Detection of one-dimensional migration of single self-interstitial atoms in tungsten using high-voltage electron microscopy

The dynamic behaviour of atomic-size disarrangements of atoms—point defects (self-interstitial atoms (SIAs) and vacancies)—often governs the macroscopic properties of crystalline materials. However, the dynamics of SIAs have not been fully uncovered because of their rapid migration. Using a combination of high-voltage transmission electron microscopy and exhaustive kinetic Monte Carlo simulations, we determine the dynamics of the rapidly migrating SIAs from the formation process of the nanoscale SIA clusters in tungsten as a typical body-centred cubic (BCC) structure metal under the constant-rate production of both types of point defects with high-energy electron irradiation, which must reflect the dynamics of individual SIAs. We reveal that the migration dimension of SIAs is not three-dimensional (3D) but one-dimensional (1D). This result overturns the long-standing and well-accepted view of SIAs in BCC metals and supports recent results obtained by ab-initio simulations. The SIA dynamics clarified here will be one of the key factors to accurately predict the lifetimes of nuclear fission and fusion materials.

have shown that the most stable SIA structures were < 011> dumbbell for Fe 15,16 but < 111> crowdion for other BCC metals, including Mo and W 17,18 . Thus, the most elementary issue on defects-the structure and dynamics of single SIAs-is an open question that needs to be experimentally clarified. Therefore, in the present study, we adopted W as a typical BCC metal, which will be an important component for radiation-resistant structural materials for future nuclear fusion reactors 19 , and we aimed to experimentally determine the migration dimension of SIAs in high-purity W. We also aimed to determine other important parameters related to the dynamics of SIAsthe activation energy for SIA migration, E m I , and the reaction radii of an SIA and a vacancy, an SIA and another SIA, and an SIA and an impurity atom, represented as r IV , r II , and r IX , respectively.
One of the most hopeful experimental methods for directly detecting the dynamics of defects within material is transmission electron microscopy (TEM), which has been successfully applied to TEM-visible nanoscale defects 20,21 . Additionally, the dynamics of even point defects have been directly detected when their migration rate was extremely slow and the specimens were extremely thin (e.g., carbon nanotubes) 22 . However, even using cutting-edge high-resolution TEM, directly tracing fast-migrating individual point defects (e.g., the jump frequency of an SIA, M I , at 16 K is estimated in this study to be greater than 2.0 × 10 7 Hz) within comparatively thick specimens is very difficult.
In this study, we propose an alternative method for detecting the dynamics of fast-migrating SIAs, in which high-voltage transmission electron microscopy (HVEM) is effectively combined with computer simulations. A schematic view of the system for the present study is shown in Fig. 1. Under high-energy electron irradiation, only isolated SIAs and vacancies are produced nearly spatially homogeneously during the primary damage process; therefore, the mesoscopic process of the clustering of SIAs, which can be directly observed by HVEM, must reflect SIA dynamics, including the migration dimension. In the current study, we extracted parameters related to SIA dynamics from the formation process of SIA clusters in the form of dislocation loops 10 by performing both HVEM experiments and object kinetic Monte Carlo (OKMC) [23][24][25][26][27] simulations of numerous point defect reactions and the resultant loop formations. In OKMC simulations, individual objects (point defects and point defect clusters) are tracked in a stochastic manner with the given input parameters. With this method, even the spatial correlation among individual objects is taken into account; hence, if the input parameters are correct, the results are expected to be correct. Needless to say, the results depend on the input parameters. In this study, using this feature of OKMC simulations, we tested a large number of parameter sets and searched for the parameter sets that would reproduce the experimental data. We will show that the range of "correct" parameter sets is strikingly narrow and that the correct parameter sets are fixed with small ambiguities. Using this system, we extracted the dynamic behaviour of fast-migrating atomic-size single SIAs, which cannot be directly traced. Within a high-voltage electron microscope, under high-energy electron irradiation, pairs of an SIA and vacancy are produced spatially homogeneously at a constant rate in a tungsten specimen. Individual SIAs migrate and react with other objects, such as immobile vacancies (leading to mutual annihilation), impurity atoms (SIA-impurity complex formation), SIA-impurity complexes (heterogeneous SIA-cluster nucleation), other SIAs (homogeneous SIAcluster nucleation), SIA clusters (SIA-cluster growth), and surfaces (SIA escape to surfaces). The formation process of nanoscale SIA clusters, which can be directly imaged by TEM, reflects numerous SIA reactions whose rates strongly depend on SIA dynamics, such as migration dimension (3D or 1D), migration frequency, and reaction radius with other objects. In the present study, the dynamics of TEM-invisible SIAs were extracted from direct observation of the formation process of TEM-visible SIA clusters. The displayed microstructure within the specimen is a snapshot (irradiation time: 331.14 s) produced by an object kinetic Monte Carlo (OKMC) simulation of SIA reactions for SIA dynamics parameters that reproduced the experimental data.
Scientific RepoRts | 6:26099 | DOI: 10.1038/srep26099 Results HVEM experiment results. Figure 2a shows TEM images of the irradiated specimen. The white dots denote loops. Figure 2b,c show the temporal variations in the average size and number density of TEM-visible loops larger than 3 nm in diameter, respectively, which were taken from the TEM images, as shown in Fig. 2a. As shown in Fig. 2b, the average diameter of the loops at an irradiation time of 1200 s is approximately 6 nm. In addition, the maximum diameter of the loops was 10 nm at an irradiation time of 1200 s.
In ref. 28, the following results, which were necessary for OKMC simulations in the present study, were obtained for the same W ingot as that used in the present study (see Supplementary Note 1 for details): Loop nucleation occurred through homogeneous nucleation via the direct combination of two SIAs and heterogeneous nucleation via the formation of embryos, which are complexes of an SIA and an impurity atom. The dimension-less concentration of impurities related to the heterogeneous nucleation of loops at 16 K, C X , was evaluated to be 3.6 × 10 −6 (case (1)) and 6.9 × 10 −6 (case (2)). The E m I values were evaluated to be 0.095 eV for case 1 and less than 0.040 eV for case 2.
OKMC simulation results. In OKMC simulations for both of the migration dimensions of the SIAs (3D and 1D), we initially tried to determine the parameter sets that would reproduce the experimental data shown in Fig. 2b. Next, from among the selected parameter sets, we searched for parameter sets that would reproduce the other experimental data shown in Fig. 2c.
First, we found that there were multiple parameter sets capable of reproducing the experimental result shown in Fig. 2b for both migration dimensions (3D and 1D). In addition, restrictions among these parameters were present. Figure 3a-c and d-f show the relationships among the parameters reproducing the experimental results shown in Fig. 2b for SIA migration dimensions 3D and 1D, respectively. Clearly, the proper parameters for r IV , r II , and r IX are restricted to the curved foil region within the (r IV , r II , r IX ) space for each fixed E m I value. The results for ≤ .   Fig. S1). Here, we touch on the origin of the correlation among the parameters shown in Fig. 3. A qualitative explanation in terms of the relationships between the individual parameters and the growth rate of the loops follows: Smaller E m I and r IV values yield larger growth rates because of the lower reaction rates of an SIA with a vacancy and consequential higher concentrations of SIAs. Larger r II values yield larger growth rates because of the larger reaction radius between an SIA with a loop. Conversely, the r IX value did not influence the growth rate of the loops within the examined range.
Next, we searched for parameter sets that reproduced the experimental results shown in Fig. 2c from among the restricted parameter sets, as shown in Fig. 3. Typical OKMC simulation results for the 3D migration of SIAs are shown in Fig. 4a for various E m I values. As shown in this figure, in the case of the 3D migration of SIAs, the densities of the loops at the initial stages were significantly higher than those in the experimental results for any of the parameter sets under the above restrictions. Thus, the 3D migration of SIAs yielded the rapid saturation of loop densities compared to the experimental results. Here, loop density tended to decrease at later stages for the case of = . E 0 040 eV m I because loops absorbed the nearby vacancies primarily owing to their significant accumulation and some of the loops shrank to less than 3 nm in diameter. Typical OKMC simulation results for the 1D migration of SIAs are shown in Fig. 4b  , the experimental result is reproduced. Strikingly, the perfect parameter sets that reproduced both of the experimental results had a narrow range as follows: SIA migration dimension: 1D, ≤ . E 0 020 eV m I (M I ≥ 2.0 × 10 7 Hz), r IV = 4.0, r II = 13.0, and r IX = 1.0 atomic-distance units. The 1D migration dimension supports the recent DFT simulation results, in which the stable structure of SIAs in W was concluded to be crowdion 17,18 . In addition, other obtained values are consistent with those in recent simulation studies, as shown in the next section. This verifies the method in the present study.

Discussion
The reason why loop formation is slower for the 1D migration of SIAs, as shown in Fig. 4, is qualitatively explained as follows. According to reaction-rate theory, the reaction rate of an SIA with spatially homogeneous arranged sinks is lower for the 1D migration of SIAs 2,5,7,9 . Therefore, the reaction rate of an SIA with another SIA or an impurity atom decreases, which makes the loop nucleation slower. From the analysis of our OKMC simulation results among all kinds of SIA reactions (see Supplementary Fig. S2), the primary reaction was that between an SIA and a vacancy. For the 1D migration of SIAs, this reaction was not homogeneous but was strongly spatially correlated and enhanced because an SIA was frequently recombined with its original counterpart vacancy, which was generated, together with the SIA, and placed within the path of the 1D-migrating SIA ( Supplementary  Fig. S2c). Enhanced reactions of an SIA and a vacancy also make loop nucleation slower. Thus, slower loop formation for the 1D migration of SIAs is attributed to both the decreased reactions of an SIA with another SIA (Supplementary Fig. S2a) or an impurity atom (Supplementary Fig. S2b) and the enhanced reactions of an SIA with a vacancy (Supplementary Fig. S2c).
Previous experimental studies on the structure of SIAs in W have deduced a < 011> dumbbell structure using the internal-friction technique 13,14 , which seems inconsistent with the results obtained in the present study. However, with the measurements of macroscopic properties, it is difficult to precisely assign the origin of internal friction peaks. In addition, in reference to previous DFT simulation studies 17,18 , the formation energy of a crowdion is smaller than that of a dumbbell by only 0.026 eV. Therefore, even if the origin of the internal friction peak is confirmed to be SIAs, the stable SIA structure may change because of the application of stress for the internal friction measurement. Here, we touch on a classical view of the SIA structure in the so-called "conversion-two-interstitial model, " 29 which has been proposed for FCC metals. In this model, crowdion was thought to be a metastable structure which is formed from irradiation at very low temperatures and is able to convert to a stable dumbbell in stage I. To the best of our knowledge, this model has been negated for FCC metals by the "one-interstitial model, " where the SIA structure was thought to be an invariably stable dumbbell 11,12 . However, the results obtained in the present study require us to consider the "conversion-two-interstitial model" again. Strictly speaking, in the present study, we do not indicate whether crowdion in W is a metastable structure realized only for low temperatures or the most stable one realized even for high temperatures because the examined temperature is limited to 16 K. To clarify this problem, further similar studies at higher temperatures and with other metals are required.
In comparatively older molecular dynamics (MD) simulations, the stable structure of an SIA has been confirmed as < 011> dumbbell, and E m I has been evaluated to be 0.37 eV 30 and 0.54 eV 31 , values that are significantly higher than the value of ≤ . E 0 020 m I eV obtained in the present study. However, in recent MD simulation studies using newer interatomic potentials, the stable structure of an SIA was < 111> crowdion, and the evaluated E m I values were comparatively lower, ranging from 0.009 eV 32 to 0.061 eV 33 . Lower E m I values have also been derived from more recent studies using a combination of DFT simulations and the Frenkel-Kontrova model (0.0026 eV) 34 and a molecular statics simulation (0.002 eV) 35 . These simulation studies imply that the E m I value was originally extremely low. Conversely, evaluations of E m I have been performed by recovery experiments of specimens irradiated at approximately 4 K (the boiling point of liquid helium). A study of electric resistance measurements yielded an E m I value of 0.054 eV 36 , and a study of field ion microscopy observation of surface SIAs yielded a value of 0.085 eV 37 , which are higher than recent E m I values determined by simulations 32,34,35 . The value of ≤ . E 0 020 m I eV obtained in the present study is more consistent with recent simulation studies.
The reaction radius values obtained in the present study are r IV = 4.0, r IX = 1.0, and r II = 13.0 atomic-distance units. There are no studies evaluating the latter two available for comparison. However, the r IV value obtained by an electric resistance recovery experiment was 3.3 38 . Referring to a molecular statics simulation study, the reaction volume for a crowdion is an ellipsoid; the semi-major radius is 6.6, and the semi-minor radius is 2.0 35 . By approximating this ellipsoid as a sphere, the radius becomes 3.0. The r IV value obtained in the present study (4.0) is close to the values obtained in the former studies 35,38 , which reflects the accuracy of the present study.
In conclusion, using a combination of HVEM experiments and OKMC simulations, we extracted the dynamics of fast-migrating single SIAs in W, which cannot be directly traced, even by cutting-edge high-resolution TEM. We revealed that the migration dimension of SIAs is not 3D but 1D, and we resolved the contradiction between the results in previous experiments and recent simulations. The SIA dynamics clarified here are key factors for predicting the lifetime of nuclear fission and fusion materials, in which SIAs are produced with high-energy particle irradiation. In addition, this study opens a new pathway for further experimental investigation of SIA dynamics for a wide range of inorganic materials and more complex point-defect dynamics such as interactions between vacancies and solute atoms, which is important to accurately understanding the responses of a wide range of metallic alloys to heat treatment.

Methods
High-voltage electron microscopy experiments. An ingot of high-purity coarse-grained polycrystalline W with a purity of 99.9999 mass% (JX Nippon Mining & Metals Co., Tokyo, Japan) was used. The impurity amounts of the ingot are shown in ref. 28. One grain of the ingot was cut and thinned to (011) disks with a thickness of 0.1 mm using spark erosion and mechanical polishing. Then, the disks were perforated at the centre by electropolishing so that the periphery of the hole became cross-sectionally wedge-shaped. The high-energy electron irradiation of the thin-foil specimens was performed within an HVEM (Hitachi H-3000) at 16 K using a liquid-helium cooling specimen holder (Oxford instruments), where thermal migration of vacancies was frozen 39 , and the radiation-induced microstructure was observed. The acceleration voltage for electron irradiation was 2000 kV. The beam flux was 3.0 × 10 22 m −2 s −1 . From Fig. 6 in ref. 40, the displacement energy for the [011] electron incidence was evaluated to be 46 eV, yielding the displacement cross-section of 30 × 10 −28 m 2 41 . Therefore, this beam flux corresponded to a displacement per atom (dpa) rate (point-defect pair generation rate) of 9.0 × 10 −5 dpa s −1 . TEM observation was carried out using the weak-beam dark-field technique 42 1 2 , with a diameter larger than approximately 3 nm were imaged. The sizes of the loops were evaluated for areas with a thickness of 48 nm, which was measured using the equal-thickness fringes 43 . The number densities of the loops were evaluated from the dependence of the areal densities on thickness, by taking into consideration the presence of denuded zones 44  incidence. In ref. 40, it was shown that the displacements of atoms toward the < 111> and < 001> directions occur much more easily than those toward the < 011> directions. Among multiple < 111> and < 001> directions, the most preferential displacement directions for the electron incidence in the direction of [011] were the [111] and [111] directions, which corresponded to a minimum recoil angle of approximately 35 degrees. Therefore, for the [011] electron incidence, the initial position of an SIA was set to be displaced from the initial position of a corresponding vacancy towards the [111] or [111] directions by the (r IV + 1) atomic distance. The dpa rate was set to be 9.0 × 10 −5 dpa s −1 , as in the experiments. 4. In the case of 3D migration, an SIA was moved in the direction of one of eight < 111> directions by one atomic distance. In the case of 1D migration, an SIA underwent a back and forth motion by one atomic distance parallel to the original displacement direction. For both dimensions, the motion direction for individual motion steps was chosen randomly. 5. SIA clusters were circular loops with b = 1/2< 111> and habit planes of {110}. The spontaneous change in b of the loops 20,45 was neglected. 6. Vacancies and impurities were immobile. 7. MD simulations and TEM studies have clarified that a loop undergoes 1D glide diffusion along its b direction by thermal energy 21,46,47 11. On the other hand, a single SIA and loop combined with impurities were immobile and thermally stable. A single SIA and loop moved when they were provided with an energy larger than E m I by momentum transferred from an incident electron. It is known that de-pinning of loops from impurities is enhanced by electron irradiation 48 . However, this effect is not significant for low irradiation temperatures and low beam fluxes, as in this experiment; therefore, it was neglected.
8. The reaction volume between an SIA and another object was approximated as a sphere with a corresponding radius. 9. Combinations of a vacancy and another vacancy, a vacancy and an impurity atom, and an impurity and another impurity did not occur. 10. The combination of two loops occurred when the minimum distance between SIAs composing each loop were equal to or smaller than r II (here, a single SIA is regarded as the minimum size loop, with b parallel to its motion direction). From MD simulations and TEM experiment studies, it was clarified that one loop basically absorbs the other one upon collision between the two loops, even those with different b values [49][50][51] . In the present study, a loop that was stationary at the moment just before the collision absorbed the other, and the mass centre of the new loop shifted to that of the absorbing loop. When both loops were immobilized by impurities, the smaller one was absorbed by the larger one and the impurity atom, which had pinned the absorbed one, became free. The combination of a loop with a surface, a vacancy, and an impurity atom similarly occurred depending on the values of 1/2r II , r IV , and r IX , respectively. 11. The range of examined parameters were as follows: migration dimension of an SIA: 3D, 1D; E m I for case (1): 0.095 eV, E m I for case (2): 0.00063, 0.0013, 0.0025, 0.0050, 0.010, 0.020, and 0.040 eV; C X for case (1): 3.6 × 10 −6 , C X for case (2): 6.9 × 10 −6 ; 1.0 ≤ r IV ≤ 9.0 (0.5 step), r II : 3.0, 5.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, and 14.0, and 0.0 ≤ r IX ≤ 4.0 (1.0 step) in atomic-distance units. The total number of examined parameter sets was approximately 500. 12. In comparisons of OKMC simulation results to experimental results, only loops larger than 3 nm in diameter were counted, as was done in the experimental measurements. 13. The judgment criteria for reproduction of the experimental results shown in Fig. 2b were as follows: (i) loops larger than 3 nm in diameter appeared within the computation box by an irradiation time of 120 s (this corresponds to a situation where the number density of TEM-visible loops was greater than 1.4 × 10 22 m −3 ), (ii) the average diameter, d, of loops at an irradiation time of 1200 s was 5.5 ≤ d ≤ 6.5 nm. Additionally, as shown in the main text, (iii) the maximum diameter of the loops was less than 10 nm even at an irradiation time of 1200 s.