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Primary defect production by high energy displacement cascades in molybdenum
Journal of Nuclear Materials 437 (2013) 19–23
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Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat
Primary defect production by high energy displacement cascades in molybdenum Aaron P. Selby a, Donghua Xu a,⇑, Niklas Juslin a, Nathan A. Capps a, Brian D. Wirth a,b a b
Department of Nuclear Engineering, University of Tennessee, Knoxville, TN 37996, USA Oak Ridge National Laboratory, P.O. Box 2008, MS6003, Oak Ridge, TN 37831, USA
a r t i c l e
i n f o
Article history: Received 6 December 2012 Accepted 24 January 2013 Available online 8 February 2013
a b s t r a c t We report molecular dynamics simulations of primary damage in molybdenum produced by high energy displacement cascades on the femto- to pico-second and Angstrom to nanometer scales. Clustering directly occurred for both interstitials and vacancies in the 1–50 keV cascade energy range explored. Point defect survival efficiency and partitioning probabilities into different sized clusters were quantified. The results will provide an important reference for kinetic models to describe the microstructural evolution in Mo under ion or neutron irradiations over much longer time and length scales. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction With a high melting point, excellent high temperature strength, and good resistance to corrosion and irradiation induced swelling in nuclear environments, molybdenum has potential to be used in new advanced fission and fusion reactors [1]. Molybdenum is also a model material in experimental mechanistic studies for BCC (body-centered-cubic) structured metals/alloys, particularly ferritic martensitic steels, since Mo does not have as severe problems as often encountered in handling the iron-based materials, for instance, oxidation and magnetization. For example, vacancy loop formation observed in ex situ TEM studies of thin Mo foils [2,3] under certain irradiations has been related to the cascade core collapse mechanism reported from a molecular dynamics (MD) simulation in pure Fe [4]. More recently, Li et al. [5] performed in situ krypton ion irradiation experiments with Mo TEM foils and observed the continuous formation and growth of dislocation loops whose nature, interstitial or vacancy type, could not be identified due to their overly small sizes. Coordinated modeling using cluster dynamics [6] and kinetic Monte Carlo [7] approaches was performed to test the consistency of various physical models (defect production mode, number of mobile defect species, mobility of defect clusters, surface sink effect, etc.) with Li et al.’s in situ experiments. In the modeling studies, defect kinetic and thermodynamic data from limited experimental and atomistic simulation literature for Mo were used, such as the formation and migration energies of a single interstitial and/or a single vacancy [8,9]. When references were not directly available for Mo, e.g., for the mobility model of different-sized interstitial clusters or the intra-cascade production probabilities of small interstitial/vacancy clusters, references were made to relevant studies in BCC iron based on which ⇑ Corresponding author. Tel.: +1 865 974 2525; fax: +1 865 974 0668. E-mail address: [email protected] (D. Xu). 0022-3115/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jnucmat.2013.01.332
certain adjustments or parameter optimization was performed [6,10–16]. While referring to studies in BCC iron can serve as a first approximation, it is necessary to extend the current atomistic database for Mo in order to fully evaluate effects of the unknown factors and to provide a more rigorous comparison and validation of the models with respect to the Mo experiments. In this paper, we focus on the primary damage production by high energy displacement cascades in Mo. Specifically, we aim to address the following two issues using molecular dynamics simulations: what defects (isolated single interstitials/vacancies and/or clusters) are directly produced; and what are the point defect survival efficiency and partitioning probabilities into different sized clusters, if clusters do form. The resulting qualitative and quantitative information is important to, and can be directly implemented in, the models of irradiation defect evolution in Mo using a cluster dynamics or kinetic Monte Carlo approach.
2. Simulation methods The MD simulations were performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) code developed by Sandia National Laboratories. A modification of the Finnis–Sinclair type potential [17] by Ackland and Thetford [18] with a ZBL (Ziegler–Biersack–Littmark) fit for short distance repulsive forces added by Salonen et al. [19] was used in the simulations. The process of simulating the cascade damage production involved setting up a simulation box of pure molybdenum, equilibrating the box at 300 K, and then assigning an arbitrary atom a velocity corresponding to a chosen energy (1, 2, 10, 20, 30, or 50 keV), which represents a primary knock-on atom (PKA). The velocity direction of the PKA was chosen randomly. For the energies of 1, 2, and 10 keV, a box size of 50a0 50a0 50a0 (250,000 atoms) was used, where a0 is the lattice parameter,
20
A.P. Selby et al. / Journal of Nuclear Materials 437 (2013) 19–23
3.1472 Å for molybdenum. For PKA energies of 20 and 30 keV, a box size of 80a0 80a0 80a0 (1,024,000 atoms) was used. An even larger box size of 120a0 120a0 120a0 (3,456,000 atoms) was used for a PKA energy of 50 keV. These box sizes were sufficiently large at the respective PKA energies to prevent self-interaction of the cascade with its periodic images. Periodic boundary conditions, a micro-canonical ensemble and a nominal time-step of 0.5 fs were used for all simulations. The time-step was allowed to reset to a lower value if an atom moved more than 0.0003a0 within a single time step. Each simulation was run sufficiently long to allow the cascade to anneal and return to thermal equilibrium. After the completion of a cascade simulation, the defects, including both interstitials and vacancies, were visualized at each output time step via Atomeye: Atomistic Configuration Viewer [20].
3. Results and discussion Consistent with previous cascade literature, such as in iron [10,21], a large number of point defects were generated within the first pico-second and most of them recombined in the next few pico-seconds as the cascade entered the annealing stage. At the end of the annealing process, only a small fraction of vacancies and interstitials survived, with the remaining vacancies mostly in isolated state whereas a sizeable portion of the remaining selfinterstitials were contained in clusters/loops. The remaining defects/clusters of different sizes (defined by the number of point defects contained) were counted, and then the total number of surviving Frenkel pairs and the partitioning probabilities of point defects into different cluster sizes were calculated for each simulation. To obtain as meaningful statistics as possible, a large number of simulations were performed for each PKA energy, namely, 80 unique simulations for 1 and 2 keV, 40 for 10, 20 and 30 keV, and 21 for 50 keV. Each simulation used a different PKA atom and incident direction of the PKA at the selected energy. The total number of surviving Frenkel pairs is plotted against the MD cascade energy in Fig. 1. Generally, due to PKA energy loss to electronic excitation, cascade energy Ecasc is only a fraction of PKA energy T that is available to generate atomic displacements by elastic collisions. As described by the Lindhard model of energy partitioning [22], the cascade energy is given by nT, where n de1 pends on T as: nðTÞ ¼ 1þ0:134Z2=3 A1=2 ð3:4 , e ¼ ð2Z2Te2 =aÞ, and e1=6 þ0:4e3=4 þeÞ
Fig. 1. The total number of surviving Frenkel pairs as a function of MD cascade energy. The red line is the power law fit of the data, i.e., 1:92E0:864 casc . The error bars correspond to one standard deviation (same meaning in Figs. 2–4). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
bohr a ¼ 0:626a , in which Z and A are the atomic number and atomic Z 1=3 mass of the matrix element, respectively, and abohr is Bohr radius. In the present classical MD simulations, however, there is no account of electronic stopping. Hence the cascade energy is simply equivalent to the PKA energy here. According to the NRT (Norgett–Robinson–Torrens) model [23], the number of Frenkel pairs generated by a cascade of energy Ecasc (determined through the Lindhard model in general) can be formulated as (Ecasc/2Ed) 0.8, where Ed is the displacement threshold (60 eV for Mo) and the 0.8 is the ‘‘displacement efficiency’’. As stated in Ref. [22], the ‘‘displacement efficiency’’ factor in the NRT model is equivalent to a combination of three factors in the half-Nelson model that account for non-hard core atomic scattering, defect recombination within a cascade, and defect recombination due to overlapping of different branches of a cascade. The NRT model states a linear dependence of the number of generated Frenkel pairs on the cascade energy. However, as shown in Fig. 1, fitting current data into a power law revealed a sub-linear dependence, namely, 1:92E0:864 casc . This observation is consistent with the sub-linear behavior previously reported in BCC iron and several face-centered-cubic (FCC) and hexagonal-close-packed (HCP) metals [10,21]. The deviation from NRT linearity indicates a cascade energy dependence of the ‘‘displacement efficiency’’ factor that is used to reduce Ecasc/2Ed. To emphasize its function of providing the quantity of point defects surviving a cascade, we will refer to this factor as ‘‘survival efficiency’’ hereafter. It is important to point out that the sub-linear (slope less than one in a double logarithmic plot) behavior in Fig. 1 is most apparent from 1 keV to 20 keV, indicating a rapid change in the survival efficiency as the cascade energy increases in this regime. From 20 keV to 50 keV, Fig. 1 shows a clear tendency to restore a linear relationship between the total number of surviving Frenkel pairs and the cascade energy, suggesting a stabilization of the survival efficiency. These features have also been reported for other metals previously [24]. The actual survival efficiency was calculated at each examined cascade energy, through NFP/(Ecasc/2Ed), where NFP is the counted number of surviving Frenkel pairs, and is plotted in Fig. 2. A rapid drop and a stabilization behavior can be clearly identified below and above 20 keV, respectively. Similar dependence of the survival efficiency on the cascade energy has also been reported in BCC iron [10]. Table 1 lists the observed vacancy or interstitial clusters and the corresponding partitioning probabilities at the six cascade energies examined, along with the survival efficiencies presented in Fig. 2. Note that the listed data were averaged over all the respective simulations at each cascade energy. The partitioning probabilities are
Fig. 2. Point defect survival efficiency as a function of MD cascade energy.
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A.P. Selby et al. / Journal of Nuclear Materials 437 (2013) 19–23 Table 1 Partitioning probabilities of point defects into clusters and point defect survival efficiency at six cascade energies. Cluster
Partitioning probabilities of point defects into clusters
Type
Size
1 keV
2 keV
10 keV
20 keV
30 keV
50 keV
Interstitial cluster
24 23 19 17 15 13 12 11 10 9 8 7 6 5 4 3 2 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.017 0.203 0.780
0 0 0 0 0 0 0 0 0 0 0 0 0.009 0.013 0.029 0.133 0.295 0.521
0 0 0 0 0 0 0 0.020 0.016 0.011 0.012 0.023 0.017 0.046 0.087 0.101 0.251 0.418
0 0 0 0 0 0.010 0 0 0 0.029 0.034 0.013 0.021 0.039 0.085 0.121 0.210 0.439
0 0 0.011 0.010 0.010 0.019 0.006 0.031 0.014 0.008 0.016 0.008 0.048 0.045 0.063 0.125 0.197 0.390
0.019 0.021 0.018 0.012 0 0 0 0.009 0.000 0.000 0.039 0.030 0.013 0.063 0.085 0.118 0.205 0.369
1 2 3 4 5 6 9
0.877 0.104 0.019 0 0 0 0
0.829 0.146 0.012 0.013 0 0 0
0.784 0.160 0.018 0.024 0.007 0.008 0
0.795 0.154 0.040 0.005 0.007 0 0
0.789 0.154 0.042 0.010 0.004 0 0
0.802 0.133 0.037 0.006 0.008 0.006 0.007
0.34
0.26
0.19
0.14
0.14
0.14
Vacancy cluster
Survival efficiency
Fig. 3. Partitioning probabilities of single interstitials into different sized clusters at six cascade energies.
further plotted in Figs. 3 and 4 against the cluster composition which is represented by the number of contained single defects. Overall, higher energy cascades produced larger interstitial/vacancy clusters. The largest interstitial cluster and the largest vacancy cluster in this study were both formed at 50 keV, the highest cascade energy examined, with the interstitial cluster containing 24 single interstitials whereas the vacancy cluster consisted of nine vacancies. Across all the examined cascade energies, only about 20% or less of the surviving vacancies existed in the form of clusters, and it is worth noting that the vacancy clus-
ters had a three dimensional rather than loop morphology. In contrast, the fraction of surviving interstitials that were contained in clusters increased from 20% at 1 keV cascade energy to 60% at 50 keV cascade energy. The sizes of the vacancy clusters are rather small and only slightly dependent on the cascade energy, whereas the sizes of the interstitial clusters are much bigger and clearly increase with the cascade energy in the range examined. Beginning at a PKA energy of 30 keV, subcascades were observed, as shown in Fig. 5. However, the frequency of subcascade formation was low. It occurred in three out of the 40 simulations
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A.P. Selby et al. / Journal of Nuclear Materials 437 (2013) 19–23
Fig. 4. Partitioning probabilities of single vacancies into different sized clusters at six cascade energies.
Fig. 6. Variation of standard deviations of partitioning probabilities with number of simulations at the 1 keV cascade energy. Note that the zero deviation for I3 in the first 40 runs is due to the fact that this cluster was not formed in these runs. Fig. 5. Dimensions are in nanometer. Big spheres represent self-interstitial atoms, small spheres vacancies.
at 30 keV and two out of the 21 simulations at 50 keV. It appears that the probability of forming subcascades increases with the PKA energy. However, determination of the subcascade formation probability would require many more simulations at the high energies. Due to the formation of subcascades, it is reasonable to infer that the primary defect production by cascades with even higher energies than those examined in this study may not significantly differ from the 30 keV or 50 keV results. Although the average values of defect survival efficiency and the partitioning probabilities reported here can be readily implemented in meso-scale kinetic models, the error bars (± a standard deviation) in Figs. 2–4 indicate that the statistical uncertainties remain rather large, despite the large numbers of simulations performed at each cascade energy. This is particularly an issue at low cascade energies of 1 and 2 keV, where the total number of
surviving point defects is low. Fig. 6 shows the variation of the standard deviations of the partitioning probabilities with number of simulations at the 1 keV cascade energy. Increasing the number of simulations does not seem to efficiently suppress the standard deviations. The large standard deviations may be more of an inherent physical aspect. Given the uncertainties in the survival efficiency and the partitioning probabilities, it may be necessary to perform error sensitivity tests on the meso-scale models when implementing these results. Finally, some configurational and mobility characteristics of the observed interstitials and interstitial clusters are worth mentioning. Single interstitials were observed in three configurations: h1 1 0i and h1 1 1i oriented dumbbells, and h1 1 1i oriented crowdions, while interstitial clusters were consisted of parallel h1 1 1i oriented dumbbells together with h1 1 1i oriented crowdions, the number of crowdions increasing with increasing cluster size. This is also consistent with the observations in MD simulations of iron
A.P. Selby et al. / Journal of Nuclear Materials 437 (2013) 19–23
[10]. Within the simulated time frames in this study, single interstitials and di-interstitial clusters both showed mixed rotation and one-dimensional gliding, while clusters with three or more interstitials only showed 1-D gliding, and their gliding frequencies appeared to decrease with increasing cluster size. However, separate simulations, with varying (longer) time scales and temperatures, are needed to fully investigate and quantify the mobilities of single interstitials and interstitial clusters. 4. Conclusions We have performed molecular dynamics simulations to study the primary defect production by high energy cascades in molybdenum. Our attention has been focused on the two quantities that are important for meso-scale kinetic modeling of irradiation damage in Mo, namely, the defect survival efficiency and the partitioning probabilities of point defects into different sized clusters. The key results can be summarized as: 1. A database of the point defect survival efficiency and partitioning probabilities has been obtained at six cascade energies. 2. 80% or more of surviving vacancies are isolated; di-vacancies are the most probable cluster for vacancies; the largest vacancy cluster observed consists of nine single vacancies; vacancy clusters have three-dimensional morphology. 3. Interstitial clustering fraction is greater than 50% for cascade energies above 10 keV; largest interstitial cluster size observed is 19 interstitials at 30 keV and 24 interstitials at 50 keV; interstitial clusters have configuration of prismatic dislocation loops in h1 1 1i direction (Burgers vector). The primary damage in Mo is qualitatively similar to that in BCC iron, however, the potential effect of their quantitative difference on the modeling of irradiation defect microstructure evolution has yet to evaluated.
23
Acknowledgements We acknowledge support by the U.S. Department of Energy, Office of Fusion Energy Sciences under Grant DOE-DE-SC0006661 and the U.S. Department of Energy, Office of Nuclear Energy’s Nuclear Energy University Programs (NEUP). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
S.J. Zinkle, N.M. Ghoniem, Fusion Eng. Des. 51–52 (2000) 55. C.A. English, M.L. Jenkins, Mater. Sci. Forum 15–18 (1987) 1003. C.A. English, M.L. Jenkins, Philos. Mag. 90 (2010) 821. N. Soneda, S. Ishino, T.D. de la Rubia, Philos. Mag. Lett. 81 (2001) 649. M.M. Li, M.A. Kirk, P.M. Baldo, D.H. Xu, B.D. Wirth, Philos. Mag. 92 (2012) 2048. D.H. Xu, B.D. Wirth, M.M. Li, M.A. Kirk, Acta Mater. 60 (2012) 4286. D.H. Xu, B.D. Wirth, M.M. Li, M.A. Kirk, Appl. Phys. Lett. 101 (2012) 101905. P.M. Derlet, D. Nguyen-Manh, S.L. Dudarev, Phys. Rev. B 76 (2007) 054107. P. Ehrhart, P. Jung, H. Schultz, H. Ullmaier, Landolt-Bornstein New Series III/25, Springer, New York, 1991. R.E. Stoller, G.R. Odette, B.D. Wirth, J. Nucl. Mater. 251 (1997) 49. B.D. Wirth, G.R. Odette, D. Maroudas, G.E. Lucas, J. Nucl. Mater. 244 (1997) 185. N. Soneda, T.D. de la Rubia, Philos. Mag. A 78 (1998) 995. J. Marian, B.D. Wirth, A. Caro, B. Sadigh, G.R. Odette, J.M. Perlado, T.D. de la Rubia, Phys. Rev. B 65 (2002) 144102. A.F. Calder, D.J. Bacon, J. Nucl. Mater. 207 (1993) 25. K. Arakawa, K. Ono, M. Isshiki, K. Mimura, M. Uchikoshi, H. Mori, Science 318 (2007) 956. P.R. Monasterio, B.D. Wirth, G.R. Odette, J. Nucl. Mater. 361 (2007) 127. M.W. Finnis, J.E. Sinclair, Philos. Mag. A 50 (1984) 45. G.J. Ackland, R. Thetford, Philos. Mag. A 56 (1987) 15. E. Salonen, T. Järvi, K. Nordlund, and J. Keinonen, J. Phys.: Condens. Matter. 15, 5845 (2003). J. Li, Mater. Sci. Eng. 11 (2003) 173. D.J. Bacon, A.F. Calder, F. Gao, V.G. Kapinos, S.J. Wooding, Nucl. Instrum. Meth. B 102 (1995) 37. M.T. Robinson, BNES Nuclear Fusion Reactors, British Nuclear Energy Society, London, 1970. M.J. Norgett, M.T. Robinson, I.M. Torrens, Nucl. Eng. Des. 33 (1975) 50. R.E. Stoller, Primary radiation damage formation, in: R.J.M. Konings (Ed.), Comprehensive Nuclear Materials, Elsevier Ltd., 2012, pp. 293–332.
Contents lists available at SciVerse ScienceDirect
Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat
Primary defect production by high energy displacement cascades in molybdenum Aaron P. Selby a, Donghua Xu a,⇑, Niklas Juslin a, Nathan A. Capps a, Brian D. Wirth a,b a b
Department of Nuclear Engineering, University of Tennessee, Knoxville, TN 37996, USA Oak Ridge National Laboratory, P.O. Box 2008, MS6003, Oak Ridge, TN 37831, USA
a r t i c l e
i n f o
Article history: Received 6 December 2012 Accepted 24 January 2013 Available online 8 February 2013
a b s t r a c t We report molecular dynamics simulations of primary damage in molybdenum produced by high energy displacement cascades on the femto- to pico-second and Angstrom to nanometer scales. Clustering directly occurred for both interstitials and vacancies in the 1–50 keV cascade energy range explored. Point defect survival efficiency and partitioning probabilities into different sized clusters were quantified. The results will provide an important reference for kinetic models to describe the microstructural evolution in Mo under ion or neutron irradiations over much longer time and length scales. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction With a high melting point, excellent high temperature strength, and good resistance to corrosion and irradiation induced swelling in nuclear environments, molybdenum has potential to be used in new advanced fission and fusion reactors [1]. Molybdenum is also a model material in experimental mechanistic studies for BCC (body-centered-cubic) structured metals/alloys, particularly ferritic martensitic steels, since Mo does not have as severe problems as often encountered in handling the iron-based materials, for instance, oxidation and magnetization. For example, vacancy loop formation observed in ex situ TEM studies of thin Mo foils [2,3] under certain irradiations has been related to the cascade core collapse mechanism reported from a molecular dynamics (MD) simulation in pure Fe [4]. More recently, Li et al. [5] performed in situ krypton ion irradiation experiments with Mo TEM foils and observed the continuous formation and growth of dislocation loops whose nature, interstitial or vacancy type, could not be identified due to their overly small sizes. Coordinated modeling using cluster dynamics [6] and kinetic Monte Carlo [7] approaches was performed to test the consistency of various physical models (defect production mode, number of mobile defect species, mobility of defect clusters, surface sink effect, etc.) with Li et al.’s in situ experiments. In the modeling studies, defect kinetic and thermodynamic data from limited experimental and atomistic simulation literature for Mo were used, such as the formation and migration energies of a single interstitial and/or a single vacancy [8,9]. When references were not directly available for Mo, e.g., for the mobility model of different-sized interstitial clusters or the intra-cascade production probabilities of small interstitial/vacancy clusters, references were made to relevant studies in BCC iron based on which ⇑ Corresponding author. Tel.: +1 865 974 2525; fax: +1 865 974 0668. E-mail address: [email protected] (D. Xu). 0022-3115/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jnucmat.2013.01.332
certain adjustments or parameter optimization was performed [6,10–16]. While referring to studies in BCC iron can serve as a first approximation, it is necessary to extend the current atomistic database for Mo in order to fully evaluate effects of the unknown factors and to provide a more rigorous comparison and validation of the models with respect to the Mo experiments. In this paper, we focus on the primary damage production by high energy displacement cascades in Mo. Specifically, we aim to address the following two issues using molecular dynamics simulations: what defects (isolated single interstitials/vacancies and/or clusters) are directly produced; and what are the point defect survival efficiency and partitioning probabilities into different sized clusters, if clusters do form. The resulting qualitative and quantitative information is important to, and can be directly implemented in, the models of irradiation defect evolution in Mo using a cluster dynamics or kinetic Monte Carlo approach.
2. Simulation methods The MD simulations were performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) code developed by Sandia National Laboratories. A modification of the Finnis–Sinclair type potential [17] by Ackland and Thetford [18] with a ZBL (Ziegler–Biersack–Littmark) fit for short distance repulsive forces added by Salonen et al. [19] was used in the simulations. The process of simulating the cascade damage production involved setting up a simulation box of pure molybdenum, equilibrating the box at 300 K, and then assigning an arbitrary atom a velocity corresponding to a chosen energy (1, 2, 10, 20, 30, or 50 keV), which represents a primary knock-on atom (PKA). The velocity direction of the PKA was chosen randomly. For the energies of 1, 2, and 10 keV, a box size of 50a0 50a0 50a0 (250,000 atoms) was used, where a0 is the lattice parameter,
20
A.P. Selby et al. / Journal of Nuclear Materials 437 (2013) 19–23
3.1472 Å for molybdenum. For PKA energies of 20 and 30 keV, a box size of 80a0 80a0 80a0 (1,024,000 atoms) was used. An even larger box size of 120a0 120a0 120a0 (3,456,000 atoms) was used for a PKA energy of 50 keV. These box sizes were sufficiently large at the respective PKA energies to prevent self-interaction of the cascade with its periodic images. Periodic boundary conditions, a micro-canonical ensemble and a nominal time-step of 0.5 fs were used for all simulations. The time-step was allowed to reset to a lower value if an atom moved more than 0.0003a0 within a single time step. Each simulation was run sufficiently long to allow the cascade to anneal and return to thermal equilibrium. After the completion of a cascade simulation, the defects, including both interstitials and vacancies, were visualized at each output time step via Atomeye: Atomistic Configuration Viewer [20].
3. Results and discussion Consistent with previous cascade literature, such as in iron [10,21], a large number of point defects were generated within the first pico-second and most of them recombined in the next few pico-seconds as the cascade entered the annealing stage. At the end of the annealing process, only a small fraction of vacancies and interstitials survived, with the remaining vacancies mostly in isolated state whereas a sizeable portion of the remaining selfinterstitials were contained in clusters/loops. The remaining defects/clusters of different sizes (defined by the number of point defects contained) were counted, and then the total number of surviving Frenkel pairs and the partitioning probabilities of point defects into different cluster sizes were calculated for each simulation. To obtain as meaningful statistics as possible, a large number of simulations were performed for each PKA energy, namely, 80 unique simulations for 1 and 2 keV, 40 for 10, 20 and 30 keV, and 21 for 50 keV. Each simulation used a different PKA atom and incident direction of the PKA at the selected energy. The total number of surviving Frenkel pairs is plotted against the MD cascade energy in Fig. 1. Generally, due to PKA energy loss to electronic excitation, cascade energy Ecasc is only a fraction of PKA energy T that is available to generate atomic displacements by elastic collisions. As described by the Lindhard model of energy partitioning [22], the cascade energy is given by nT, where n de1 pends on T as: nðTÞ ¼ 1þ0:134Z2=3 A1=2 ð3:4 , e ¼ ð2Z2Te2 =aÞ, and e1=6 þ0:4e3=4 þeÞ
Fig. 1. The total number of surviving Frenkel pairs as a function of MD cascade energy. The red line is the power law fit of the data, i.e., 1:92E0:864 casc . The error bars correspond to one standard deviation (same meaning in Figs. 2–4). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
bohr a ¼ 0:626a , in which Z and A are the atomic number and atomic Z 1=3 mass of the matrix element, respectively, and abohr is Bohr radius. In the present classical MD simulations, however, there is no account of electronic stopping. Hence the cascade energy is simply equivalent to the PKA energy here. According to the NRT (Norgett–Robinson–Torrens) model [23], the number of Frenkel pairs generated by a cascade of energy Ecasc (determined through the Lindhard model in general) can be formulated as (Ecasc/2Ed) 0.8, where Ed is the displacement threshold (60 eV for Mo) and the 0.8 is the ‘‘displacement efficiency’’. As stated in Ref. [22], the ‘‘displacement efficiency’’ factor in the NRT model is equivalent to a combination of three factors in the half-Nelson model that account for non-hard core atomic scattering, defect recombination within a cascade, and defect recombination due to overlapping of different branches of a cascade. The NRT model states a linear dependence of the number of generated Frenkel pairs on the cascade energy. However, as shown in Fig. 1, fitting current data into a power law revealed a sub-linear dependence, namely, 1:92E0:864 casc . This observation is consistent with the sub-linear behavior previously reported in BCC iron and several face-centered-cubic (FCC) and hexagonal-close-packed (HCP) metals [10,21]. The deviation from NRT linearity indicates a cascade energy dependence of the ‘‘displacement efficiency’’ factor that is used to reduce Ecasc/2Ed. To emphasize its function of providing the quantity of point defects surviving a cascade, we will refer to this factor as ‘‘survival efficiency’’ hereafter. It is important to point out that the sub-linear (slope less than one in a double logarithmic plot) behavior in Fig. 1 is most apparent from 1 keV to 20 keV, indicating a rapid change in the survival efficiency as the cascade energy increases in this regime. From 20 keV to 50 keV, Fig. 1 shows a clear tendency to restore a linear relationship between the total number of surviving Frenkel pairs and the cascade energy, suggesting a stabilization of the survival efficiency. These features have also been reported for other metals previously [24]. The actual survival efficiency was calculated at each examined cascade energy, through NFP/(Ecasc/2Ed), where NFP is the counted number of surviving Frenkel pairs, and is plotted in Fig. 2. A rapid drop and a stabilization behavior can be clearly identified below and above 20 keV, respectively. Similar dependence of the survival efficiency on the cascade energy has also been reported in BCC iron [10]. Table 1 lists the observed vacancy or interstitial clusters and the corresponding partitioning probabilities at the six cascade energies examined, along with the survival efficiencies presented in Fig. 2. Note that the listed data were averaged over all the respective simulations at each cascade energy. The partitioning probabilities are
Fig. 2. Point defect survival efficiency as a function of MD cascade energy.
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A.P. Selby et al. / Journal of Nuclear Materials 437 (2013) 19–23 Table 1 Partitioning probabilities of point defects into clusters and point defect survival efficiency at six cascade energies. Cluster
Partitioning probabilities of point defects into clusters
Type
Size
1 keV
2 keV
10 keV
20 keV
30 keV
50 keV
Interstitial cluster
24 23 19 17 15 13 12 11 10 9 8 7 6 5 4 3 2 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.017 0.203 0.780
0 0 0 0 0 0 0 0 0 0 0 0 0.009 0.013 0.029 0.133 0.295 0.521
0 0 0 0 0 0 0 0.020 0.016 0.011 0.012 0.023 0.017 0.046 0.087 0.101 0.251 0.418
0 0 0 0 0 0.010 0 0 0 0.029 0.034 0.013 0.021 0.039 0.085 0.121 0.210 0.439
0 0 0.011 0.010 0.010 0.019 0.006 0.031 0.014 0.008 0.016 0.008 0.048 0.045 0.063 0.125 0.197 0.390
0.019 0.021 0.018 0.012 0 0 0 0.009 0.000 0.000 0.039 0.030 0.013 0.063 0.085 0.118 0.205 0.369
1 2 3 4 5 6 9
0.877 0.104 0.019 0 0 0 0
0.829 0.146 0.012 0.013 0 0 0
0.784 0.160 0.018 0.024 0.007 0.008 0
0.795 0.154 0.040 0.005 0.007 0 0
0.789 0.154 0.042 0.010 0.004 0 0
0.802 0.133 0.037 0.006 0.008 0.006 0.007
0.34
0.26
0.19
0.14
0.14
0.14
Vacancy cluster
Survival efficiency
Fig. 3. Partitioning probabilities of single interstitials into different sized clusters at six cascade energies.
further plotted in Figs. 3 and 4 against the cluster composition which is represented by the number of contained single defects. Overall, higher energy cascades produced larger interstitial/vacancy clusters. The largest interstitial cluster and the largest vacancy cluster in this study were both formed at 50 keV, the highest cascade energy examined, with the interstitial cluster containing 24 single interstitials whereas the vacancy cluster consisted of nine vacancies. Across all the examined cascade energies, only about 20% or less of the surviving vacancies existed in the form of clusters, and it is worth noting that the vacancy clus-
ters had a three dimensional rather than loop morphology. In contrast, the fraction of surviving interstitials that were contained in clusters increased from 20% at 1 keV cascade energy to 60% at 50 keV cascade energy. The sizes of the vacancy clusters are rather small and only slightly dependent on the cascade energy, whereas the sizes of the interstitial clusters are much bigger and clearly increase with the cascade energy in the range examined. Beginning at a PKA energy of 30 keV, subcascades were observed, as shown in Fig. 5. However, the frequency of subcascade formation was low. It occurred in three out of the 40 simulations
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Fig. 4. Partitioning probabilities of single vacancies into different sized clusters at six cascade energies.
Fig. 6. Variation of standard deviations of partitioning probabilities with number of simulations at the 1 keV cascade energy. Note that the zero deviation for I3 in the first 40 runs is due to the fact that this cluster was not formed in these runs. Fig. 5. Dimensions are in nanometer. Big spheres represent self-interstitial atoms, small spheres vacancies.
at 30 keV and two out of the 21 simulations at 50 keV. It appears that the probability of forming subcascades increases with the PKA energy. However, determination of the subcascade formation probability would require many more simulations at the high energies. Due to the formation of subcascades, it is reasonable to infer that the primary defect production by cascades with even higher energies than those examined in this study may not significantly differ from the 30 keV or 50 keV results. Although the average values of defect survival efficiency and the partitioning probabilities reported here can be readily implemented in meso-scale kinetic models, the error bars (± a standard deviation) in Figs. 2–4 indicate that the statistical uncertainties remain rather large, despite the large numbers of simulations performed at each cascade energy. This is particularly an issue at low cascade energies of 1 and 2 keV, where the total number of
surviving point defects is low. Fig. 6 shows the variation of the standard deviations of the partitioning probabilities with number of simulations at the 1 keV cascade energy. Increasing the number of simulations does not seem to efficiently suppress the standard deviations. The large standard deviations may be more of an inherent physical aspect. Given the uncertainties in the survival efficiency and the partitioning probabilities, it may be necessary to perform error sensitivity tests on the meso-scale models when implementing these results. Finally, some configurational and mobility characteristics of the observed interstitials and interstitial clusters are worth mentioning. Single interstitials were observed in three configurations: h1 1 0i and h1 1 1i oriented dumbbells, and h1 1 1i oriented crowdions, while interstitial clusters were consisted of parallel h1 1 1i oriented dumbbells together with h1 1 1i oriented crowdions, the number of crowdions increasing with increasing cluster size. This is also consistent with the observations in MD simulations of iron
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[10]. Within the simulated time frames in this study, single interstitials and di-interstitial clusters both showed mixed rotation and one-dimensional gliding, while clusters with three or more interstitials only showed 1-D gliding, and their gliding frequencies appeared to decrease with increasing cluster size. However, separate simulations, with varying (longer) time scales and temperatures, are needed to fully investigate and quantify the mobilities of single interstitials and interstitial clusters. 4. Conclusions We have performed molecular dynamics simulations to study the primary defect production by high energy cascades in molybdenum. Our attention has been focused on the two quantities that are important for meso-scale kinetic modeling of irradiation damage in Mo, namely, the defect survival efficiency and the partitioning probabilities of point defects into different sized clusters. The key results can be summarized as: 1. A database of the point defect survival efficiency and partitioning probabilities has been obtained at six cascade energies. 2. 80% or more of surviving vacancies are isolated; di-vacancies are the most probable cluster for vacancies; the largest vacancy cluster observed consists of nine single vacancies; vacancy clusters have three-dimensional morphology. 3. Interstitial clustering fraction is greater than 50% for cascade energies above 10 keV; largest interstitial cluster size observed is 19 interstitials at 30 keV and 24 interstitials at 50 keV; interstitial clusters have configuration of prismatic dislocation loops in h1 1 1i direction (Burgers vector). The primary damage in Mo is qualitatively similar to that in BCC iron, however, the potential effect of their quantitative difference on the modeling of irradiation defect microstructure evolution has yet to evaluated.
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Acknowledgements We acknowledge support by the U.S. Department of Energy, Office of Fusion Energy Sciences under Grant DOE-DE-SC0006661 and the U.S. Department of Energy, Office of Nuclear Energy’s Nuclear Energy University Programs (NEUP). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
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