.9. By comparing Fig. 6 and Fig. 7 (a), we observe that the quality of the t in the high dpU region (> 0.4 dpU) is signicantly improved after separating the contributions of perfect loops and lines. The decrease in disorder observed after 0.4 dpU can therefore be ascribed to the reduction of the loop density at the expense of the perfect lines. The disorder constants are given in Table II. A feature worth putting forward from these constants is the dierent sensitivity of the two techniques, in terms of estimated disorder, to the nature of the dislocation loops. While RBS/C is more sensitive to perfect loops, with a r F /r p loop ratio of 0.8, the XRD phenomenon is more aected by Frank loops, with a corresponding ratio of 1.4. The reason for this dierence is not entirely clear. We can note however that none of the RBS/C-derived and XRD-derived disorder values used here are absolute. As briey above-mentioned, it is known that the backscattering yield depends on the probing-ion energy, and this dependence varies with the defect nature. Similarly, the Debye-Waller factor in XRD is a complex function of the scattering vector Q, and defects of dierent nature exhibit specic Q dependencies. A careful inspection of both disorder parameters may hence allow to get very detailed and complementary information about the nature of the defects present in the MD cells but also in actual samples.

Examination of the constants in Table II provides another important result: dislocation lines have the largest inuence on the disorder, as compared to the other defects, including perfect dislocation loops. This nding illustrates the complementarity between strain and disorder for the evaluation of the level of damage in irradiated materials. Indeed, because of the very nature of these two types of defects, namely perfect dislocation loops and lines, their eect on the disorder and on the strain is dierent. Dislocation loops are quasi 0dimensional defects, so their strain elds have a nite spatial extension and the long-rang order (LRO) of the crystal is weakly aected [START_REF] Krivoglaz | Theory of X-Ray and Thermal-Neutron Scattering by Real Crystals[END_REF]. On the contrary, dislocation lines are 1-dimensional defects with a diverging ln(r) term in the displacement eld that signicantly alters the LRO of the crystal. The fact the strain kinetic reaches its maximum earlier (in terms of dpU) than the disorder can be entirely explained by this dierent sensitivity to the defect type. As a nal comment, it is worth mentioning that the number of damage metrics, two in the current work, namely the strain and the disorder, denes the number of defect transformations that can be studied. If there were an additional type of defect, say stacking faults or precipitates, then another metric would be needed to monitor the defect evolution with time. In the present case, we limited the use of XRD data to the intensity and the strain. However, additional parameters, such as the reection broadening also conveys information regarding the defect structure. This will the topic of a forthcoming paper. Another solution to deal with a higher number of defect types would be to implement additional techniques, such as Raman [START_REF] Roma | Modeling the contribution of point defects to the Raman spectrum of crystalline materials[END_REF] or positron annihilation spectroscopy [START_REF] Wiktor | Coupled experimental and DFT + U investigation of positron lifetimes in UO2[END_REF] for which it is possible to generate computed signals.

Conclusions

Summarizing, we generated experimental-like signals of RBS/C and XRD techniques using the same MD dataset produced with the Frenkel pair accumulation method. We then computed disordering kinetics, with disorder and strain parameters to monitor the damage accumulating in the MD cells. We demonstrated that, on the one hand, both techniques can be used interchangeably to quantify the disorder in defective cells and in actual materials. On the other hand, because they have a dierent sensitivity on the probed defects, their combination can provide complementary information about the defect nature. Despite discrepancies in terms of absolute values of uence range and levels of strain and disorder, on a relative scale, the computed strain and disordering kinetics agree well with those determined from actual experiments, which demonstrates the validity of the integrated approach. Additional work is in progress with other materials to provide further support and improve the methodology. Nonetheless, the results provided in this paper can be transposed to materials exhibiting the same disordering behaviour. The inset shows part of the kinetics using a linear scale for the x axis. (In order to avoid a dense overlap, only one value in every ve simulation results is shown when the uence is larger than 1.0 dpU.) Figure 7: Fits of the disordering kinetics assuming proportionality between defect quantities and the induced disorder and separating the perfect loop and line contributions. The blue dots represent the averaged disorder computed from a RBS/C simulations (RMSD = 6.5 × 10 -2 ) and b XRD simulations (RMSD = 3.2 × 10 -2 ). Insets show part of the kinetics using a linear scale for the x axis. (In order to avoid dense overlap, only one value in every ve simulation results is shown when the uence is larger than 1.0 dpU.) Tables Table I: Relaxation volumes, V rel , of point defects and strain constants, S, of dislocations. The errors are obtained from the non-linear least squares tting algorithm.

Fig. 1 (

 1 Fig.1(b) depicts several examples of RBS/C spectra generated with RBSADEC using MD cells of dierent dpU levels. Blue lines represent the RBS/C spectra of 3.085 MeV He ions on UO 2 in a random conguration, and orange lines represent the RBS/C spectra along the <001> aligned direction. Unlike usual experimental spectra measured for bulk targets, the spectra in Fig.1(b) are peak-shaped, since the target thickness is very small (∼22 nm). In this case, it is reasonable to assume that, essentially, direct scattering events contribute to the yield, and the dechanneling component can be neglected. This assumption facilitates the determination of the (RBS/C) disorder fraction, f d , in each of the MD cells, which can be calculated as follows:f d = y a -y p y r -y p (2)

Figures

  Figures

Figure 1 :

 1 Figure 1: Characteristic microstructures obtained by MD simulations and corresponding RBS/C and XRD signals generated from the UO 2 MD cells. a Snapshots of UO 2 MD cells analyzed by the DXA method with dierent damage levels: 0.1 dpU, 0.3 dpU and 3.0 dpU. Only dislocations including Frank loops (blue), perfect dislocations (green) and Shockley partial dislocations (orange) in the cells are visualized. (Other types of dislocations including stair-rod dislocations and ill-dened dislocations are represented by red lines.) b RBS/C spectra of 3.085 MeV He on the [001]-oriented UO 2 MD cells presented in a. Blue lines represent random spectra, and orange lines represent aligned spectra. c RSMs calculated in the vicinity of the [004] Bragg reection of the UO 2 MD cells presented in a. Units of H and L are reciprocal lattice units (r.l.u) along the x and z directions, respectively. Intensities indicated by the color bar are represented on a logarithmic scale.

Figure 2 :

 2 Figure 2: Defect quantities as a function of uence (dpU). a Concentrations of vacancies and of interstitials, b Densities of 1/3<111> Frank loops, of 1/6<112> Shockley partial dislocations and of 1/2<110> perfect dislocations.

Figure 3 :

 3 Figure 3: Disordering kinetics obtained from: a The XRD simulations on the UO 2 MD cells, b the RBS/C simulations on the UO 2 MD cells, c RBS/C experiments on UO 2 crystals irradiated with 500 keV La ions. In the simulations, the MD cells were characterized along three major directions: [001] (blue lines), [010] (orange lines) and [100] (green lines). Insets show part of the kinetics using a linear scale for the x axis.

Figure 4 :

 4 Figure 4: Elastic strain kinetics obtained from: a The XRD simulations on the UO 2 MD cells, b XRD experiments of 20 keV He on UO 2 crystals, c XRD experiments of 500 keV Ce on UO 2 samples. In the simulations, the MD cells were characterized along three major directions: [001] (blue lines), [010] (orange lines) and [100] (green lines). Samples in experiments were characterized along <001> directions. Insets show part of the kinetics using a linear scale for the x axis.

Figure 5 :

 5 Figure5: Fit of the elastic strain kinetics. The blue dots represent the elastic strain computed from the XRD simulations and averaged over the three <001> directions. The solid line represents the total t by combining contributions of the dierent defects (dashed lines). The RMSD value equals to 1.7 × 10 -1 . The inset shows part of the kinetics using a linear scale for the x axis. (In order to avoid a dense overlap, only one value in every ve simulation results is shown when the uence is larger than 1.0 dpU.)

Figure 6 :

 6 Figure 6: Fit of the disordering kinetics assuming proportionality between defect quantities and the induced disorder. The blue dots represent the disorder computed from the RBS/C simulations and averaged over the three <001> directions. The solid line represents the total t by combining the contributions of the dierent defects (dashed lines).The inset shows part of the kinetics using a linear scale for the x axis. (In order to avoid a dense overlap, only one value in every ve simulation results is shown when the uence is larger than 1.0 dpU.)

  S F (Å 2 ) S p (Å 2 ) 0.4 ± 0.1 1.0 ± 0.1 75.1 ± 3.5 10.3 ± 2.4

  , the displacement threshold energies, E d , for U and for O can signicantly aect the ratio of d U over d t . When E d (U ) equals to 40 eV and E d (O) equals to 20 eV[START_REF] Soullard | High voltage electron microscope observations of UO 2[END_REF], for experimental data used at Section 3 (20 keV He ions, 500 keV La ions and 500 keV Ce on UO 2 ), the ratios of d U over d t are similar, which are around 0.253 at the position of maximum damage. So 1 dpa is equivalent to 0.760 dpU at that position.

Table II :

 II Disorder constants, r, for defects used in the ts. a RBS/C disordering kinetics, b XRD disordering kinetics. The errors are obtained from the non-linear least squares tting algorithm.r I r F (×10 -17 m 2 ) r loop p (×10 -17 m 2 ) r line p (×10 -17 m 2 )

	RBS/C 4.31 ± 0.96	0.75 ± 0.07	0.92 ± 0.02	2.10 ± 0.08
	XRD	2.60 ± 0.47	0.97 ± 0.03	0.67 ± 0.08	1.71 ± 0.05

Acknowledgments X. Jin is grateful to SIMME Doctoral School (Université de Limoges) and PHENIICS Doctoral School (Université Paris-Saclay) for the nancial support of his PhD research work. Frédérico Garrido and Lionel Thomé are acknowledged for fruitful discussions. Part of this research work has been funded within the NEEDS-CNRS program.

Competing interests

The authors declare no conict of interest.