C.-M Arvhult
email: arvhult@kth.se

S Poissonnet

D Menut

S Gossé

C Guéneau

Thermodynamic Assessment of the Fe-Te System. Part I: Experimental study

Keywords: "nuclear reactor materials", "thermal analysis", "scanning electron microscopy, SEM", "metallography", "X-ray diffraction", "phase transitions"

A thermodynamic description of the Fe-Te system needs to be developed in order to model internal corrosion by fission products in fuel pins of Generation IV nuclear reactors. In preparation for a thermodynamic assessment of the system, an experimental study has been performed in order to clarify some unknown or conflicting phase diagram data. New phase diagram data have been obtained using Differential Thermal Analysis and isothermal heat treatments followed by electron microscopy with EDS and WDS analysis. The DTA analysis revealed new phase boundary data, and confirmed a very steep Fe-rich liquidus, supporting the possibility of a liquid miscibility gap in the Fe-FeTe region. The analyses also confirmed the probable eutectoid reaction at 523 . The invariant arrests of the unknown phase were consistent with information available in literature, but the phase was not identified via XRD of samples at its postulated composition. However, metallography of the samples revealed an unexpected microstructure pertaining to the phase, which might be the phase, and is discussed in this paper. The monoclinic space group is proposed for the phase based on XRD. The collected data will be used together with that available in literature to perform a thermodynamic Calphad assessment in a subsequent paper Part II: Thermodynamic modelling.

Introduction

Most commercial light water reactors (LWR) operate in a thermal neutron spectrum, with zirconium alloys as the fuel encapsulation (cladding). Among Generation IV nuclear reactors currently under development are the Sodium cooled Fast neutron Reactors (SFR), operating with a fast-neutron spectrum and a liquid sodium coolant. The MOX fuel (mixed oxide of uranium and plutonium) pellets are contained in stainless steel cladding. Post irradiation examination of MOX fuel pins from SFR reactors have revealed an internal corrosion process where some fission products initiate Fuel-Cladding Chemical Interaction (FCCI) and Fission-Product induced Liquid Metal Embrittlement (FPLME) [START_REF] Adamson | Chemical thermodynamics of Cs and Te fission product interactions in irradiated LMFBR Mixed-Oxide Fuel pins[END_REF][START_REF] Naganuma | High burnup irradiation performance of annylar fuel pins irradiated in fast reactor PFR[END_REF][START_REF] Boidron | Fast neutron reactors claddings corrosion[END_REF]. This corrosion is facilitated by the release and migration of these volatile fission products from the fuel, into the fuel pin gap, and cesium and tellurium are key elements in this process.

Thermodynamic studies of the effect of Cs and Te on stainless steel have concluded that the hyperstoichiometry of the oxide fuel during burnup results in a sufficiently high tellurium potential to enable corrosion [START_REF] Saha | Thermochemistry and phase relationships in M-Te (M=Fe, Cr and Ni) systems of relevance to fuel-clad interactions in fast breeder nuclear reactors -a high-temperature mass spectrometric study[END_REF][START_REF] Mathews | Thermochemistry of transition metal tellurides of interest in nuclear technology[END_REF][START_REF] Sai Baba | Thermodynamic and Phase Diagram Studies on Metal-Tellurium Systems Employing Knudsen Effusion Mass Spectrometry[END_REF]. Pulham and Richards [START_REF] Pulham | Chemical reactions of caesium, tellurium and oxygen with fast breeder reactor cladding alloys. Part I -The corrosion by tellurium[END_REF][START_REF] Pulham | Chemical reactions of caesium, tellurium and oxygen with fast breeder reactor cladding alloys. Part II -The corrosion by caesium-oxygen mixtures[END_REF][START_REF] Pulham | Chemical reactions of caesium, tellurium and oxygen with fast breeder reactor cladding alloys. Part III -The effect of oxygen potential on the corrosion by caesium-tellurium mixtures[END_REF][START_REF] Pulham | Chemical reactions of caesium, tellurium and oxygen with fast breeder reactor cladding alloys. Part IV -The corrosion of ferritic steels[END_REF][START_REF] Pulham | The effect of gaseous Cs-Te mixtures on nickel[END_REF] made experimental studies on the corrosion of Cs, Te and (Cs,Te) mixtures with varying oxygen potential on the candidate cladding alloys Nimonic PE16, austenitic M316 and ferritic FV448 and DT2203Y05. They concluded that the corrosion process, including steel matrix attack and inter-granular corrosion, is sensitive to different Cs/Te ratios and oxygen potentials, which change the mode of corrosion. In most cases, transition metal tellurides and oxides were formed, deposited in scales composed mainly of Cr-Cs and Cr-Te compounds at low oxygen potentials, and Fe-Te, Cs-Fe-O and Cs-Cr-O compounds at high potentials.

In order to predict whether or not this internal corrosion of the fuel pins during operation will be a life-limiting factor of the reactor, a corrosion model has to be implemented. As input data for that model, a thermodynamic database of the Fe-Cr-Ni-Cs-Te-O system must be developed. In order to assess such a complex system, all the binary, ternary and higher-order sub-systems need to be assessed.

This paper presents an experimental study of the Fe-Te phase diagram, to be included in the thermodynamic assessment presented in Part II: Thermodynamic modelling. Section 2 summarizes the state of the art of the Fe-Te system. Section 3 then describes the experimental methods used, followed by a discussion of the results of metallographic studies on annealed samples in section 4.1, and an analysis of heat curves from DTA in section 4.2. Finally, section 5 summarizes the main conclusions and gives suggestions for future work.

State of the art on the Fe-Te system

In their work, Grønvold et al. [START_REF] Grønvold | Phase and Structural Relations in the System Iron Tellurium[END_REF] characterized most known phases of the iron-tellurium system, i.e. the , , and phases. This was later completed with the discovery of another high-temperature phase, , by Røst and Webjørnsen [START_REF] Røst | A High Temperature Iron Telluride with Rhobohedral Structure[END_REF]. A number of researchers then published experimental phase diagrams and homogeneity ranges of the Fe-Te system [START_REF] Chiba | The Magnetic Properties and Phase Diagram of Iron Tellurium System[END_REF][START_REF] Llewellyn | The Magnetic Susceptibility of Iron Ditelluride: I[END_REF][START_REF] Geiderikh | Thermodynamic Properties of Iron-Tellurium Alloys in the Solid State[END_REF][START_REF] Abrikosov | Iron-Tellurium System[END_REF][START_REF] Brostigen | Compounds with the Marcasite Type Crystal Structure V. The crystal structure of FeS2, FeTe2 and CoTe2[END_REF]. Ipser et al. [START_REF] Ipser | Transition Metal-Chalcogen Systems, V.: The iron-Tellurium Phase Diagram[END_REF] observed inconsistencies between those diagrams, and revised the phase diagram based on isopiestic, Thermal Analysis (TA), Differential Thermal Analysis (DTA) and X-ray diffraction (XRD) measurements, as shown in Figure 1a. Chattopadhyay and Bharadwaj [START_REF] Chattopadhyay | Evaluated phase diagrams of binary metal-tellurium systems of the D-block transition elements[END_REF] reviewed the system and accepted the phase diagram by Ipser et al., adding a liquid miscibility gap suggested by Mann and Van Vlack [START_REF] Mann Och | Fe1.2Te-MnTe Phase Relationships in the Presence of Excess Iron[END_REF] in their partial diagram shown in Figure 1b. Okamoto and Tanner [START_REF] Okamoto | The Fe-Te (Iron-Tellurium) System[END_REF] published a separate review, accepting a similar phase diagram including an austenite loop.

Figures 1a and 1b: Fe-Te phase diagrams by Ipser et al. [19] (1a) and Mann and Van Vlack [21] (1b).

Table 1 presents the available crystallographic data of the system, as well as their approximate chemical formulae. The designation of Greek letters by Grønvold et al. will be used throughout this text. Several XRD studies are available on the crystal structures of the , and phases [START_REF] Grønvold | Phase and Structural Relations in the System Iron Tellurium[END_REF][START_REF] Ipser | Transition Metal-Chalcogen Systems, V.: The iron-Tellurium Phase Diagram[END_REF][START_REF]X-Ray Investigation of Manganese Selenide, Iron Telluride, Nickel-Zinc, and Platinum-Zinc[END_REF][START_REF] Tengnér | Diselenides and Ditellurides of Iron, Cobalt and Nickel[END_REF][START_REF] Thompson | Frohbergite, FeTe2: A New Member of the Marcasite Group[END_REF]. As shown in Figure 1a, Ipser et al. proposed the existence of and ' single-phase regions with an intermediate two-phase region. The CdI 2 -type ordering of the hexagonal phase was confirmed by Mössbauer and neutron diffraction studies [START_REF] Howe | 57Fe Mössbauer diffusional line-broadening and neutron diffraction studies of Fe1.33Te2[END_REF][START_REF] Reddy | Mössbauer Studies on the Fe-Te System[END_REF]. While the monoclinic phase is related to the NiAs structure family, the exact space group remains unknown. In the Ni-Te system, the Ni-rich phase is of the NiAs structure, experiencing a 2 nd order transition into the CdI 2 -type with increased Te-content [START_REF] Coffin | The observation of an order-disorder transition in the ni-Te system by neutron diffraction[END_REF][START_REF] Norén | An Electron and X-Ray Diffraction Investigation of Ni1+xTe2 and Ni1+xSe2 CdI2/NiAs Type Solid Solution Phases[END_REF]. Modelling has supported the possibility of a order-disorder transformation in the Fe-Te system, but it has not been experimentally confirmed [START_REF] Ipser | Nonstoichiometry and the NiAs structure[END_REF].

The ordered nature of the phase has been confirmed by neutron diffraction studies as an intermediate state between the Cu 2 Sb and PbO type structures, with half of the tetrahedral sites filled with Fe atoms, and excess Fe in the octahedral sites [START_REF] Leciejewicz | A Neutron Diffraction Study of Magnetic Ordering in Iron Telluride[END_REF][START_REF] Fruchart | Structure antiferromagnetique de Fe1.125Te accompagnee d'une deformation monoclinique[END_REF]. The high-temperature phase was observed by Røst and Webjørnsen as similar to the (160) Fe 1.5 Ni 1.5 Te 2 phase found by Stevels [START_REF] Stevels | Phase Transitions in Nickel and Copper Selenides and Tellurides[END_REF], a ternary phase that was later re-characterized as (166) by Åkesson and Røst, known as Fe 0.28 Ni 0.28 Te 0.44 [START_REF] Røst | A High Temperature Iron Telluride with Rhobohedral Structure[END_REF][START_REF] Åkesson | The Crystal Structure of Fe.28Ni.28Te.44[END_REF]. Only a few reflections were noted, with qualitative intensities, but all shared with the Fe 0.28 Ni 0.28 Te 0.44 structure.

Reflections of a phase of unknown crystal structure were found between the and phase, called the phase, believed to exist at 54.2 at% Te (Figures 1) [START_REF] Grønvold | Phase and Structural Relations in the System Iron Tellurium[END_REF][START_REF] Ipser | Transition Metal-Chalcogen Systems, V.: The iron-Tellurium Phase Diagram[END_REF]. Although much work has been done to study the Fe-Te system, some questions remain that may significantly affect the thermodynamic modeling of the system. Mann and Van Vlack [START_REF] Mann Och | Fe1.2Te-MnTe Phase Relationships in the Presence of Excess Iron[END_REF] suggested a liquid miscibility gap in the Fe-FeTe region (Figure 1b) which was accepted by Okamoto and Tanner [START_REF] Okamoto | The Fe-Te (Iron-Tellurium) System[END_REF], while no Fe-rich liquidus data nor activity data exist to support this. The Te-rich phase boundary of the phase remains largely uncertain, as well as the transformation and the solvus lines of the and phases.

In the present work, we performed isothermal heat treatments with metallographic studies and DTA in order to determine these unknown or uncertain phase boundaries of the Fe-Te system, as well as gain more insight into the nature of the and phases. An explanation of the experimental methods employed follows in section 3.

Experimental methods

Materials preparation

The samples were prepared using 99.9 % pure iron wire pieces from Balzers whose surfaces were polished to remove any surface oxide, rinsed with ethanol, dried, cut and weighed before being collected in silica ampoules together with 99.999 % pure nuggets of tellurium from Goodfellow Cambridge Ltd. Before filling, ampoules were cleaned in an ultrasonic bath, rinsed with ethanol and dried.

Samples for isothermal heat treatments had a total mass of around 1 gram and samples for DTA-TGA studies had a total mass of around 80-110 mg to ensure a strong signal of weak thermal arrests. Ampoules were then backfilled with argon and sealed under primary vacuum. Further details of the isothermal heat treatments and DTA procedures follow in paragraphs 3.2-3.3.

Isothermal heat treatments and sample characterization

For the isothermal heat treatments, an array of two to four sealed ampoules was rested at an incline inside a horizontal tube furnace (

). The sample batch was held for two to four hours above the predicted melting temperature of the alloy, then held for one to four weeks at a selected temperature ( to . At the end of the treatment, the sample array was pulled out of the furnace and quenched in cold water. A decision was made not to crush the ampoules upon retrieval in order to a) avoid destroying the microstructure of the very brittle samples, b) limit oxidation; with these factors in mind, the possibility of a non-equilibrium state was considered during analysis. Retrieved samples were cold set in resin pellets, which after polishing were coated with a thin carbon layer by carbon fiber wire flash pulse prior to Scanning Electron Microscopy (SEM) analysis. The polishing was done with grit 800, 1200 and 2000 SiC paper, followed by diamond suspensions of 9, 6 and finally 1 µm particle size.

Samples were then studied via Light Optical Microscopy (LOM) and SEM (Zeiss LEO 1450VP), and phase compositions were measured with Energy Dispersive Spectroscopy (EDS). More accurate tieline determination was performed using a Cameca SX50 microprobe with Wavelength Dispersive Spectroscopy (WDS) detectors. When the solid phase composition seems uniform, it is regarded as equilibrated and the average of the flat bulk composition is taken as the solubility limit. In case the composition profile has a linear shape, it is not at equilibrium, but treated as a diffusion couple and the phase boundary evaluated from extrapolation of the composition to the interface. Most samples showed no to slight reaction with the silica ampoules, which is expected from liquid tellurium, and the alloys held isothermally in equilibrium with liquid had clearly reacted. Therefore, in all WDS analyses oxygen and silicon contaminations were also searched for; silicon contamination above the detection limit was never found. When uniform oxygen traces were observed, the composition was normalized to the Fe-Te content to best approximate the binary solubility.

Differential Thermal Analysis

A Setaram Setsys 16/18 DTA-TGA machine was used to confirm phase boundaries of the Fe-Te system, support a study of the nature of the gamma phase, and to find the believed to be eutectoid decomposition ofinto -. Since the Fe-rich liquidus had not yet been found, a sample was prepared to attempt identifying that as well.

The temperature was calibrated using the melting points of lead, zinc, silver and gold standards from SETARAM placed in alumina crucibles. For silver and zinc, platinum lids covered the crucibles to limit mass loss. For each run with alloys, sealed silica ampoules were placed in the bottom of open alumina crucibles: one containing an Fe-Te sample, and one an empty reference ampoule. To check the delay in heat conduction through the silica ampoule to the bottom of the alumina crucible, a measurement was performed on gold at 5, 3 and 1 K/min; a rather constant temperature offset from the calibration measurement of 0.4 K was found and thus subtracted from all subsequent measurements on alloy samples.

All alloy samples were melted in the machine for 30 to 40 minutes followed by cycles of heating and cooling programmed at rates of 5, 3 and 2 or 1 K/min. Argon was used as the carrier-and furnace protection gas together with a closed coolant circuit of water kept at 20 °C. Due to a lack of calibration standards that do not experience significant supercooling, the apparatus was not calibrated for cooling cycles. The methodology recommended by Boettinger et al. was followed for the sampling of onsets and offsets of reactions [START_REF] Boettinger | DTA and Heat-Flux DSC Measurements of Alloy Melting and Freezing[END_REF].

X-ray Diffraction and structure refinement

X-ray Diffraction (XRD) was performed on some of the samples in order to verify present phases, as well as attempt to identify the crystal structure of the phase and the space group of the phase. FT51_T2, FT54_S and FT55_X were machine-milled and the others milled by mortar and pestle. Powders were then filtered using a 20µm test sieve. A Brucker D8 Advance and a PANalytical X'Pert PRO diffractometer were used with asetup. Samples were spinned in order to improve the measurement statistics of the XRD patterns. For the first few samples a Cu X-ray tube was used. A knife was placed by the sample to mitigate diffusion at low angle, and the 1-dimensional linear Lynxeye detector cutoff energy was set above the energy of the Fe fluorescence line. Subsequently, a Co X-ray tube was used with an X'Celerator 1-dimensional detector. Both X-ray sources had a intensity ratio of 2:1.

Rietveld refinement was performed using the MAUD software [START_REF] Lutterotti | Total pattern fitting for the combined size-strain-stress-texture determination in thin film diffraction[END_REF][START_REF] Lutterotti | Rietveld texture analysis from diffraction images[END_REF], with an isotropic size (Gaussian distribution)-strain (lorentzian contribution) model considering a Pseudo-Voigt line profilefunction with axial divergence [START_REF] De Keijser | Use of the Voigt function in a single-line method for the analysis of X-ray diffraction line broadening[END_REF]. Due to absorption by Te, the peak intensity decreases with higher angle; therefore, Fe site fractions could not be well refined, and were instead fixed at probable compositions particularly deduced by structure relaxation via DFT computation. Moreover, due to residual texturing effects including the observation of platelet phases, the Le Bail method [START_REF] Lebail | Whole Powder Pattern Decomposition Methods and Applications: A Retrospection[END_REF] was used as structure factor model.

Results and discussion

Phase diagram data from heat treatments

Table 2 presents the prepared compositions and expected phases of the samples for isothermal heat treatments. The sample names identify two things: first, FT# implies an Fe-Te sample of # at% Te. Second, suffixes denote purpose: T for tie-line determination, S for single-phase composition and X for crystal structure determination via XRD. A final number was added if several samples of the same composition were prepared. As can be seen in Table 2, some samples were heat treated for 4 weeks to ensure thermodynamic equilibrium at such low temperatures. Table 3 summarizes phase boundaries of two-phase samples evaluated from the composition analyses via EDS and WDS. The respective tie-lines are shown in Figure 2, imposed on the calculated phase diagram compared with published phase boundary data (see Part II: Thermodynamic modelling). Filled symbols represent the prepared average sample compositions (Table 2), and empty symbols represent the evaluated phase boundary compositions (Table 3). The tie-lines are overall consistent with available solubility data.

Most heat treated samples showed no to small interaction with the silica ampoule. Samples treated in equilibrium with liquid showed a thin, soot-like, dark layer on the inner wall of the ampoule; while a reaction had clearly taken place, the extent was deemed small enough to not affect the state of the bulk sample and the ampoules remained intact. The largest extent of reaction was seen in sample FT58_S2, being cratered with large semi-spherical bubbles. The outer rim of the sample showed a clear and uniform, about 5 µm thick, region of phase composition. Analysis of the deposits on the ampoule revealed tellurium oxide particles. In all two-phase samples, oxygen was concentrated in the most Fe-rich phase, i.e. the phase with more vacant interstitial sites.

A discussion of the composition analysis of the individual heat treated samples follows, divided into different regions of the phase diagram with descending tellurium content.

Samples of 77 to 65 at% Te

As seen in Figure 5, sample FT77_T consists of primary phase precipitates and a quenched liquid phase that solidified into dendrites and tellurium, in agreement with the phase diagram. A lack of grain boundaries in the quenched matrix phase signifies a possibly retained amorphous state. Precipitates of pure tellurium <0.4 µm in size were also observed embedded in the matrix. FT77_T showed slightly more reaction with the silica than other samples, with the exception of FT58_S2, due to both the higher tellurium content, and the partially liquid state of FT77_T during heat treatment. The sample-ampoule reaction resulted in a tellurium depletion about 10 µm into the sample, which was deemed not to have significantly affected the average solid phase composition. The XRD powder pattern could not be refined due to very high noise as well as an oscillating background, possibly from amorphous tellurium. Reflections from phase and traces of pure Te were identified, but also traces of phase, the reason of which cannot be readily explained; it could be due to the interaction between liquid solution and the ampoule.

As seen in Figure 8, FT65_T contained large bands of pores as well as cracks delineating almost all phase boundaries. The areas were completely smooth while was cratered with surface damage from polishing (Figure 5 and Figure 8).

In both samples, EDS and WDS showed on average flat composition profiles in the phase interfaces, and they were regarded as equilibrated with a maximum data spread of ±0.5 at% Te for the phase and ±0.3 at% Te for the phase (Table 3). No oxygen was found in the phase but there were detectable levels in ; oxygen signal peaks were mostly related to surface damage, which might explain the larger error in equilibrium bulk concentration of .

Samples of 54 to 51 at% Te

FT54_S was prepared with the aim to obtain a pure phase; instead, what seemed to be a twophase equilibrium of was obtained. This may be due to the difficulty of nucleating , as has been noted before [START_REF] Ipser | Transition Metal-Chalcogen Systems, V.: The iron-Tellurium Phase Diagram[END_REF]. However, a foreign microstructure was observed in the phase region with composition consistent with the phase, shown in back-scattered electron imaging of Figure 3. The stripes seem to be aligned along the grain orientation (See Figure S1, Supplementary material), and were about 1 µm wide, as can be seen in Figure S2. Due to a stripe width of about the same size as the affected zone of EDS and WDS analysis (1 µm), discrete composition differences between the regions could not be accurately determined, but the minimum and maximum compositions found from 36 points in EDS were 55.3 at% Te and 59.7 at% Te; the former is close to the phase (54.2 at% Te), the latter seems consistent with the phase. The formation of on quenching is possible since the phase has previously been found to easily decompose into and on cooling [START_REF] Ipser | Transition Metal-Chalcogen Systems, V.: The iron-Tellurium Phase Diagram[END_REF]. FT51_T2 was prepared at a higher temperature in the region, but again no phase region with a composition of the phase was found. Here the phase fraction was much smaller, with fewer and finer Te-rich stripes, appearing to precipitate in and emanate from the phase boundaries (Figure 4). No such stripes were found in the two samples FT58_S1 and FT58_S2, prepared to be pure phase. This absence means that the striped microstructure may not merely be the partial decomposition of into , since that would also have been observed in FT58_S1 and FT58_S2.

It is interesting to note that the phases in samples FT54_S and FT51_T2 grow in µm size sheets, similarly to what is usually called 2D crystals (see Figures S3-S8, Supplementary material). This is common for the NiAs structure family of transition metal-chalcogenides that tend to grow in layers [START_REF] Burton | Prediction of entropy stabilized incommensurate phases in the system MoS2-MoTe2[END_REF][START_REF] Coleman | Two-Dimensional Nanosheets Produced by Liquid Exfoliation of Layered Materials[END_REF]. The actual crystal thickness cannot be deduced from the SEM images, since they may be thinner than is visible (See Figure S9 for magnification). The BSE image of Figure S10 shows that the striped structure discussed above is not related to individual layers of phase, since they are here seen to cross over several crystals. This indicates that the stripes probably formed later than the precipitation of phase. With the above discussion in mind, it is possible that the phase is a NiAsrelated structure, and forms by diffusion over short distances in the phase. This could explain the preferred formation of the structure only in phase regions, and the parallel orientation.

The high average oxygen content of FT54_S, shown in Table 3 to be about twice the detection limit of around 0.2 wt%, is not negligible. Indeed, oxygen seems associated with the phase. All strong oxygen signals are found at surface effects, e.g. cracks, pits, and scratches. The composition analyses of the interfaces of both FT54_S and FT51_T2 showed a small composition gradient 5-10 µm into the phase followed by a flattening of the profile into the bulk of the phase, with a spread of ±0.5 at% Te. This gradient may have two explanations; either it is a result of iron diffusion during precipitation of the observed striped microstructure, or the sample never reached thermodynamic equilibrium. The normalized average of the flat portion of the profile was used as the phase boundary of the phase, with large accepted error, and they were rather consistent with the tielines by Ipser et al. [START_REF] Ipser | Transition Metal-Chalcogen Systems, V.: The iron-Tellurium Phase Diagram[END_REF].

The diffraction patterns of these samples showed high levels of noise, peak broadening and decaying peak intensity due to absorption by Te (see figures S11 and S12, Supplementary material). Rietveld refinement could match most reflections with the and phases, with rather low accuracy due to the broadening (see Table 4 for the results). The a parameter of the phase is consistent with literature [START_REF] Grønvold | Phase and Structural Relations in the System Iron Tellurium[END_REF], while the c parameter deviates. The space group of was deduced from the rietveld refinement of sample FT55_X, as will be detailed below. Broadening was not completely accounted for in FT54_S, since crystal size would not converge well given the tricky distribution of data points in the pattern. Additional low-angle peaks were found at 22.6 and 24.9 , exclusively present in all machine-milled samples. They seem to be a convolution of both discrete and broad peaks. The structure that best matches the additional peaks is of space group (Figure S12); the sample was probably oxidized during milling. One small peak at 27.5 remains unmatched. Sample FT54_S had one abnormally intense peak associated with the (200) reflection of the phase.

Sample of 55 at% Te

Sample FT55_X, of a composition in the region close to the tentative phase boundary, was treated isothermally at 575 for 4 weeks, in order to deduce if the sample contains hexagonal or monoclinic phase. This sample was more efficiently quenched by breaking the ampoule on retrieval.

WDS analysis of the sample was inconclusive due to bad focus resulting in a very low sum of wt%. As seen in Table 3 this gave a standard deviation of 5 at% Te. The tie-line, as shown in Figure 2, can therefore not be trusted. Such a shape of the phase diagram could be explained if the invariant of phase formation were in fact the eutectoid transition; however, Ipser et al found the phase to be stable below this temperature.

The XRD analysis showed mostly phase, and a possible trace amount of phase, as also seen in the LOM image of Figure 9. A possible monoclinic modification of the hexagonal NiAs type structure is the space group, and an initial refinement found that an ordered superstructure of the space group can also fit the pattern. The latter unit cell was relaxed via DFT, as further detailed in the modelling Part II, resulting in more realistic atomic positions. This structure was further refined via rietveld analysis, and compared with the and space groups as shown in Figure A1 (Appendix); space group fit identically well as , and is therefore left out. While and can equally well fit the same peaks, accounts for additionally two otherwise unmatched peaks, one of which is the one at 27.5 also found in FT51_T2 and FT54_S. As seen in Table 4, the weighted profile R-factors ( ) show that although the space groups fit the pattern almost equally well, gives the better fit. Note that the high noise, and the large unmatched peaks makes the Rwp less useful than a simple qualitative comparison. The final crystallographic data on the phase of sample FT55_X are detailed in (Table A1, Appendix).

Samples of 47 to 25 at% Te

FT25_T1 and FT25_T2 were prepared in theregion. As seen in Figure 6 and Figure 10, the analyzed samples showed cores of pure iron with thick phase layers. With the assumed limited diffusion of Te through Fe, the samples have not reached uniform composition, and are therefore not at equilibrium. Although the phase has not reached an equilibrium iron content, the sample may be treated as a diffusion couple for the determination of Te solubility in -. The last points in the phase of the composition profiles before theinterface were rather consistent over several line scans, and taken as the solubility limit, consistent with the tie-line published by Ipser et al (Figure 2).

In addition to these samples, FT47_S1 and FT47_S2 were prepared in the ' single-phase region. Retaining some phase was attempted, even though failure was expected. Judging by the EDS analysis FT47_S1 had a uniform composition. These samples also contained sheet crystals of µm size, similarly to the phases, which is expected from PbO structure chalcogenides [START_REF] Oyler | Chemical Synthesis of Two-Dimensional iron Chalcogenide Nanosheets: FeSe, FeTe, Fe(Se,Te), and FeTe2[END_REF]. SEM and LOM images revealed layered growth in the entire sample, as shown in Figure 7. Both samples showed parallel cracks in the metallographic analyses, probably aligned with the 2D crystal planes since the ordered phase should have weak bonds between the Te-Te layers lacking interstitial iron. Metallography of FT47_S2 showed a similar appearance and uniform composition. As seen in Figure S13 (Supplementary material), XRD analysis of FT47_S2 showed complete decomposition into the phase, as expected. The lattice parameters deduced from rietveld refinement (Table 4) correspond well with available literature [START_REF] Grønvold | Phase and Structural Relations in the System Iron Tellurium[END_REF].

Phase transition data from DTA

Thermograms of Fe-Te alloys, normalized to linear baselines, are presented in Figure 11 superimposed over the calculated phase diagram. The baselines are positioned roughly at the corresponding sample compositions, and red markers indicate where the onset of reaction (deviation from baseline) or end of reaction (local maximum of heat curve) were sampled. All invariant reactions are consistently found a few degrees lower than previous DTA studies. This could be due to changing standards of sampling; earlier it was common to use the extrapolated onset of reaction, as is done for pure elements, while presently it is accepted to sample the first deviation from the baseline as the onset of reaction [START_REF] Boettinger | DTA and Heat-Flux DSC Measurements of Alloy Melting and Freezing[END_REF], as was done in this work.

Table 5 shows the observed reactions from the DTA measurements. All measurements show good agreement with the phase diagram proposed by Ipser et al. (Figure 1a), indicating that the sample compositions were successfully prepared. Chemical analysis found no silicon dissolved in the samples; the results however over-predicted the composition by 1.56-4.37 at% Te with a relative error of 3 %. Those compositions are highly improbable, since the very clear thermal effects would not have been observed if the chemical analyses had been accurate, indicating a possible contaminant or anomaly during the analyses. If this contaminant were oxygen, either from remaining oxide of the pure materials or reaction with the silica ampoules, the good agreement with previous experiments indicate that it does not significantly offset the reaction temperatures from the binary phase diagram boundaries.

While Ipser et al. did not find the iron rich liquidus, this work raised the studied temperature of sample FT47_D; thus a small but clear local maximum was observed before returning to the baseline, i.e. the liquidus (for a magnification, See Figure S14 in Supplementary material). Thermal effects on heating were found consistent with the β' phase peritectoid formation, solvus and peritectic melting. The solidus could not be clearly determined, since the diffuse onset of melting was difficult to quantify (labeled too weak in Table 5). This experiment, and all subsequent ones, showed periodic oscillations in the TG and heat curves, effectively hiding any small thermal effect in thermograms of cycles at 1 K/min. Modification of the setup to use tap water to cool the furnace mitigated the issue. This change of setup was not found detrimental to the calibration.

FT51_D experienced several weak steps and bumps in the baseline on heating and cooling, one of which was consistent with the suggested formation of the γ phase [START_REF] Ipser | Transition Metal-Chalcogen Systems, V.: The iron-Tellurium Phase Diagram[END_REF]. This roughly lambda-shaped peak was clear in FT54_D. Chiba believed that this peak, found in FeTe and FeTe 1.08 , was an anomaly related to the strange shape of the phase solvus (Peak B in [START_REF] Chiba | The Magnetic Properties and Phase Diagram of Iron Tellurium System[END_REF]). Chiba related the peak in FeTe 1.33 to this anomaly, but it was actually the solvus line. The invariant arrests of γ phase and δ phase decomposition were not separated on heating in FT51_D, although after several runs peak separation was observed on cooling. They separated inconsistently between cycles, and therefore the additional peaks could be irreversible artifacts. The FT54_D measurements were also focused on finding the δ' to δ transition without success. As a result, FT58_D was prepared at the hypothesized eutectoid composition of said reaction, but the baseline was completely flat in the region. It was, however, noted that all samples at lower compositions than the predicted eutectoid showed a second onset in the peak assigned to the just mentioned reaction.

Returning to the analysis of reactions above and below the phase, customized silica ampoules were created to fit directly to the DTA thermocouples in order to obtain a more highly resolved signal, and FT54_D2 was thusly analyzed. Figure 12 shows the slowest cycle of heating and cooling, normalized to linear baselines. It can be seen in Figure 12 that the final local maximum on heating, and the first onset of reaction on cooling, i.e. the liquidus, coincide well. Subsequent reactions on cooling, however, do not coincide, thus indicating supercooling; all DTA samples exhibited this, at least to some extent. The second onset is very clearly seen, on peak (1), as well as the heat of phase formation, peak (2); the higher-temperature peaks were now separated into the decomposition of and phases on heating (Figure 13). The apparatus was not yet calibrated for these crucibles, so in order to identify the separated peaks, the heating spectrum was normalized between four well known reactions in the phase diagram: onset (1) in Figure 12 was fixed to the known invariant; onset (2) was fixed to the invariant; onset (3) in Figure 13 was fixed to the eutectoid reaction; and local maximum [START_REF] Pulham | Chemical reactions of caesium, tellurium and oxygen with fast breeder reactor cladding alloys. Part I -The corrosion by tellurium[END_REF] was fixed to the liquidus. This normalization was sufficient to give an approximate linear correlation between measured and real temperature. As a result, peak (6) lined up approximately with the invariant , and peak (5) aligned with our other measurements of the invariant. The respective magnitudes of peak ( 5) and ( 6) also support this finding. This leaves offset (4) unassigned to any presently known reaction; while it is small, it is clearly preceded by a short interval at baseline level, then turning into a linear slope that may be extrapolated towards the local maximum of the liquidus (see dashed line, Figure 13). This looks characteristic of going from a single-phase region, through a series of two-phase regions divided by two invariant arrests, before eventually completely melting. This may suggest that the phase is not stoichiometric, but has a narrow homogeneity range, and that this sample happens to pass through a single-phase region between peak (3) and ( 4), after which one briefly passes through a two-phase region before decomposes at peak [START_REF] Mathews | Thermochemistry of transition metal tellurides of interest in nuclear technology[END_REF]. Another possible explanation is that the phase somehow extends to be stable above the phase. This is not probable, however, since a rather high-temperature phase solvus was found in FT58_D. One should note how weak the formation peak ( 2) is, which also supports the possibility that the phase is similar to the phase. FT58_D, FT61_D, FT63_D and FT64_D were consistent with the liquidus, solidus and peritectic invariants found by Ipser et al. [START_REF] Ipser | Transition Metal-Chalcogen Systems, V.: The iron-Tellurium Phase Diagram[END_REF]. The rightmost solvus of the δ phase was found in FT63_D and FT64_D, while the onset was very weak in FT61_D, and therefore ignored. These data agree more with the tie-line determined by Ipser et al. and the tie-line of sample FT65_T than the phase boundary determined by Chiba [START_REF] Chiba | The Magnetic Properties and Phase Diagram of Iron Tellurium System[END_REF]. 4) Unknown. ( 5) . ( 6) . ( 7) . With these new DTA and tie-line data, this work has contributed to an improved characterization of the phase boundaries of the Fe-Te system. We lowered most invariant arrests by a few degrees, found a point on the Fe-rich liquidus, found the probable eutectoid reaction between the and phases to lie very close to the eutectoid formation temperature, as well as adding and phase boundaries. We showed indications that the phase may have a narrow homogeneity range, and metallographic studies note that the phase may have a crystal structure related to the phase. If the striped microstructures found in our samples do relate to the phase, the phase seems to form either in cooperative growth with the phase, or out of a parent phase. Regardless of the nature of the phase, the related invariant arrests certainly exist. We highlight some key points for future studies of the system.

Conclusions

1) The phase boundaries of the phase were approximated from DTA measurements of invariant arrests [START_REF] Ipser | Transition Metal-Chalcogen Systems, V.: The iron-Tellurium Phase Diagram[END_REF], while less accurate solubility data estimated from XRD [START_REF] Røst | A High Temperature Iron Telluride with Rhobohedral Structure[END_REF] suggest a congruent transition. We recommend for future studies to confirm the Fe-rich phase boundaries of the and phases, with rigorous sample preparation ensuring equilibrium. In addition, we recommend performing In-situ high-temperature XRD to confirm if the phase is indeed the same crystal structure as the ternary phase , as has been proposed in this work.

2) It would be useful to further verify that the space group of the phase is using better, high-energy diffraction methods, or for example Raman spectroscopy or EBSD.

3) The phase seems to be related to the NiAs ( ) structure family. It could be useful to use Ab initio methods to suggest a stable crystal structure. In addition, a large number of samples for isothermal heat treatment can be prepared in order to metallographically analyze the striped microstructure further, e.g. via EBSD. 4) Furthermore, it would be useful to the modeling to design experiments to identify the presence of a liquid miscibility gap in the Fe-FeTe composition interval. This might prove challenging, since samples need to be contained in vessels that can withstand the reactive tellurium as well as up to 10-100 kPa of vapor pressure.

The data gathered in this paper will be used in our subsequent publication Part II: Thermodynamic modelling for a critical thermodynamic assessment of the system.

Figure 2 :

 2 Figure 2: Calculated Fe-Te phase diagram from Part II: Thermodynamic modelling, compared with tie-lines from isothermal heat treatments, phase boundaries from DTA, and selected published data

Figure 3 :

 3 Figure 3: BSE image of the striped microstructure in sample FT54_S. Image shows stripes of varying direction in different grains around a pore.

Figure 4 :

 4 Figure 4: BSE image of boundary between (dark gray) and (light gray) phases in sample FT51_T2. Here the stripe precipitates seem to emanate from the phase boundary.

Figure 5 :

 5 Figure 5: BSE image of a corner of sample FT77_T. phase has been in equilibrium with liquid, which has on cooling solidified into dendrites and tellurium.

Figure 6 :

 6 Figure 6: BSE image of FT25_T1, magnified on the boundary between the iron core and the surrounding scale of phase.

Figure 7 :

 7 Figure 7: SEM image taken between precipitates in sample FT47_S1, showing the layers of the phase.

Figure 8 :

 8 Figure 8: LOM image of sample FT65_T. The phase has precipitated between bands of phase, with significant porosity and intergranular cracking.

Figure 9 :

 9 Figure 9: LOM image of FT55_X. Bright regions are or phase, small gray precipitates phase.

Figure 10 :

 10 Figure 10: SEM image of sample FT25_T2. A piece of iron wire is surrounded by phase, and a separate piece entirely consists of phase.

Figure 11 :Figure 12 :

 1112 Figure 11: Calculated Fe-Te phase diagram with superimposed thermograms from the DTA heating cycles. Flat baselines mark sample compositions. Red marks show the average sampled temperature of reactions.

Figure 13 :

 13 Figure 13: Magnification of the last group of heat signals during heating of FT54_D2 at 2 K/min. Dashed blue line is to aid heat curve interpretation. Probable reactions: (3). (4) Unknown. (5) . (6) . (7) .

  Tie-lines of the Fe-Te phase diagram have been determined by EDS and WDS measurements on isothermally heat-treated samples. Phase boundaries have been quantified from DTA measurements. Rietveld refinement of XRD powder patterns corroborated with DFT calculations (Part II: Thermodynamic Modelling) lead us to propose a new space group of the phase.

Table 1 :

 1 Crystallographic data of Fe-Te phases. Lattice parameters available in literature is given for phases relevant to this work. lt: low-temperature phase, ht: high-temperature phase, hp: high-pressure phase. *: Crystal structure not confirmed. **: Suggested by Chevreton[START_REF] Chevreton | Etude structurale des séléniures et tellurures de métaux de transition, en particulier du chrome[END_REF], without evidence; this work later finds that the phase may be of this space group, however not this prototype.

			Pearson		Lattice parameters
		Phase	symbol	Space group	a	b	c		Prototype	Ref.
									W
									Cu
	-	* (ht)			4.013	-	20.96	-	Fe 0.28 Ni 0.28 Te 0.44	[34]
					4.013	-	20.96	-		[13]
	-	(lt)			3.82	-	6.28	-	Cu 2 Sb	[12,31]
					3.82	-	6.28	-	PbO	[12,31]
	( -	lt)								[31,32]
	-	* (ht)	N/A	N/A			N/A		N/A	[12]
	-	* (ht2)		N/A	3.846	-	5.641	90.2	N/A	[12,19]
	Suggested:				This work
						This work		Cr 3 S 4 **	[22]
	-	(ht1)			3.80	-	5.662	-	CdI 2	[26,36]
			(		3.80	-	5.662	-	NiAs)	[12]
	-	(lt)			5.26	6.268 3.876 -	FeS 2 (Marcasite)	[18]
	( -	hp)							FeS 2 (Pyrite)	[37]
					4.466	-	5.919	-	-Se	[38]

Table 2 :

 2 A priori properties of isothermally heat treated Fe-Te samples. *: Rounding error in prepared composition are all <0.007 at% Te.

		Prepared composition	Heat treatment
	Sample ID	at% Te*	wt% Te	Mass [mg]	T [K]	Time [h]	Expected phase(s)
	FT77_T	76.83	88.34	1093	866	672	Liq
	FT65_T	65.33	81.15	1125	866	672
	FT58_S1	58.37	76.21	1167	954	331
	FT58_S2	58.41	76.24	1118	1043	330
	FT55_X	55.13	73.74	1588	848	672
	FT54_S	54.09	72.92	928	954	331
	FT51_T1	50.99	70.39	865	1142	336	Liq
	FT51_T2	51.01	70.41	888	1043	330
	FT47_S1	47.05	67.00	1083	1142	336
	FT47_S2	47.12	67.06	1047	1142	168
	FT25_T1	25.58	43.99	918	954	331
	FT25_T2	24.80	42.97	911	1043	330

Table 3 :

 3 Condensed results of WDS composition analysis for tie-line determination. *: Sample machine milled and partially oxidized, probably into Fe 3 O 4 and Fe 2 O 6 Te

	Sample ID	Region	Boundary: composition [at% Te]	WDS wt% oxygen mean max	mean sum-wt 1 %	Phases found by XRD
	FT77_T		: 68.3±0.3	: N/A	2		0	0.24	100.44	, trace
	FT65_T		: 63.3±0.5	: 66.8±0.3	0.04	0.28	99.79	Not analyzed
	FT55_X		: 49.43±5	: 59.88±5	0.2	0.44	93.82	, trace *
	FT54_S		: 48.5±0.5	: 57.0±0.7 3	0.41	1.52	101.27	*
	FT51_T2		: 48.4±0.5	: 55.2±0.5	3	0.22	1.11	101.3	*
	FT25_T1		-: 0.3±0.05	: N/E	4		0.22	1.43	102.04	Not analyzed
	FT25_T2		-: 0.7±0.4	: N/E		0.10	1	102.85	Not analyzed

Table 4 :

 4 Results of powder pattern refinement on selected samples, with optimized lattice parameters and weighted profile R-factor (Rwp). angles were fixed, since the parameter does not converge with the given quality of measurement.

	Sample ID,
	target	Space group	a [Å]	b [Å]	c [Å]	[ ]	%
	FT47_S2, Cu		3.821 ±2E-4	-	6.266 ±2E-4	-	48.8
	FT51_T2, Co		3.84 ±6E-4	-	6.33 ±0.002	-	40.1
			6.70 ±0.005	4.10 ±6E-4	10.94 ±0.006	90.65
			4.66 ±0.001	-	9.39 ±0.004
	FT54_S, Co		3.82 ±0.002	-	6.45 ±0.005	-	92.1
			6.70 ±0.003	4.08 ±8e-4	10.94 ±0.006	90.65
			4.67 ±7E-4	-	9.39 ±0.006
	FT55_X, Co		6.70 ±0.007	3.86 ±0.005	11.23 ±0.005	90.65	42.3
			3.876 ±4E-4	6.648 ±6E-4	5.585 ±2E-6	90.2	44.4
			3.851 ±0.001	-	5.603 ±0.003	-	44.5
			4.65 ±0.001		9.36 ±0.006

Table 5 :

 5 Thermal effects of the Fe-Te system found in this work via DTA. Reference composition only shown when different from that of this work; irrelevant for invariant reactions. *: Two invariant reactions very close together, peaks not separated. **: Maximum rounding error in prepared composition.

			Measured	Ref. T [K]	Type of	Phase transformation
	Sample ID	at% Te**	T±1 [K]	(at% Te) [19]	reaction	on cooling
	FT47_D	47.65±0.07	1345	N/A	Liquidus
			1191	1187	Peritectic
			1092	1093 (47.6)	Solvus
			1071	1073	Eutectoid
	FT51_D	51.64±0.10	1165	1161 (51.65)	Liquidus
			1081	1082	Peritectoid*
			1072	1073	Eutectoid
			910	909	Eutectoid
			797	N/A	Eutectoid
			790	792	Eutectoid
	FT54_D	54.83±0.10	1089	1090 (54.90)	Liquidus
			1078	1082	Peritectoid
			909	909	Eutectoid
			795	N/A	Eutectoid
			789	792	Eutectoid
	FT58_D	58.84±0.06	1075	1075 (59.05)	Liquidus
			1037	Not comparable	Solidus
			787	792	Eutectoid
	FT61_D	60.81±0.08	1062	1061 (61.25)	Liquidus
			1037	1039	Peritectic
			1020	1020 (61.25)	Solidus
			787	792	Eutectoid
	FT63_D	63.25±0.14	1047	1052 (62.45)	Liquidus
			1038	1039	Peritectic
			996	994 (62.45)	Solidus
			856	N/A	Solvus
			790	792	Eutectoid
	FT64_D	64.28±0.07	1033	1038 (64.50)	Liquidus
			961	N/A	Solidus
			886	N/A	Solvus
			789	792	Eutectoid

The sum includes Fe, Te, Si and O

The dendritic region, was not analyzed in FT77_T.

Average composition of two-phase microstructure that used to be phase

Not in equilibrium.

Acknowledgements

The authors would like to thank the Swedish Research Council (Vetenskapsrådet) for the funding of the PhD project, which contributes to the SAFARI project in the framework of CEA-SRC collaboration of R&D for the ASTRID reactor. Furthermore, C.-M. A. is grateful for the resources provided by CEA to perform the experimental work, the scientists and engineers that taught him how to use them and aided him in metallographic studeis: Sylvie Chatain for teaching him how to calibrate and use the DTA apparatus. Kevin Ginestar for teaching how to prepare samples and use the SEM. Christophe Bonnet for aiding with sample preparation for XRD. Dominique Gosset and Justine Roubaud for performing XRD measurements and performing the initial analysis of sample FT55_X. He also thanks Prof. Malin Selleby for supervision and oversight during the work at KTH, and his colleagues at CEA and KTH for all the advice provided for the analysis of microstructures.

Finally, C.-M. wishes to thank Professor Herbert Ipser for discussions regarding the peculiarities of the Fe-Te system, and especially for personally lending him a copy of his thesis.

Appendix


Kai Nordlund
email: kai.nordlund@helsinki.fi

Steven J Zinkle

Andrea E Sand

Fredric Granberg

Robert S Averback

Roger Stoller

Tomoaki Suzudo

Lorenzo Malerba

Florian Banhart

William J Weber

Francois Willaime

Sergei Dudarev

David Simeone

Improving atomic displacement and replacement calculations with physically realistic damage models

Improving atomic displacement and replacementcalculations with physically realistic damagemodels

Introduction

Quantification of the amount of displacement damage introduced by energetic particle interactions in matter is important for a broad range of fundamental science and applied engineering applications ranging from semiconductor physics to nuclear energy generation. [START_REF] Seitz | Displacement of atoms during irradiation[END_REF] Kinchin and Pease 2 developed the basis for an early model to calculate displacements per atom (dpa) by considering kinetic energy transfers above a threshold material-specific "displacement energy". The current de facto international standard for quantifying atomic displacement levels in irradiated materials is based on the more than 40 year old binary collision computer simulations of ion collisions in solids [START_REF] Robinson | Computer simulation of atomic-displacement cascades in solids in the binary-collision approximation[END_REF][START_REF] Norgett | A proposed method of calculating displacement dose rates[END_REF] . The predicted number of atomic displacements (Nd) as a function of cascade energy, or the damage function, is given in this model by

( ) = ⎣ ⎢ ⎢ ⎢ ⎡ 0 , < 1 , < < . . , . < < ∞ ⎦ ⎥ ⎥ ⎥ ⎤ ( 1
)
where Td is the damage energy, i.e. the kinetic energy available for creating atomic displacements. The damage energy for a single ion is given by the total ion energy minus the energy lost to electronic interactions (ionization). Typical values of Ed for different materials range from 20 to 100 eV. [START_REF] Jung | Atomic Defects in Metals, Landolt-Börnstein-Group III Condensed Matter[END_REF][START_REF] Averback | Atomic displacement processes in irradiated metals[END_REF] This is essentially the Kinchin-Pease model, except that the original kinetic energy term was replaced by the damage energy to account for ionization effects and a factor of 0.8 was introduced to account for more realistic interatomic potentials.

The importance of the calculated dpa parameter is that it is the starting point for calculations of virtually all radiation effects in solid materials, and it facilitates quantitative comparisons of different materials irradiated with the same irradiation source as well as materials irradiated in different irradiation sources such as electron, ion and neutron irradiation [START_REF] Seitz | Displacement of atoms during irradiation[END_REF][START_REF] Kinchin | The displacement of atoms in solids[END_REF][START_REF] Robinson | Computer simulation of atomic-displacement cascades in solids in the binary-collision approximation[END_REF][START_REF] Norgett | A proposed method of calculating displacement dose rates[END_REF][START_REF] Jung | Atomic Defects in Metals, Landolt-Börnstein-Group III Condensed Matter[END_REF][START_REF] Averback | Atomic displacement processes in irradiated metals[END_REF][START_REF] Farrell | An evaluation of low temperature radiation embrittlement mechanisms in ferritic alloys[END_REF][START_REF] Jones | Surveillance of reactor pressure vessel (RPV) embrittlement in Magnox reactors[END_REF] facilities. Estimation of the damage is also important in modern materials processing by focused ion beams, or when irradiating nanomaterials [START_REF] Mayer | TEM sample preparation and FIBinduced damage[END_REF][START_REF] Krasheninnikov | Ion and electron irradiation-induced effects in nanostructured materials[END_REF] . However, it has been recognized for several decades that the dpa value calculated from equation (1) for energetic cascades in pure metals on the one hand overestimates the number of stable defects by a factor of 3 to 4 (Refs. 11-14), and on the other hand underestimates the amount of atomic mixing (atoms permanently displaced from their initial lattice position to replace an atom in another position) [START_REF] Zinkle | Analysis of displacement damage and defect production under cascade damage conditions[END_REF][START_REF] Kirk | Atomic rearrangements in ordered fcc alloys during neutron irradiation[END_REF][START_REF] Kim | Low-temperature ion beam mixing in metals[END_REF] that takes place as a result of the cascade. Even though the initial effect is on the nanometric scale, it has also been estimated that it can lead to macroscopic consequences such as a five-year underestimation of the lifetime of a nuclear reactor pressure vessel exposed to a very high thermal flux [START_REF] Caro | Spectral effects on defect production in the reactor pressure vessel of a pressurized heavy-water reactor[END_REF] . Similar trends have also been reported for intermetallic alloys [START_REF] Diaz De La Rubia | Radiation effects in FCC metals and intermetallic compounds -A molecular-dynamics computer-simulation study[END_REF] and ceramics [START_REF] Zinkle | Defect production in ceramics[END_REF][START_REF] Gao | Computer simulation of disordering and amorphization by Si and Au recoils in 3C-SiC[END_REF][START_REF] Devanathan | Dynamic annealing of defects in irradiated zirconia-based ceramics[END_REF] . Figure 1 illustrates the time-dependent evolution of a displacement cascade based on molecular dynamics (MD) simulations in a typical metal (see Methods section). Such a displacement cascade can be induced by a passing neutron or other highenergy (MeV or more) particle. The first lattice atom to receive a recoil energy is called the "primary knock-on atom". Note how initially, when the atoms are highly excited, many of them are displaced from their lattice sites. However, as the cascade begins to thermally equilibrate with its surroundings, nearly all atoms regain positions in the perfect lattice structure. It is because of these two so-called 'heat spike' effects 22,[START_REF] Nordlund | Coherent displacement of atoms during ion irradiation[END_REF] (also known in parts of the literature as 'thermal spike') that the amount of final defects generated is much smaller, and the number of atoms replacing other atoms (atomic mixing) much larger than the prediction from simple linear collision cascade models such as the NRT-dpa model (see Figure 2). The physical reasons to this are discussed in detail in the following two subsections (building upon an earlier review work by us [START_REF] Nordlund | Primary Radiation Damage in Materials -Review of Current Understanding and Proposed New Standard Displacement Damage Model to Incorporate in Cascade Defect Production Efficiency and Mixing Effects[END_REF] ), which also present improved functional forms and tests of these against experimental and new simulation data.

Results

In-cascade recombination effects on defect production

The physical basis for the overprediction by the NRT-dpa model of the defect production at high energies is the enhanced recombination of defects in close proximity in energetic displacement cascades. The binary collision simulations used as the basis of the NRT-dpa model 3 focused on the collisional phase of the displacement cascade and did not consider the dynamics of cascade evolution as atomic velocities fell to the speed of sound (~5 eV) and lower, when many-body interactions become relevant. In energetically dense cascades, like that shown in Fig. 1, local melting clearly plays an important role in defect retention and structure. With increasing primary knock-on atom energy, the displacement event produces progressively more Frenkel defects (pairs of vacancies and interstitials [START_REF] Ehrhart | Properties and interactions of atomic defects in metals and alloys, volume 25 of Landolt-Börnstein[END_REF] ) that are spatially close to other defects. The ~10-100 jumps occurring per atom during the 1-10 ps cascade cooling phase [START_REF] Stoller | Comprehensive Nuclear Materials[END_REF] can induce significant additional recombination events as the cascade atom energies decrease, following the collisional phase, from Ed to the threshold value for atomic migration

Actual damage production NRT-dpa damage model Actual atom replacements

Figure 2 a) summarizes the defect production as a function of primary knock-on atom energy as determined from experiments performed in Cu near 4 K (where long range thermally activated defect motion is impossible [START_REF] Ehrhart | Properties and interactions of atomic defects in metals and alloys, volume 25 of Landolt-Börnstein[END_REF] ). The predicted defect production and number of replaced atoms obtained from MD simulations are also shown. The figure shows that the actual defect production is sublinear with respect to damage energy between ~0.1 and 10 keV (Ref. 23), becoming about 1/3 of the NRT-dpa prediction. At energies > 10 keV corresponding to the onset of subcascade formation [START_REF] Stoller | Comprehensive Nuclear Materials[END_REF][START_REF] Averback | Displacement damage in irradiated metals and semiconductors[END_REF][START_REF] Muroga | The effect of recoil energy spectrum on cascade structures and defect production efficiencies[END_REF] , the defect production increases linearly with damage energy but maintains the factor of ~3 lower defect production compared to the NRT-dpa value.

The physical basis for the reduction in surviving defects, with respect to the NRT model, with increasing knock-on atom energy can be understood by considering the following simplified derivation.

The ultimate survival of initially created Frenkel defects requires physical separation of the interstitial and vacancy beyond a minimum distance known as the spontaneous recombination distance ( L).

Atomic collisions along close-packed directions (known as "recoil collision sequences") are one example of a method to efficiently transport interstitial atoms to the periphery of a displacement cascade, leaving the associated vacancy near the cascade interior. Molecular dynamics simulations 22 indicate atom transport from the displacement cascade interior may be associated with a supersonic shock-front expanding from the primary recoil event during the early stages of the cascade evolution. At low energies (below the subcascade formation regime [START_REF] Stoller | Comprehensive Nuclear Materials[END_REF] ) the displacement cascades are roughly spherical with radius R, and form a liquid-like zone of dense collisions (the heat spike described above).

It is further assumed that only interstitials transported to the cascade outer periphery defined by R-L to R will result in stable defects, whereas Frenkel pairs created in the cascade interior (0 to R-L) will experience recombination. The fraction of initially created NRT-dpa defects that survive is therefore given by the ratio of the outer spherical shell volume to the total cascade volume:

= = ( ) =3 -3 + ≈3 ( 2 )
for L << R. This "surviving defect production fraction" thus tells which fraction of defects predicted by the NRT-dpa model without any recombination survives. The cascade radius R can be, within the regime of spherical cascades, estimated from classical theory of nuclear stopping power [START_REF] Lindhard | Range concepts and heavy ion ranges[END_REF][START_REF] Ziegler | The Stopping and Range of Ions in Matter[END_REF] . In practise, we used the SRIM code which implements an integral calculation to obtain mean range tables, based on cross sections from the widely used Ziegler-Biersack-Littmark (ZBL) interatomic potential [START_REF] Ziegler | The Stopping and Range of Ions in Matter[END_REF] .

We found that low-energy (less than or of the order to 10 keV) recoils of damage energy have an average movement distance (range) R that is proportional to , where the exponent x is ~0.4-0.6 for the metals considered in this study. Since ∝ , this further gives

( ) . survive = . 3 ∝ . 3 d ∝ d 1- ( 3 )
This simple model thus provides an intuitive explanation for why cascade damage production is sublinear with damage energy in the heat spike regime. Physically realistic molecular dynamics simulation studies [START_REF] Stoller | Comprehensive Nuclear Materials[END_REF][START_REF] Bacon | The primary damage state in fcc, bcc and hcp metals as seen in molecular dynamics simulations[END_REF] have reported that defect production rates up to the onset of subcascade formation in a variety of metals can be well described by Nd ~ (Td) 1-x , where x is between 0.2 and 0.3. These x values are slightly larger than the value obtained in our simplified model because real cascades are not perfectly spherical and some defects form small clusters, reducing the recombination probability.

However, it is well known that at high energies cascades break up into subcascades [START_REF] Nordlund | Primary Radiation Damage in Materials -Review of Current Understanding and Proposed New Standard Displacement Damage Model to Incorporate in Cascade Defect Production Efficiency and Mixing Effects[END_REF][START_REF] Nordlund | Defect production in collision cascades in elemental semiconductors and FCC metals[END_REF][START_REF] Backer | Subcascade formation and defect cluster size scaling in high-energy collision events in metals[END_REF] , after which damage production becomes linear with energy. Hence the surviving defect fraction factor ( ) that accounts also for subcascade breakdown should have the feature of being a power law at low energies, but becoming a constant c at high ones. A function which fulfils both criteria is

( ) = + ( 4 )
where b < 0 is consistent with the damage production efficiency reducing with increasing energy Td and the desired limit ( ) → when → ∞. This thus gives a total damage production

( ) = . ′ d + = . d 1+ + . ( 5
)
Note that here the exponent b is not the same as x, since the latter function is not a pure power law. The prefactor is defined by demanding the function to be continuous, i.e. (2 d /0.8) =

1.

Taken together, this derivation leads us to propose (based in part on review work done within an OECD Nuclear Energy Agency group [START_REF] Nordlund | Primary Radiation Damage in Materials -Review of Current Understanding and Proposed New Standard Displacement Damage Model to Incorporate in Cascade Defect Production Efficiency and Mixing Effects[END_REF] ) a modified defect production model, the athermal recombination corrected displacements per atom (arc-dpa).

, ( ) = ⎣ ⎢ ⎢ ⎢ ⎡ 0 , < 1 , < < . . ( ) , . < < ∞ ⎦ ⎥ ⎥ ⎥ ⎤ ( 6 )
with the new efficiency function ( ) given by

( ) = ( / . ) + ( 7
)
Here Ed is the average threshold displacement energy [START_REF] Nordlund | Molecular dynamics simulations of threshold energies in Fe[END_REF] which is the same as in the NRT-dpa and barcdpa and carcdpa are material constants, that need to be determined for a given material from MD simulations or experiments. The overall form (Eq. 5) and the constant 0.8 are retained for direct comparison with the NRT-dpa model; in particular making it easy to modify computer codes that now use the NRT-dpa by simply multiplying with the function ( ).

Figure 3 compares the derived arc-dpa expression for Fe and W with several recent MD simulation results used for the fitting. We tested that if the fit is limited to energies < 10 keV, one also can fit the data well with a power law with an exponent of ~0.7-0.8, i.e. the data is consistent with MD reports of power law dependencies. However, the arc-dpa form has the major improvement that it can also describe the saturation. Even though there is some variation in the MD data (due to differences in interatomic potentials), all of the MD results give damage production well below the =1 value predicted by the NRT-dpa model for cascade energies > 1 keV. The arc-dpa fit to the composite data gives a reasonable averaging description of the decreasing trend in up to ~10 keV and the expected approach to a constant value at higher cascade energies.

Replacements-per-atom (rpa) model

Since the NRT-dpa model deals with production of defects that are not on perfect lattice sites, it cannot predict the number of atoms that are transported from their initial lattice site to a new lattice site, i.e. replace another atom in a perfect crystal site (right panel in Figure 2 c). This number of atom replacements is experimentally measurable via so called radiation mixing experiments. Typically, an ion beam is used to bombard a thin marker layer inside a material, and the resulting broadening of the marker layer is measured [START_REF] Kim | Low-temperature ion beam mixing in metals[END_REF][START_REF] Klatt | Ion beam mixing in Ag-Pd alloys[END_REF] . For ordered alloys, it can also be conveniently measured by electrical resistivity [START_REF] Kirk | Atomic rearrangements in ordered fcc alloys during neutron irradiation[END_REF] . Via an analogy with random walk atom diffusion, it is possible to relate this measured broadening to the actual number of atom replacements per ion inside the material [START_REF] Diaz De La Rubia | Role of thermal spikes in energetic collision cascades[END_REF] . Analysis of neutron and ion beam radiation mixing data has shown that the actual number of replaced atoms can be more than an order of magnitude larger than the number of displacements predicted by the NRT-dpa model [START_REF] Kirk | Atomic rearrangements in ordered fcc alloys during neutron irradiation[END_REF][START_REF] Nordlund | Mechanisms of ion beam mixing in metals and semiconductors[END_REF][START_REF] Nordlund | Role of electron-phonon coupling on collision cascade development in Ni, Pd and Pt[END_REF][START_REF] Zhang | Radiation-induced mixing between metals of low solid solubility[END_REF] . A correct estimation of this number can be of enormous importance in predicting the effects of irradiation on phase stability [START_REF] Jaouen | Crystalline to amorphous transformation in NiAl: ion irradiation studies in relation to cascade parameters[END_REF][START_REF] Gao | Cascade overlap and amorphization in 3C-SiC: Defect accumulation, topological features, and disordering[END_REF] and the associated mechanical properties of materials. Nanostructured materials, such as nanolaminates and nanoscale oxide-dispersion strengthened (ODS) steels are particularly sensitive to these errors owing to their small length scales.

The superlinear increase in the number of replaced atoms with increasing knock-on atom energy can be understood by a model considering the spatial extent of a collision cascade. We consider first low energies (in the keV regime) and dense materials, where cascades are normally compact. As noted abo ve , low-e ne rgy c asc ade s are ro ugh ly sphe ri c al . A fte r the balli sti c ph ase o f a c asc ad e , MD simulations show (cf. Fig. 1) that the lattice breaks down and a liquid-like region forms. In this region all atoms are free to move and hence are almost certain to lead to one or more replacements during the thermal spike phase (as illustrated in Fig 2 c), right frame). The number of atoms N in a spherical cascade of radius R is proportional to the sphere volume, i.e. ∝ , and (as already noted for the arcdpa model, ∝

. We thus find that the number of replaced atoms ∝ . Since x > 1/3, this simple consideration gives an intuitive explanation for why the number of replaced atoms increases superlinearly with energy at low energies, when cascades are compact. At high energies, when cascades split into subcascades [START_REF] Backer | Subcascade formation and defect cluster size scaling in high-energy collision events in metals[END_REF][START_REF] Calder | On the origin of large interstitial clusters in displacement cascades[END_REF] , the behaviour can be expected to change to a linear dependence with damage energy. Similarly to the arc-dpa function, we thus arrive at a form

( ) ∝ ( 8 )
The proportionality prefactor is again set to ensure continuity, (2 /0.8) = 1. To augment the NRT equation in order to predict the number of replaced (mixed) atoms, we propose another improved damage function, the "replacements-per-atom" (rpa) equation. The correction factor is applied in Equation 5 as for the arc-dpa, but now it has the form

( ) = ( / . ) +1 ( 9 )
Here brpa and crpa are the new material constants. This form of ( ) is constructed to be consistent with the derivation above. Since the NRT equation already is proportional to Td, with this form the prediction is that the number of replaced atoms increases at low energies with Td as = = , i.e. = 3 -1. At high energies, when Td >> brpa, the form becomes linear with energy, as expected when cascades are split into subcascades. In this functional form, brpa has a physical meaning as the average energy for subcascade breakdown in terms of number of replaced atoms. Moreover, similar to the arc-dpa form, Eq. 9 fulfils the same conditions of continuity and compatibility with the NRT-dpa model.

Discussion

We first reiterate the physical meaning of the newly introduced material constants. brpa is related to the subcascade formation energy, and has energy units. The unitless exponents barcdpa and are associated with the dependence of the ion range with energy. Finally, the unitless quantity carcdpa is associated with the saturation value of damage recombination with heat spike size.

As a summary of the arc-dpa and rpa models, Figure 3 compares the obtained rpa and arc-dpa curves with the NRT dpa prediction for Fe and W, as a function of damage energy. Note that the NRT-dpa damage production equation does not, except at the very lowest energies near 100 eV, describe correctly either the surviving defects or the amount of radiation mixing in energetic displacement cascades. The newly introduced arc-dpa correction factor for the primary damage leads to a calculated surviving defect fraction about a factor of 3 lower than the NRT-dpa prediction, and the rpa correction factor leads to the calculated number of atom replacements to be about 30 times higher than the predicted NRT displacement value. Both the correction factors agree very well with MD simulation results over more than 3 orders of magnitude in energy for all elements, giving confidence that the derived functional forms are well motivated. We note that when additional and more accurate MD or experimental data becomes available, the models (eqs. ( 6) and ( 9)) could be refined for a better description e.g. near the threshold.

We also considered the dependence of the results on the ambient temperature. Several previous studies have shown that the effect of ambient temperature on primary damage production or atom replacements at ps time scales is insignificant or weak up to temperatures around roughly half the melting point [START_REF] Stoller | Comprehensive Nuclear Materials[END_REF][START_REF] Phythian | A comparison of displacement cascades in copper and iron by molecular dynamics and its application to microstructural evolution[END_REF][START_REF] Paine | Ion beam mixing: basic experiments[END_REF] .

For this work, we also simulated two of the data points for W at an elevated temperature, 800 K. The results (solid circles in Fig. 3 b) show that both the damage and replacements is the same within the statistical uncertainty as those at low temperature for the same potential. We note that given sufficiently large and statistically accurate data sets for a range of higher elevated temperatures, it would be possible to make the arc-dpa and rpa model parameters temperature-dependent.

Table 1 gives results of the arc-dpa and rpa model fit parameters for several metals based on MD data. The metals were chosen as those for which a sufficiently wide MD database was available for the fitting.

We note that, in spite of its failure to predict damage production correctly, the original NRT-dpa standard remains useful for comparing scaled radiation dose (exposure), as it is essentially proportional to the collision-relevant portion of radiation energy deposited per volume. The correct use of this standard as the first variable by which to compare radiation damage levels in different environments remains, therefore, strongly recommended. The new models, arc-dpa and rpa, developed here extend the usefulness of the dpa in that, while they can also be used to compare different irradiations, they in addition give accurate predictions of primary damage production and radiation mixing. They are deliberately constructed in such a way that they are easy to implement in existing radiation effects software: they add only four additional parameters (or two, if only one of the two new models is used) for each element. The arc-dpa and rpa equations thus enable a significant increase in physical relevance with minimal increase in computational efficiency or complexity. Of course, however, the practical application of arc-dpa and rpa requires the construction of suitable databases of parameters for each material: not only Ed but also the b and c constants. Collectively, the new models represent an important step toward improved quantification of the primary damage state during irradiation of materials.

In general, these damage models are expected to be relevant for many other materials besides the elemental metals discussed here. Studies of damage in metal alloys indicate the arc-dpa model will be directly applicable to both dilute 42 and concentrated [START_REF] Granberg | Mechanism of radiation damage reduction in equiatomic multicomponent single phase alloys[END_REF] metal alloys. In non-metallic materials, the arcdpa function may not be universally relevant, as damage production involves effects such as amorphization [START_REF] Ehrhart | Properties and interactions of atomic defects in metals and alloys, volume 25 of Landolt-Börnstein[END_REF] that cannot be captured by any simple equation. However, some ceramic materials are known to undergo significant in-cascade recombination [START_REF] Zinkle | Defect production in ceramics[END_REF][START_REF] Gao | Computer simulation of disordering and amorphization by Si and Au recoils in 3C-SiC[END_REF][START_REF] Devanathan | Dynamic annealing of defects in irradiated zirconia-based ceramics[END_REF] , and for these arc-dpa can be useful. On the other hand, the rpa function can be expected to be relevant in any material where heat spikes are significant (i.e. all dense materials), since in all of these the atomic mixing will be enhanced by collective atomic motion. The formation of the arc-dpa and rpa also motivate systematic experimental and simulation studies to understand better the primary state of damage in non-metallic systems, where (with the exception of Si) studies are scarce.

Prior to concluding, we emphasize that the arc-dpa and rpa models deal with the primary damage state only, i.e. the damage produced during the first few ps after a collision cascade initiated. Already at room temperature, thermally activated defect migration is known to be significant, and can reduce the damage production significantly from the arc-dpa value due to recombination effects, or enhance atom mixing from the rpa value. They also do not describe defect clustering or damage overlap effects [START_REF] Granberg | Mechanism of radiation damage reduction in equiatomic multicomponent single phase alloys[END_REF][START_REF] Gao | Computer simulation study of cascade overlap effects in alpha-iron[END_REF][START_REF] Nordlund | Point defect movement and annealing in collision cascades[END_REF] . However, even for these cases the new functions can be useful, as a starting point for e.g. kinetic Monte

Carlo or rate theory calculations of high-dose irradiation effects [START_REF] Granberg | Mechanism of radiation damage reduction in equiatomic multicomponent single phase alloys[END_REF] (where cascades overlap) or conditions where thermal defect migration recombines defects.

In conclusion, the new arc-dpa and rpa models introduced here allow, in a very simple and efficient way, to incorporate the improved understanding on radiation defect generation mechanisms gained during the last 4 decades into software calculating primary radiation damage generation rates in macroscopic reactor components. The arc-dpa model accounts for the enhanced recombination active in pure metals as well as in many alloys, which strongly reduces the number of point defects present in the primary damage, compared to the traditional NRT-dpa model. The rpa model provides a measure of the volume of the irradiated material directly affected by the cascade, which is important e.g. for phase stability considerations. In calculations of radiation damage effects where the dpa measure is used as a starting point, these new functions provide improved accuracy in a simple analytical form. This allows, e.g., differentiating between irradiation conditions dominated by either low-or highenergy recoils, and perhaps even more importantly, introduces the possibility to quantify analytically the very large (about 2 orders of magnitude) difference between damage production and atom relocation effects.

Methods

Molecular dynamics simulations

The new data used for the fits of the arc-dpa and rpa functions were obtained from molecular dynamics simulations following Refs 31, 36, 37, 48, 49. The Fe damage data is from Ref. Periodic boundary conditions were used in all directions, to correspond to high-energy ion or neutron effects deep inside a material, and the lattice constant was set to the equilibrium value at 0 K (previous works show that 0 and 300 K primary damage results are identical within the statistical uncertainty in transition metals). After equilibration, an atom was selected randomly near the center of the simulation cell, and given a recoil energy in a random direction in 3 dimensions. The central parts of the simulation cell were simulated in the NVE ensemble, while excess energy was removed from the system using temperature scaling towards room temperature in the outermost 1 unit cell thick regions of the cell. To account for energy loss to ionizations, the ZBL96 electronic stopping power [START_REF] Ziegler | The Stopping and Range of Ions in Matter[END_REF] was applied as a frictional force on all atoms with a kinetic energy higher than 10 eV, and the damage energy Td was calculated as the difference between the initial recoil energy and the total sum of energy lost to electronic stopping. The damage was analyzed using the Wigner-Seitz cells approach [START_REF] Nordlund | Defect production in collision cascades in elemental semiconductors and FCC metals[END_REF] that is space-filling and hence allows for a unique determination of whether a defect is vacancy-or interstitial-type.

Data availability

The new molecular dynamics data that has been produced for this paper, and comprises the source data for the parameters given in Table 1, is freely available for download at http://urn.fi/urn:nbn:fi:csc-kata20180125132021651079.

Figure 1 .Figure 2 .

 12 Figure 1. Collision cascade.A cross-sectional view of a collision cascade induced by a 10 keV primary knock-on atom in Au obtained from typical molecular dynamics simulations. The individual dots show atom positions. Blue circles illustrate atoms with low temperature and red and whitish atoms have high kinetic energies, with the energy scale given to the right. Note how initially, when the atoms are hot (high kinetic energy), a large number of atoms are displaced from their lattice sites. However, as the cascade cools down, almost all atoms regain positions in the perfect lattice sites. It is because of these two so called 'heat spike' effects that the number of atoms replacing other atoms is much larger and the amount of final defects generated much smaller than the prediction from simple linear collision cascade models like the NRT-dpa model.

Figure 3 .

 3 Figure 3. Improvement with arc-dpa and rpa. Illustration of the improvement obtained with the new arc-dpa and rpa equations for a) Fe and b) W. The W data also includes two data points simulated at 800 K with the DD potential (solid circles). The references are: [A98]: Ref. 26, [Z93]: Ref. 13. The Fe damage data is from Refs. 14 (Stoller) and 42 (AMS, MEA-BN, DD-BN). The Fe replacement data and all W data is original work for this publication, see Methods section. Error bars are given in s.e.m.

  48 or the references indicated in the figure, and the Ni, Pd and Pt data from Ref. 49, and other data previously unpublished. In all cases, a crystalline simulation cell in either the face-centered cubic or body-centered cubic crystal structure was first constructed and equilibrated by a short (few picoseconds) MD simulation at room temperature. The interatomic interactions were modelled with equilibrium reactive interatomic potentials to which the Ziegler-Biersack-Littmark repulsive potential 36 was joint at small separations to realistically mimic high-energy interactions. The interatomic potentials indicated by abbreviations in the figures are for Cu: CEM: Ref. 50, S-L: Ref. 51 and for W: DD: Ref. 52, JW: Ref. 53, AT: Ref. 54, AH: Ref. 55.

Acknowledgements

We thank the OECD/NEA for setting up the initial working group that initiated this work. This work was sponsored in part by the U.S. Department of Energy, Office of Fusion Energy Sciences, SJZ, RES, Office of Basic Energy Sciences, RSA, WJW. AS acknowledges support from the Academy of Finland through project No. 311472. Grants of computer time from CSC -the Finnish IT Center for Science as well as the Finnish Grid and Cloud Infrastructure (persistent identifier urn:nbn:fi:research-infras-2016072533) are gratefully acknowledged. This work has been also partly carried out within the framework of the EUROfusion Consortium and has also received partial funding from the Euratom research and training programme 2014-2018 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.

Author contributions

K.N. was the lead author of the theory and modelling parts, guided the computational simulations, derived the rpa equation, and did the fitting of the arc-dpa and rpa models to the plots. S.Z. derived the arc-dpa equation, performed the analyses of experimental data and stimulated activities on the importance of rpa. A.S. simulated all the new MD data. R. S. A. and S. Z. were lead authors on the discussion of experiments and R. S. A on the mixing parts. All authors contributed to data interpretation and to the manuscript preparation and approved the final version of the manuscript.

Additional information

The authors declare no competing financial interests. A cross-sectional view of a collision cascade induced by a 10 keV primary knock-on atom in Au obtained from typical molecular dynamics simulations. The individual dots show atom positions. Blue circles illustrate atoms with low temperature and red and whitish atoms have high kinetic energies, with the energy scale given to the right. Note how initially, when the atoms are hot (high kinetic energy), a large number of atoms are displaced from their lattice sites. However, as the cascade cools down, almost all atoms regain positions in the perfect lattice sites. It is because of these two so called 'heat spike' effects that the number of atoms replacing other atoms is much larger and the amount of final defects generated much smaller than the prediction from simple linear collision cascade models like the NRT-dpa model. the concepts and quantities used in deriving the new arc-dpa and rpa equations. c) Schematic illustration of the damage predicted by the three different damage models for the case of ~1 keV damage energy in a typical metal. For illustration purposes, the damage is illustrated as if all damage were produced in the same 2-dimensional plane. Blue circles illustrate atoms in original lattice positions, yellow-brown denotes atoms that are in a different lattice position after the damage event, red atom pairs denote two interstitial atoms sharing the same lattice site, and empty lattice positions denote vacancies. Left: Damage production predicted by the NRT-dpa model. Middle: actual damage production, addressed by the new arc-dpa equation. Right: actual atom replacements, addressed by the new rpa equation, agreeing better with experimental data on number of replaced atoms (ion beam mixing). Note that in real 3 dimensional systems, the difference is even larger than in this 2D schematic.