Nuclear Instruments and Methods in Physics Research B 286 (2012) 154–158

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Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb

Atomistic simulations of the radiation resistance of oxides A. Chartier a,⇑, L. Van Brutzel a, J.-P. Crocombette b a b

CEA-Saclay, DEN/DANS/DPC/SCP, 91191 Gif-Sur-Yvette, France CEA-Saclay, DEN/DANS/DMN/SRMP, 91191 Gif-Sur-Yvette, France

a r t i c l e

i n f o

Article history: Received 22 August 2011 Received in revised form 12 October 2011 Available online 10 January 2012 Keywords: Molecular dynamics Radiation damages Oxides

a b s t r a c t Fluorite compounds such as urania and ceria, or related compounds such as pyrochlores and also spinels show different behaviors under irradiations, which ranges from perfect radiation resistance to crystalline phase change or even complete amorphization depending on their structure and/or their composition. Displacement cascades – dedicated to the understanding of the ballistic regime and performed by empirical potentials molecular dynamics simulations – have revealed that the remaining damages of the above mentioned oxides are reduced to point defects unlike what is observed in zircon and zirconolite, which directly amorphize during the cascade. The variable behavior of these point defects is the key of the various responses of these materials to irradiations. This behavior can be investigated by two specific molecular dynamics methodologies that will be reviewed here: (i) the method of point defects accumulation as a function of temperature that gives access to the dose effects and to the critical doses for amorphization; (ii) the study Frenkel pairs life-time – i.e. their time of recombination as function of temperature – that may be used as a tool to understand the results obtained in displacements cascades or to identify the microscopic mechanisms responsible for the amorphization/re-crystallization during the point defects accumulations. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Urania UO2, ceria CeO2 or other structurally related fluorite compounds such as pyrochlores A2B2O7 (A being trivalent and B tetravalent cations), along with spinels AB2O4 (A being divalent and B trivalent cations) are ceramics of utmost interest for nuclear industry. All of them undergo the effects of radiation damages, whether during their life-time in nuclear reactor, or in long term storage. Depending on the structure (herein fluorite-like or spinels) and their chemistry, they show very different behaviors under irradiation. Their responses to irradiations vary from almost perfect resistance to complete amorphization, with possible phase transitions. Different regimes of irradiations lead to different effects in the compounds: electron irradiations produce electronic disorders up to local ions displacements depending on their kinetic energies, low energetic heavy ions irradiations also induce ballistic interactions and therefore displacement cascades in the compounds, swift heavy ions release their kinetic energy in compounds by electronic losses, which impulse extreme local heating. Indeed, a fundamental understanding of the responses to irradiations of the above mentioned oxides is a key-point for their current and potential use in the nuclear industry. Many experiments and simulations ⇑ Corresponding author. Tel./fax: +33 1 6908 3168. E-mail address: [email protected] (A. Chartier). 0168-583X/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2012.01.002

have been performed in order to establish reliable general rules that would help to understand and eventually to predict the behaviors of some family of oxides submitted to different types of irradiations. The pyrochlore family A2B2O7 has focused many efforts in the past 20 years, and general rules based on the ionic radii ratio between the A and B cations have been extracted [1]; the closer the ionic radii of A and B are, the more stable is the disordered fluorite structure (A,B)4O7, and the more resistant to irradiation the pyrochlore compound is. Such a ground state criterion relies on the knowledge that pure fluorite structures – such as UO2 for example – are very resistant to irradiations. This criterion also applies to a large scale of irradiations conditions. Some refinements based on the iono-covalent character of the cation–oxygen bonds [2] may be added to account for deviations from this general rule. Despite the a priori similarity between pyrochlores A2B2O7 and spinels AB2O4 in the sense that two cations are present, the spinel family seems to ask for more refined analysis in regard to its behavior under irradiations. The ionic radii ratio – inspired from the one used in the pyrochlore family – does not lead to a reliable picture of their radiation resistance. Part of the reason lies in the still on-going discussion about the type of first transition produced by irradiations [3,4]: is it an order–disorder transition, or a phase transition towards a rock-salt structure? This question still remains and takes its root in the multiple choices of interstitial sites the cations can sit in when they are displaced by irradiations.

A. Chartier et al. / Nuclear Instruments and Methods in Physics Research B 286 (2012) 154–158

Overall, the current understanding and modeling of the radiation resistance of oxides with fluorite-like or spinel structure mainly relies on ground state energetic considerations [1] and thus neglects the inherent out of equilibrium character of irradiating conditions. The energetic considerations one can find in literature may be based (i) on point defects formation energies such as anti-site formation energies to describe the easiness they demonstrate to accommodate the disorder [1] or (ii) on phase diagrams [5] with the underlying idea that irradiations and temperature are similar kinds of entropy sources, in the sense of the effective temperature promoted by Martin [6]. In the meantime, Weber [7] tentatively summarized and rationalized the experimental results using kinetic models. These models inferred the existence of an effective healing mechanism that is not yet fully identified. We focus in the present work on the effect of ballistic damages from the point of view given by empirical potentials molecular dynamics simulations. We recall the main results obtained by displacement cascades that specifically reproduce the elementary state of damage after cascades. The use of point defects accumulations method is then emphasized for both fluorite-like and spinel oxides. It addresses both the amorphization in pyrochlores and the epitaxial and homogeneous re-crystallization induced by irradiations in spinels. The underlying atomic mechanisms for the response to irradiation can be identified on the fly during the point defects accumulations simulations. Dedicated point defects lifetimes studies – that include the related threshold displacement energies and the recombination of Frenkel pairs – prove to help the identification of the microscopic mechanisms.

2. Primary ballistic damages: displacement cascades The behavior of oxides under irradiations depends strongly on the nature of the irradiation that is responsible for the damage creation and on the nature of the oxides. The balance between damage creation and annealing rates determines the resulting state of the irradiated oxides. In some oxides, the balance is in favor to the damage creation rate [22]. These oxides become amorphous above a critical dose of irradiation. The critical dose usually increases with temperature, until a critical temperature TC above which no amorphization is observed. The annealing rate is inferred to be thermally activated, but the very details of the mechanisms are still unknown. In this part, we focus on the damage creation term. The annealing term and their balance will be considered separately later on. The irradiations by low energy heavy ions produce mainly ballistic damages induced by elastic collisions between irradiating ions and oxides. This type of damages can be conveniently simulated by means of displacement cascades in the framework of molecular dynamics (MD) simulations with empirical potentials (see Refs. [8–11] for details). As the damages created by irradiations range from disordering down to complete amorphization, the empirical potentials used in the MD simulations have to reproduce all the phases that could appear during irradiations (see Refs. [11–13] for details). Selected results for fluorite-like oxide structure (UO2, CeO2, La2Zr2O7 and Gd2(Zr,Ti)2O7), spinel MgAl2O4 and for zircon ZrSiO4 and zirconolite CaZrTi2O7 are reported in Table 1. Two main trends may be extracted from those results. We observe that displacement cascades induce direct impact amorphization in zircon ZrSiO4 and zirconolite CaZrTi2O7, while only point defects are created for fluorite-like oxides and spinels. Both types of results are in agreement with experiments, and may be rationalized using different

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criterions such as the topological disorder [1], the iono-covalency of the bonds [2] or the re-crystallization rates [7] of each oxide. Each type of primary state of post-cascade damage implies different atomistic mechanisms for the counteracting annealing process mentioned above. In the case of a direct impact amorphization, the amorphous pockets created by the displacement cascades may anneal by epitaxial re-crystallization. For point defects, the annealing may be insured by recombination of these point defects. The annealing rate depends on the concentration of point defects and on the recombination rate. The latter is customarily thought to be related to the diffusion of vacancy V and interstitial I and to the recombination radius in which V and I recombine instantaneously. Our calculations show that displacement cascades answer the question about the nature of the primary damage: do cascades produce direct impact amorphization or point defects only? The last result offers the opportunity of further MD simulations as we will see below. 3. Beyond primary events: the point defects accumulation The complete description of the irradiations in oxides must include the dose and the flux effect, and hence more than one single displacement cascade event. Cascade overlap studies have been performed in UO2 recently [19] and evidence that the Schottky defects gather and create voids. An alternative method is to skip the computationally very expansive methodology of cascade overlap, and to focus on the evolution of the remaining damage obtained by displacement cascade event. This type of study has been conducted by Aidhy et al. [20]. They follow the kinetic evolution of vacancies and interstitials in UO2 and observe that vacancies cluster into Schottky defects and oxygen interstitials form cuboctahedral oxygen clusters. However, neither of those methods includes the flux effect. In order to take into account the flux effect, whenever displacement cascades produce point defects only, we re-introduced [17] the point defects accumulation methodology proposed a while ago in Lennard-Jones systems [21]. Such a methodology (described elsewhere [17]) consists in a continuous accumulation of cation Frenkel pairs in the framework of MD simulations. Indeed it proved sufficient to accumulate cation defects, the oxygen disorder being tightly driven by the disorder on the cation sublattice. The dose effect – i.e. the number of displacement per cation (dpc) – and also the flux effect as a function of temperature can be investigated. Simulations of point defects accumulations involve drastic compromise between the size of the Super-cells and the time of simulation [17]. Hence, the use of small Super-cells (less than thousand of ions) with high rates of Frenkel pairs creation (as much as 12 magnitudes higher than the 103 dpa/s attainable experimentally) is required. This avoids the thermal diffusion of any ion during the simulations: only the short-range V–I recombination is allowed. The method of point defects accumulation is therefore a tool to investigate the role of the concentration of defects and their recombination, without thermal diffusion contribution to the annealing term. It is therefore more reliable to consider cases where the thermal diffusion can be neglected compared to the V–I recombination, as shown for example in ceria [26]. 4. Amorphization of pyrochlores A2B2O7 by irradiation Simulations of point defects accumulation in the framework of MD were performed in the lanthanum pyrozirconate La2Zr2O7 [17]

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Table 1 Summary of the displacement cascades and point defects accumulation results obtained by MD simulations in different oxides.

ZrSiO4 CaZrTi2O7 UO2 La2Zr2O7 Gd2Zr2O7 Gd2Ti2O7 Gd2(ZrxTi1x)2O7 MgAl2O4

Displacement Cascades

Refs

Point defects accumulations

Refs

Amorphization Amorphization + point defects Point defects Point defects Point defects Point defects – Point defects

[14] [15] [8] [9] [10] [10] – [18]

– – Point defects Phase transitions with amorphization function of temperature Phase transitions without amorphization function of temperature Phase transitions with amorphization function of temperature Critical temperature for amorphization function of chemistry i.e. function of x Phase transitions without amorphization function of temperature

– – [16] [17] [12]

and in the gadolinium pyro-titano-zirconate solid solution Gd2(Zr,Ti)2O7 [12]. During the simulations, the structure of the zirconates rich pyrochlores transits first to the disordered fluorite structure (where cations are randomly distributed in the cation sublattices as (A,B)4O7) before an eventual amorphization. The titanate rich pyrochlores Gd2Ti2O7 and Gd2Zr0.5Ti1.5O7 showed an incomplete transition to the fluorite structure, and readily become amorphous. The critical doses for amorphization D as a function of temperature were extracted by fitting the MD results [12,17] using a rate equation proposed by Weber [7]. We thus determined [12,17] the activation energy Eact. that will be used below in the Zr–Ti pyrochlore series. The high rate of point defect accumulation in the simulation calls for a rescaling to be comparable with experiments. Thus, the critical temperatures for amorphization TC – defined as the temperature above which there is no amorphization anymore and extracted from the MD critical doses for amorphization as a function of temperature – have to be corrected, assuming that the recovery processes are of thermal origin. After Weber [7], TC reads as:

TC ¼

Eact: kB lnðC mater =UÞ

ð1Þ

Cmater is a constant that represents the ratio between the annealing rate and the damage cross section and is independent of the irradiation flux U. The critical temperatures for amorphization TC – obtained by MD simulations and corrected for the flux – are in very good agreement with the experimental determinations [22] (see Fig. 1). One observes – as expected – that TC increases with ionic radius ratio rA/rB between cations A and B in A2B2O7. It means that the methodology of point defects accumulation in the framework of MD simulations captures the main mechanisms responsible for the radiation behaviors of pyrochlores. This implies that the concentra-

Fig. 1. Critical temperatures for amorphization as a function of the ionic radius ratio rA/rB between cations A and B in A2B2O7. The inset compares the experimental and MD critical temperatures for amorphization, and the line corresponds to y = x.

[11]

tion of point defects and their recombination radius are responsible for the response of pyrochlores to irradiations. Let us focus on the Gd2(Zr,Ti)2O7 solid solution where the change of nature of the B cation (in A2B2O7) is the only parameter. The substitution of Ti by Zr changes the behavior from amorphisability up to around TC = 1200 K for Gd2Ti2O7 gradually to no amorphization for Gd2Zr2O7 (TC close to 0 K). Point defect accumulation allows going beyond the ionic radius ratio criterion proposed by Sickafus et al. [1]. The variation along the Ti–Zr series is in fact related to the differences in configurations of the titanium and zirconium interstitials. Zirconium interstitials readily recombines with vacancies, and eventually form anti-sites when the vacant site is a former site for Gd. This induces the transition of the pyrochlore to the fluorite structure observed in zirconate pyrochlore Gd2Zr2O7. The initiation of a similar transition is observed in Gd2Ti2O7, with creation of antisites. However, for each antisite created the titanium interstitials recombine with the titanium setting in antisite position and stabilizes in some specific configurations – called dumbbells (D) in [12]. They are made of corner-sharing TiO5–TiO6 polyhedra or of edge-sharing TiO5–TiO5 polyhedra (see Fig. 2, for an example). These configurations preserve the fivefold and sixfold coordination preferred by Ti. They constitute the nucleus for the amorphization of gadolinium titanate pyrochlores at low temperature.

5. Irradiation induced crystallization of amorphous MgAl2O4 We recall first the results in pristine spinel MgAl2O4. Point defects accumulation [23] showed two phases transitions under point defects accumulations. The normal spinel – i.e. ordered MgAl2O4 – first transits towards a disordered spinel where the cations are randomly distributed between their sites leaving the oxygen sublattice unchanged. This disordered spinel then further transforms to a rock-salt like structure – emptying the tetrahedral (originally Mg) site and filling the interstitial octahedral site. The critical dose for the disordered to rock-salt structure transition was observed to increase with temperature. No amorphization could be obtained even with high doses (68 dpc) at low temperature (30 K) [23]: MgAl2O4 keeps its rock-salt like structure. Similarly to the pyrochlores, the recombination of the cation Frenkel pairs was identified as the annealing process. The cation interstitials first recombine with vacancies in the cation sublattice, leading to a disordered spinel. Second, the cation interstitials smoothly stabilize (recombine) in the interstitial octahedral sites rather than in the tetrahedral ones (whether former Mg sites or interstitial tetrahedral site), and drive the disordered spinel to the rock-salt like structure. Starting from the amorphous state of MgAl2O4, we applied the point defects accumulation procedure in order to investigate the homogeneous (H) crystallization induced by irradiation [11]. We used two different point defects accumulation fluxes, with a difference of one order of magnitude (see Fig. 3) at temperatures ranging from 30 K to 1800 K. Point defects accumulation induces a crystallization of the amorphous spinel. The spinel stabilizes in the rock-salt

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Fig. 2. Configuration of the edge-shared TiO5–TiO5 polyhedra – i.e. Ti–Ti D dumbbell – in Gd2Ti2O7 projected in the 1 1 0 plane.

Fig. 3. Critical doses for homogeneous and epitaxial crystallization as a function of temperature for MgAl2O4. Filled and open symbols represent high and low point defects accumulation fluxes, respectively. The diagonal dashed line is drawn to separate homogeneous (H) from epitaxial (E) results. The dose is expressed in displacement per cation (dpc).

like structure in agreement with the experimental observations [24]. We observed that the doses for crystallization decrease with temperature (Fig. 3) for a given flux. The increase of the flux induces a decrease of the crystallization dose. We also investigated the epitaxial (E) crystallization induced by point defects accumulation [11]. Half of the Super-cell is the amorphous state of MgAl2O4. The other half is the spinel in its rock-salt like structure, which is the steady state of MgAl2O4 under point defects accumulation. Again, two different fluxes were considered, at temperatures ranging from 30 K to 1000 K. The amorphous part of MgAl2O4 crystallize, at doses that decrease with temperature (see Fig. 3), similarly to what is observed for the homogeneous case. But unlike the homogeneous crystallization, the doses for epitaxial crystallization increase with the flux. These behaviors can be understood after Heera et al. [25] by considering that the rate of crystallization is proportional to the number of point defects. The number of point defects is controlled by the balance between the damage creation and the annealing rates.

At a given temperature, we also observe in Fig. 3 that the doses for crystallization are one order of magnitude higher for the homogeneous crystallization than for the epitaxial one. The processes for crystallization are therefore different. In the homogeneous case, the crystallization was related to the local reorganization around each defect [25]. Finally, the effect of the flux is opposite between homogeneous and epitaxial crystallization. The increase of the flux proved to help the homogeneous crystallization as more point defects are available for crystallization to occur [11]. The methodology of point defects accumulation presented above allows to access to the irradiation dose and flux, for the particular case where point defects are produced by single irradiation event. Such a methodology can be used to investigate the underlying mechanisms that drive the amorphization or the epitaxial/ homogeneous crystallization under irradiations. The results on pyrochlores and spinels emphasize the role played by the shortrange recombination/the local reorganization of point defects in the understanding of radiation resistance of oxides as a function of temperature.

6. Threshold displacement and Frenkel pair recombination As the point defects behavior is a key of radiation resistance of oxides, it is natural to dedicate part of the effort to their study. We present below the recombination of Frenkel pairs in four fluorite structure compounds, namely Li2O, CaF2, CeO2 and UO2 and pyrochlore [16,26–28,30]. This is a complimentary way to tackle the recombination radius at which the interstitial–vacancy (I–V) anneals. The main results can be summarized as follows. First, MD simulations evidence that the recombination volume of I–V is not isotropic. The recombination volume strongly depends upon the point symmetry of the considered defects. For example, the oxygen interstitial that is the seventh next nearest neighbor to oxygen vacancy can recombine, whereas the third one cannot in CeO2 [28]. This consequently invalidates the recombination radius hypothesis usually done. Second, different types of recombination are observed. The recombination processes are either direct or occur by replacement

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sequences [27]. The sequences of recombination of Frenkel pair give an explanation to the order–disorder phase transitions in pyrochlore or spinel compounds. Indeed, the recombination processes by means of replacement sequences kick cations off their perfect positions to anti-sites configurations [29]. One may also observes direct IA–VB annealing (IA being an interstitial of type A and VB a vacancy of type B) which also creates anti-sites. Some recombination processes operate instantaneously (within 1 ps) and are not temperature dependent [16,26,27]. Other recombination processes are thermally activated. Their life-time s can be determined as a function of temperature and proves to vary as:

s ¼ s0 exp

  Eact: kB T

ð2Þ

The fitting of the calculations [16,26,27] allow to extract the pre-factor s0 and the activation energy Eact.. The obtained activation energies range from 2  102 to 1.3 eV. For instance, in the case of Gd2Ti2O7, the calculated activation energy for Ti–dumbbell recombination is equal to 0.25 eV. This value is very close to the value of 0.33 eV extracted from the critical doses for amorphization obtained by MD [12]. The origin of the very small activation energies obtained in experimental measures of amorphization doses as a function of temperature was puzzling to many authors (e.g. [30]). Indeed they are way too small to be related to migration energies which are consistently larger. Our calculations show that these small activation energies are in fact related to the shortrange recombination of FP and not to long range diffusion as it was usually thought. Dedicated studies on the recombination of Frenkel pairs were used to refine the analysis of the 80 keV displacements cascades in urania UO2 [16]. In this MD simulation, very few (almost none) close Frenkel pairs (for I–V distance less than 8 Å) survive the displacement cascades. This was easily related to the instantaneous recombination processes of Frenkel pairs. The urania lattice shows in fact a rapid reconstruction after the cascades, helped by the recombination processes. The Frenkel pairs recombination processes and their life-times offer also the opportunity to define more precisely the threshold displacement energies (TDE) of ions. The TDE is usually defined as the minimum kinetic that must be given to one ion so that it stabilizes as a close Frenkel pair. The life-time and temperature at which this TDE is valid is seldom defined. From the results of the recombination Frenkel pairs, we shall define the 30 K/1 ps as the most reliable temperature/life-time couple. Such a couple excludes most of the recombination processes that are thermally activated, and keeps active the instantaneous ones. 7. Conclusion The molecular dynamics (MD) simulations of irradiations in oxides presented here concerns ballistic damages and to a certain extent damages produced by electron irradiations. Despite the limitation inherent to the use of empirical potentials that cannot explicitly account for the electronic effects, many insights can be extracted at the atomic scale, following the procedure below: (i) Displacement cascades answer the question of the direct impact amorphisation or amorphisation by point defects accumulation.

(ii) Point defects accumulation – if the results of displacement cascades are point defects only – accesses the role of the irradiation dose and dose rate as function of temperature. (iii) Point defects recombination/threshold displacement energies identify the microscopic mechanisms that drive the system under irradiation. Using such a procedure for fluorite oxides, pyrochlores and spinel, we have demonstrated the key role of point defects for the resistance of oxides against irradiation damages in the ballistic regime. We have shown that the knowledge of the microscopic processes of recombination/re-organization of the point defects cannot be circumvented for the understanding the response of oxides under irradiations. The recombination/re-organization of the point defects are short-range, often thermally activated, dependent on chemical species and on point symmetry related mechanisms. They proved to pilot the response of oxides to irradiations. The understanding and the prediction of the response of oxides under irradiation thus rely on the knowledge of the very details of the short-range recombination/re-organization of point defects. References [1] K.E. Sickafus, L. Minervini, R.W. Grimes, J.A.V. Ishimaru, F. Li, K.J. McLellan, T. Hartmann, Science 289 (2000) 748. [2] G.R. Lumpkin, M. Pruneda, S. Rios, K.L. Smith, K. Trachenko, K.R. Whittle, N.J. Zaluzec, J. Solid State Chem. 180 (2007) 1512. [3] K.E. Sickafus, N. Yu, M. Nastasi, J. Nucl. Mater. 304 (2002) 237. [4] D. Simeone, C. Thiriet-Dodane, D. Gosset, P. Daniel, M. Beauvy, J. Nucl. Mater. 300 (2002) 151. [5] K.E. Sickafus, R.W. Grimes, J.A. Valdez, A. Cleave, M. Tang, M. Ishimaru, S.M. Corish, C.R. Stanek, B.P. Uberuaga, Nat. Mater. 6 (2007) 217. [6] G. Martin, Phys. Rev. B30 (1984) 1424. [7] W.J. Weber, Nucl. Instr. Meth. Phys. B 98 (2000) 166–167. [8] L. Van Brutzel, M. Rarivomanantsoa, D. Ghaleb, J. Nucl. Mater. 354 (2006) 28. [9] A. Chartier, C. Meis, J.-P. Crocombette, L.R. Corrales, W.J. Weber, Phys. Rev. B67 (2003) 174102. [10] J.A. Purton, N.L. Allan, J. Mater. Chem. 12 (2002) 2923. [11] A. Chartier, T. Yamamoto, K. Yasuda, C. Meis, S. Matsumura, J. Nucl. Mater. 378 (2008) 188. [12] A. Chartier, G. Catillon, J.-P. Crocombette, Phys. Rev. Lett. 102 (2009) 155503. [13] A. Chartier, C. Meis, J.-P. Crocombette, W.J. Weber, L.R. Corrales, Phys. Rev. Lett. 94 (2005) 025505. [14] R. Devanathan, L.R. Corrales, W.J. Weber, A. Chartier, C. Meis, Nucl. Instr. Meth. Phys. Res. B228 (2005) 299. [15] L. Veiller, J.-P. Crocombette, D. Ghaleb, J. Nucl. Mater. 306 (2002) 61. [16] L. Van Brutzel, A. Chartier, J.-P. Crocombette, Phys. Rev. B78 (2008) 24111. [17] J.-P. Crocombette, A. Chartier, W.J. Weber, Appl. Phys. Lett. 88 (2006) 51912. [18] R. Smith, D. Bacorisen, B.P. Uberuaga, K.E. Sickafus, J.A. Ball, R.W. Grimes, J. Phys.: Condens. Matter. 17 (2005) 875. [19] G. Martin, P. Garcia, C. Sabathier, L. Van Brutzel, B. Dorado, F. Garrido, S. Maillard, Phys. Lett. A374 (2010) 3038. [20] D.S. Aidhy, P.C. Millet, T. Desai, D. Wolf, S.R. Phillpot, Phys. Rev. B80 (2009) 104107. [21] D.F. Pedraza, J. Mater. Res. 1 (1986) 2175. [22] R.C. Ewing, W.J. Weber, J. Lian, J. Appl. Phys. 95 (2004) 5949. [23] T. Yamamoto, A. Chartier, K. Yasuda, C. Meis, K. Shiiyama, S. Matsumura, Nucl. Instr. Meth. Phys. Res. B266 (2008) 2676. [24] N. Yu, K.E. Sickafus, M. Nastasi, Mater. Chem. Phys. 46 (1996) 161. [25] V. Heera, T. Henkel, R. Kögler, W. Skopura, Phys. Rev. B52 (1995) 15776. [26] A. Guglielmetti, A. Chartier, L. Van Brutzel, J.-P. Crocombette, K. Yasuda, C. Meis, S. Matsumura, Nucl. Instr. Meth. Phys. Res. B266 (2008) 5120. [27] N. Pannier, A. Guglielmetti, L. Van Brutzel, A. Chartier, Nucl. Instr. Meth. Phys. Res. B267 (2009) 3118. [28] K. Shiiyama, T. Yamamoto, T. Takahashi, A. Guglielmetti, A. Chartier, K. Yasuda, S. Matsumura, K. Yasunaga, C. Meis, Nucl. Instr. Meth. Phys. Res. B268 (2010) 2980. [29] J.-P. Crocombette, A. Chartier, Nucl. Instr. Meth. Phys. Res. B250 (2006) 24. [30] W.J. Weber, R.C. Ewing, C.R.A. Catlow, T. Diaz de la Rubia, L.W. Hobbs, C. Kinoshita, Hj. Matzke, A.T. Motta, M. Nastasi, E.K.H. Salje, ER. Vance, S.J. Zinkle, J. Mater. Res. 13 (1998) 1434.