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First-principles study of noble gas atoms in bcc Fe
Journal of Nuclear Materials 492 (2017) 134e141
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First-principles study of noble gas atoms in bcc Fe Pengbo Zhang a, b, *, Jianhua Ding a, Dan Sun a, Jijun Zhao a, ** a b
Key Laboratory of Materials Modification by Laser, Ion and Electron Beams (Dalian University of Technology), Ministry of Education, Dalian 116024, China Department of Physics, Dalian Maritime University, Dalian 116026, China
h i g h l i g h t s Interstitial noble gas atoms tend to stay together with each other by self-trapping. H/He prefers to locate interstitial sites nearby Ne atom than other interstitials. Noble gas atoms like Ne can act as a trapping site for H/He impurities in bcc Fe.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 January 2017 Received in revised form 17 May 2017 Accepted 17 May 2017 Available online 19 May 2017
We investigate the energetics and clustering trend of noble gas atoms (He, Ne, and Ar) in bcc Fe, and their interactions with vacancy or H/He impurities using first-principles calculations. We determine the formation energy of single and double noble gas atoms inside Fe host lattice as well as the resulted volume changes. The Ne/Ar formation energy is two and three times that of He. The attraction between Ne/Ar and vacancy is stronger than He-vacancy, indicating higher dissolution energy of Ne/Ar. The interstitial Ne-Ne/Ar-Ar pairs have stronger attractions (1.91 eV/1.40 eV) than He-He (0.37 eV), forming stable <110> configurations. Such strong attraction means that He/Ne/Ar tend to aggregate, which can be well explained by the lower electron density induced by interstitial noble gas atoms and its strong repulsion with Fe atoms. Moreover, H/He energetically prefers to occupy the tetrahedral sites nearby Ne/Ar atom. The attraction energies of He-Ne/He-Ar pairs (1.01 eV/-0.85 eV) are much stronger than those of H-Ne/ H-Ar (0.22 eV/0.10 eV) and their charge density differences are discussed. The distinct attraction strengths by various noble gas atoms provide a preliminary explanation for the difference in irradiation effects on Fe solid by He, Ne, Ar, and HeþH/NeþHe. These findings improve our understanding about the behavior of noble gas atoms and gas bubble formation in iron under irradiation. © 2017 Elsevier B.V. All rights reserved.
Keywords: Noble gas Fe Interaction Irradiation First principles
1. Introduction High-strength Fe-based alloys of bcc phase have been widely used in nuclear reactors [1e3]. Large amounts of He and H impurities are produced in the neutron-irradiated Fe solid and alloys via (n, a) transmutation reactions, which play an important role in microstructure evolution and degradation of mechanical properties, known as the irradiation damage effect [4e7]. Thus, understanding the effect of He atoms in metals and alloys is crucial for
* Corresponding author. Key Laboratory of Materials Modification by Laser, Ion and Electron Beams (Dalian University of Technology), Ministry of Education, Dalian 116024, China. ** Corresponding author. E-mail addresses: [email protected] (P. Zhang), [email protected] (J. Zhao). http://dx.doi.org/10.1016/j.jnucmat.2017.05.022 0022-3115/© 2017 Elsevier B.V. All rights reserved.
development of materials resistant to irradiation under fusion and fission environments. Experimentally, multi-beam irradiations by He/Ne/Ar ions have been intensively carried out to investigate He embrittlement and irradiation damage in metals and alloys [8e15]. Previous TEM experiments [8,11,13] observed formation of noble gas bubbles of He, Ne, or Ar in pure Ni, W metals and Fe alloys. He atoms can be deeply trapped in vacancies and grain boundaries, leading to bubble formation [16,17]. The volume change associated with void swelling is quite large (over 10%) [18]. Until now, experimental data on defect energies of the noble gas atoms in bcc Fe solid and its alloys, especially the formation/binding energy of noble gas atoms and their binding energies with vacancy/H/He, are generally scarce. Density functional theory (DFT) calculations have been extensively performed to investigate the defect formation energy, diffusion and clustering of He atoms in bcc Fe [19e28]. Fu and
P. Zhang et al. / Journal of Nuclear Materials 492 (2017) 134e141
Willaime examined the stability of He and He-vacancy clusters, their results showed that He migration barrier is very small (about 0.06 eV) and the strength of He-He attraction in a-Fe lattice is 0.43 eV [19]. Seletskaia reported that magnetic interactions influence the behavior of He in bcc Fe [20] and revealed that strong hybridization between He and the surrounding Fe atoms results in high defect formation energy. Sakuraya et al. investigated the effects of H, He, C, and N impurities in a-Fe [24] and found the magnetic moment of Fe slightly increases after He insertion. Our recent studies demonstrated that interstitial He greatly reduces the vacancy formation energy in bcc Fe [22] and interstitial He-He has strong attraction in bcc Fe, Cr, Mo and W metals, suggesting that He clustering may occur by self-trapping [29]. Kong revealed He clustering in W host lattice by self-trapping [30]. However, the effects of other noble gas atoms and their impact on H/He behaviors in bcc Fe have received much less attention in terms of atomic simulations. Owing to the accumulation of He and other noble gas atoms in pure Fe, steels, and other metals, accurate determination of defect formation energies and their interaction strengths are important for developing predictive models for the quantification of irradiation damage. In this paper, we investigate the energetics and clustering trend of noble gas atoms (He, Ne, and Ar), as well as their interactions with vacancy and H/He impurities in bcc Fe solid using first-principles calculations. Firstly, we determine the occupying site, volume changes and formation energy of single and dual noble gas atoms in Section 3.1. We then calculate binding energy between a noble gas atom and a vacancy in Section 3.2. We further calculate binding energy between two noble gas atoms (Ne/Ar and H/He) and discuss their electronic structures in Section 3.3 and 3.4. We find that interstitial noble gas atoms tend to stay together with each other and are able to attract He/H impurities. The present results provide vital insights into the behavior of noble gas atoms and gas bubble formation in iron under irradiation. 2. Computational methods and models All calculations were performed using spin-polarized density functional theory and the plane-wave pseudopotential approach [31,32], as implemented in the Vienna Ab initio Simulation Package (VASP 5.3.5) [33,34]. We adopted the generalized gradient approximation (GGA) with the Perdew and Wang (PW91) functional [35] for the exchange-correlation interaction and the projector-augmented wave (PAW) potentials [36,37] for the ionelectron interaction. A 128-atom bcc supercell (4 4 4 unit cells) of Fe was used and the cutoff energy of plane-wave basis was set as 350 eV. For accurately describing the interaction between two noble atoms and the defect formation energy, single-pointenergy calculations were carried out with a higher energy cutoff of 500 eV. During relaxations, the Brillouin zone integration was achieved using a Methfessel-Pazton smearing width of 0.1 eV. The Brillouin zones were sampled with 3 3 3 k points by Monkhorst-Pack scheme [38]. The atomic positions were fully relaxed at constant volume until the force on each atom is less than 0.005 eV/Å. The climbing image nudged elastic-band (CI-NEB) method [39,40] was used to determine the diffusion barriers of noble gas atom in bcc Fe solid. Five images between the initial and final configurations were considered and all images were relaxed until the force on each atom is less than 0.01 eV/Å. Formation energy of an interstitial and substitutional impurity X (X ¼ H, He, Ne, Ar) in Fe host solid can be defined by Refs. [19,41]: f
E ðXinterstitial Þ ¼ Eð1X; NFeÞ E½NFe EðXisolated Þ;
Ef ðXsubstitutional Þ ¼ Eð1X; NFeÞ ðN 1ÞE½NFe=N EðXisolated Þ; (2) respectively. Here E(1X, NFe) and E[NFe] are the energies of supercell with and without a X atom, respectively; E(Xisolated) is the energy of an isolated noble X atom in vacuum, while E(Hisolated) is half of the energy of a H2 molecules in vacuum (3.40 eV from our calculations). By definition, a positive energy denotes endothermic process, while a negative energy denotes exothermic. Binding energy between two defects, i.e., A and B, can be defined by:
Eb ðA; BÞ ¼ Ef ðA þ BÞ Ef ðfar A þ BÞ:
Table 1 The lattice constants (a), bulk modulus (B), magnetic moment mB, and formation energies (eV) of intrinsic defects (mono-/di-vacancy and self-interstitial) and H defects in bcc Fe in comparison with the available experimental and theoretical results [19,42e48,50,52,53]. Type
This work
Theor.
Exp.
a (Å)
2.83 177 2.16 2.16 4.12 4.10 4.96 3.85 4.53 0.30 0.46 0.16 0.62
2.83a, 2.88b 174a, 180b, 160d 2.20a, 2.32d 2.14e, 2.12f,2.17g 4.13f, 4.08h 4.04f, 4.01h 4.97f, 4.64b 3.93f, 3.64b 4.58f, 4.34b 0.27i 0.45i 0.13g 0.58g
2.87c 168c 2.22c 2.0 ± 0.2c
B (GPa)
mB
Ef (Vac) Ef1nn (Vac2) Ef2nn (Vac2) Ef<100>(SIA) Ef<110>(SIA) Ef<111>(SIA) Ef(HT-site) Ef(HO-site) Ed(HO-T) Eb(H-vacancy) a b c d e
g h
and
(3)
Here Ef(AþB) is the formation of energy for coexistence of A and B defects, Ef(far AþB) is the formation of energy for A and B defects far away from each other. Obviously, a negative binding energy means attractive interaction between two defects, while a positive one means repulsive interaction. Using the present computational scheme, we first determine the bulk properties of bcc Fe and the intrinsic defect formation energies of monovacancy, divacancy and self-interstitials (SIA) in Fe host lattice in comparison with the available experimental data and theoretical results, as summarized in Table 1. The calculated lattice constants (a), bulk modulus (B) and magnetic moment mB of bulk Fe solid are 2.83 Å, 177 GPa and 2.16 mB, respectively, in good agreement with the experimental values (2.87 Å, 168 GPa and 2.22 mB) [42] and previous DFT results (2.83 Å, 174 GPa and 2.20 mB [43]; 2.88 Å 180 GPa and 2.31 mB [44]), respectively. The calculated monovacancy formation energy of 2.16 eV coincides with the experimental data of 2.0 ± 0.2 eV [42,45] and previous DFT values of 2.14 eV [46] and 2.12 eV [47]. The divacancy formation energy (Vac2) as first (1nn) and second (2nn) nearest neighbor are 4.12 eV and 4.10 eV, respectively, in line with previous theoretical results (4.13 eV and 4.04 eV [47], 4.08 eV and 4.01 eV [19,44]). For selfinterstitials, the <110> configuration is most stable with the formation energy of 3.85 eV relative to both <100> and <111> configurations, which accords with other DFT results of 3.93 eV [47]
f
(1)
135
i
DFT calculations (PAW-GGA-VASP) from Ref. [43]. DFT calculations (PBE-GGA-SIESTA) from Refs. [19,44]. The experimental values from Refs. [42,45]. DFT calculations (PW91-GGA-VASP) from Ref. [53]. DFT calculations (PW91-GGA-VASP) from Ref. [46]. DFT calculations (PBE-GGA-VASP) from Ref. [47]. DFT calculations (PAW-GGA-VASP) from Ref. [48]. DFT calculations (PBE-GGA-PLATO) from Refs. [50,52]. DFT calculations (PBE-GGA-GPAW) from Ref. [24].
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and 3.64 eV [19,44]. The difference in calculated H formation energies between tetrahedral (0.30 eV) and octahedral (0.46 eV) interstitial sites is 0.16 eV, also consistent with 0.18 eV from Sakuraya's DFT calculations [24] and 0.13 eV from Ohsawa's DFT calculations [48]. Overall speaking, for a given type of defect, the calculated energy differences between different configurations also coincide well with the previous DFT values. The small differences between ours and previous results may originate from the computational details. 3. Results and discussion 3.1. Formation energy of single/double noble gas atoms Noble gas atoms with closed-shell electronic configurations are typically insoluble in most metals. To find the most stable site for each kind of atom, we first consider three possible occupation sites in a perfect bcc Fe lattice: tetrahedral (T-site), octahedral (O-site), and substitutional (S-site). The formation energies and volume changes of He, Ne, and Ar at these three sites are calculated and summarized in Table 2. Energetically, the interstitial T-site for He, Ne and Ar atom is more stable than the O-site, corresponding to the formation energies of 4.67 eV, 9.03 eV, and 11.57 eV, respectively. For the three elements, the energy difference between O-site and Tsite is nearly identical, i.e., about 0.20 eV. The substitutional site for a He/Ne/Ar atom is the lowest-energy configuration with the formation energies of 4.38 eV, 5.76 eV, and 8.03 eV, respectively. Overall, the formation energy increases rapidly with increase of the atomic number, while the values for Ne and Ar are two and three times that for He. Upon optimization of the simulation supercell, the interstitial noble gas atom induces strong relaxation in the Fe lattices (see Table 2). The volume expansion of 0.61% for HeT-site increases to 1.60% for NeT-site and 1.97% for ArT-site, while the substitutional defect causes less lattice distortion by about 0.01e0.80%. We also analyze local lattice expansion (stress field) [24]. For He insertion, the local lattice expansion of T-site, O-site and S-site configurations is about 11%, 17% and 0%, respectively. The local expansion for Ne insertion increases to about 28%, 40% and 2% for the T-site, O-site and S-site configurations, respectively. For Ar insertion, the expansion increases to about 31%, 45%, and 4% for the three configurations, respectively. After comparison, we find that Ne and Ar insertions cause larger lattice expansion. As a result, larger lattice expansion should be reasonable for the higher formation energy of Ne/Ar in Fe with regard to He. The calculated migration barriers of He, Ne and Ar are 0.06 eV, 0.10 eV and 0.16 eV, respectively, which is identical to the previous DFT data by Fu and
Willaime for He (Em¼0.06 eV) [19]. Besides, after insertion of noble gas atom, the bulk modulus (B) of Fe lattice increases with the atomic number of noble gas. For Fe crystal with a He/Ne/Ar atom at the T-site of 128-atom supercell, the bulk modulus (B) is predicted as 184 GPa, 188 GPa, and 197 GPa, respectively, which is obvious higher than that of perfect Fe crystal (177 GPa). This indicates that incorporation of noble gas atoms make iron solid harder (higher bulk modulus). However, it is noteworthy that such effect would be much weaker in the realistic situation, since the concentration 1:127 of noble gas atoms to iron in our supercell model is too high. To understand the clustering trend and stability of noble gas atoms related to possible bubble formation, we first determine the formation energies for various Ne-Ne and Ar-Ar configurations in bcc Fe lattice, and the representative results are displayed in Fig. 1. Overall, the interstitial noble gas atoms of Ne/Ar always prefer to stay together rather than being separated in Fe host lattice, similar to the behavior of He atoms [19,29]. Energetically, the <110> configurations are most stable for Ne-Ne and Ar-Ar pairs, and followed by the <111>, <100> and far (separated) configurations. This trend is the same as previously reported for He-He [29,49] and selfinterstitial configurations [44,50]. The formation energy of Ar-Ar pair (21.71e23.05 eV) is markedly higher than that of Ne-Ne pair (16.38e18.29 eV) by over 5.0 eV. Upon optimization, the Ne-Ne distance (1.92e1.94 Å) is always smaller than the Ar-Ar distance (2.53e2.66 Å) by about 0.6 Å, compared with the even shorter HeHe distance of 1.65 Å [29]. Meanwhile, there are large volume expansions of about 3.0% for Ne-Ne configurations and 4.0% for Ar-Ar configurations, compared to 1.2% for He-He configurations. Clearly, larger lattice expansion caused by noble gas atom corresponds to higher defect formation energy. To explain the physical origin of the higher defect formation energies and relative stability to He, we calculate the local densities of states (LDOS) on selected Ne/Ar and Fe atoms for the Ne-Ne/ArAr configurations. Fig. 2 displays the d-projected DOS of Fe atom and the p-projected DOS of noble gas atom. Apparently, the overall similarity in the shapes of Fe-d and Ne-p/Ar-p states indicates the strong hybridization between them. As for a closed-shell atom in metal, it is well known that stronger hybridization and larger distortion on the LDOS of the surrounding lattice atoms corresponds to the higher formation energy [41]. The Ar-Fe hybridization is much stronger than Ne-Fe, which means higher formation energy of Ar-Ar pair and stronger repulsion between Fe and Ar. For Ne-Ne configurations, Ne as <110> pair causes relatively less distortion on the LDOS of the nearby Fe atoms than the <111> and <100> pairs (see Fig. 2a). At the same time, the height of Ne p states as <110> configuration is substantially lower than the latter two
Table 2 The defect volume changes and formation energies of He, Ne, and Ar at the T-site, Osite and S-site of bcc Fe in comparison with pervious DFT results [19,20,24,44,46]. The last two columns list migration energy of noble gas atom and bulk modulus of Fe within a tetrahedral noble gas atom.
DV(%) HeT-site HeO-site HeS-site NeT-site NeO-site NeS-site ArT-site ArO-site ArS-site a b c d
DFT DFT DFT DFT
0.61 0.64 0.01 1.60 1.63 0.41 1.97 2.22 0.80
calculations calculations calculations calculations
Ef (eV) 4.67 4.87 4.38 9.03 9.22 5.76 11.57 11.76 8.03
Ref. a
b
c
d
4.56 , 4.39 , 4.91 , 4.60 4.75a, 4.57b, 5.11c, 4.37d 4.34a, 4.22b, 4.08d e e e e e e
(PAW-GGA) from Ref. [46]. (PBE-GGA) from Refs. [19,44]. (PBE-GGA from Ref. [24]. (PAW-GGA) from Ref. [20].
Em (eV)
B (GPa)
0.06 (0.06b) e e 0.10 e e 0.16 e e
184 e e 188 e e 197 e e
Fig. 1. Formation energies of Ne-Ne and Ar-Ar pairs for the <100>, <110>, <111> and far configurations in bcc Fe lattice, respectively.
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Fig. 2. Local DOS for (a,c) Fe atom, (b) Ne and (d) Ar atom with three Ne-Ne/Ar-Ar pairs. The black lines denote the DOS of pure Fe. The blue, pink and green lines denote the DOS of Fe with Ne-Ne/Ar-Ar as the <100>, <110> and <111> configurations, respectively. The Fermi energy of the supercell with the Ne/Ar defect pair is 0.00 eV. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
configurations at the Fermi level. Similar results are also observed in the three Ar-Ar configurations. Ar as the <111> and <100> pairs cause stronger distortion in DOS than <100> pair (see Fig. 2c and d). Therefore, stronger hybridization between noble gas atom and lattice Fe atoms and larger DOS distortion should be responsible for higher formation energy and larger volume expansion. 3.2. Binding energy between a noble gas atom and a vacancy Vacancies as trapping sites play a significant role in bubble formation. Here we consider a noble gas atom trapped by a vacancy (Vac) and obtain the binding energies of 2.45 eV, 5.41 eV, 5.83 eV for He-Vac, Ne-Vac, and Ar-Vac configurations, respectively. In the case of He, the dissolution energy via interstitial diffusion is the summation of He-Vac binding energy and He migration energy: 2.45 þ 0.06 ¼ 2.51 eV, in reasonable agreement with pervious DFT calculation (2.36 eV) [19]. Apparently, the binding energy considerably increases from He-Vac to Ne-Vac and Ar-Vac, indicating much stronger attraction of Ne or Ar atom to a vacancy defect. The calculated migration barriers for Ne and Ar atom are 0.10 eV and 0.16 eV, respectively; thus Ne and Ar have much higher dissolution energies from vacancies (5.51 eV and 5.99 eV) than He. Since the atomic radius and covalent radius of Ne and Ar are larger than He, the trend of binding energies of these noble gas atoms in Fe host lattice might be understood by their atomic radius and covalent radius. Fig. 3 shows the local densities of states (LDOS) on selected He/Ne/Ar and Fe atoms surrounding vacancy. Ne or Ar atom at vacancy causes larger DOS distortion of Fe than He, and the hybridization between the Fe-d and Ne-p/Ar-p states is stronger than Fe-He hybridization. The Ne/Ar atom occupying vacancy center reduces more in DOS distortion of Fe and hybridization between Fe and Ne/Ar (repulsion energy with Fe) with respect to He because of the stronger repulsion of Ne-Fe/Ar-Fe, thus resulting in stronger attraction of vacancy to Ne/Ar. 3.3. Binding energy between two noble gas atoms To understand the clustering behavior of noble gas atoms inside
Fig. 3. Local DOS for Fe atom, He, Ne and Ar atom at vacancy. The black lines denote the DOS of pure Fe. The blue, pink and green lines denote the DOS of Fe nearby the noble atom and the DOS of He/Ne/Ar, respectively. The Fermi energy of the supercell with the He/Ne/Ar defect at vacancy is 0.00 eV. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fe host lattice, we calculate the binding energy between two noble gas atoms as a function of the interatomic distance, as shown in Fig. 4. General speaking, the binding energies of Ne-Ne (1.91 eV) and Ar-Ar (1.40 eV) pairs are stronger than that of He-He (0.37 eV from our calculation, 0.43 eV [19] and 0.37 eV [29] from previous studies), which correspond to larger equilibrium distances of 1.92 Å (Ne-Ne) and 2.58 Å (Ar-Ar) than 1.57 Å for He-
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pair is higher than that of Ne-Ne pair. By comparison, we observe that the binding nature of Ne-Ne and Ar-Ar pairs is similar to that of He-He in Fe host lattice [29], while their interaction strengths are different. Combining with the volume changes and DOS analysis above (Section 3.1), the stability and clustering behavior can be nicely explained by the attractive interaction originated from the reduced valence-electron density by interstitial noble gas atoms and the repulsive interaction between the gas atom and Fe atom. 3.4. Binding energy between noble gas atoms and H/He
Fig. 4. Binding energy of He-He/Ne-Ne/Ar-Ar pairs as a function of the interatomic distance.
He. Within the distances of 1.90e4.00 Å, all binding energies of NeNe and Ar-Ar pairs are negative, meaning that the interaction between two noble gas atoms is attractive and the Ne-Ne attraction is stronger. Beyond 4.00 Å, the binding energy gradually converges to zero with increasing the interatomic distance. Such strong attraction suggests that noble gas atoms can aggregate in Fe host lattice by self-trapping under irradiations [30], while the effects of Ne and Ar atoms are more pronounced. We turn to discuss the nature of attractive interaction between two Ne/Ar atoms. The charge density difference of Ne-Ne/Ar-Ar pair in Fe host lattice is shown in Fig. 5. The presence of interstitial NeNe/Ar-Ar pair in Fe lattice induces strong perturbation of local charge density distributions, forming the low electron density area surrounding the noble gas atoms. Lower electron density corresponds to lower defect formation energy [51]; thus this formed low density region can act as a preferable site for noble gas atoms. Since Ne and Ar atoms have a closed-shell electronic configuration, stronger polarization of charge densities would cost more extra energy. This can explain well why the formation energy of Ar-Ar
To gain an insight into the effect of noble gas impurities on H/He retention properties in Fe, we determine the formation energies and stable configurations of single H/He atom near and far away from the interstitial Ne or Ar atom, as shown in Fig. 6. Energetically, H and He impurities prefer to occupy interstitial sites in the region of interstitial Ne/Ar atom. The <111> configurations are most stable for both H-Ne and He-Ne pairs with the formation energies of 0.10 eV and 3.76 eV, respectively, which are obviously lower than the far T-sites (0.32 eV for H and 4.77 eV for He). Similar results are observed for H and He nearby Ar atom, corresponding to the lower formation energies of 0.13 eV and 3.80 eV than the far T-sites, respectively. By comparison, the equilibrium distance of H-Ar/HeAr is generally larger than H-Ne/He-Ne by about 0.2e0.8 Å. To obtain their attraction strength, we calculate the binding energy of H-He/H-Ne/H-Ar/He-Ne/He-Ar pairs as a function of the interatomic distance. As shown in Fig. 7, the binding energies of He-Ne (1.01 eV) and He-Ar (0.85 eV) pairs are larger than those of H-He (0.03 eV), H-Ne (0.22 eV) and H-Ar (0.10 eV) in bulk lattice of Fe, indicating that the former pairs have much stronger attractive interaction than the latter ones. Within the distances of 1.7e3.0 Å, we observe a strong attraction region for He-Ne and HeAr pairs, corresponding to the binding energy of 1.01 ~ 0.25 eV. Such strong attraction indicates that He and Ne/Ar atoms can trap with each other in absence of other defects. By contrast, a weak attraction of H-He/H-Ne/H-Ar pairs is limited in a relatively
Fig. 5. Charge density difference of Ne-Ne pairs and Ar-Ar pairs with the <100>, <110> and <111> configurations in Fe host lattice, respectively. Blue contour denotes region of charge depletion, and yellow contour denotes region of charge accumulation (isovalue is 0.005 e/Å3). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 6. Formation energies and atomic configurations of H and He near and far away from an interstitial Ne or Ar atom, respectively.
Ne/Ar as well HeþH/NeþH irradiations [10,12]. 4. Conclusions First-principle calculations have been carried out to investigate the energetic and aggregation tendency of noble gas atoms (He, Ne, Ar), and their interactions with vacancy, H and He defects inside bcc Fe lattice. Interstitial noble gas atoms tend to stay together with each other and can attract He/H impurities. The electronic structure factor and atomic sizes play major roles in the defect formation energy and the clustering behavior of noble gas atoms. Our main conclusions are summarized as follows: Fig. 7. Binding energy of H-He/H-Ne/H-Ar and He-Ne/He-Ar pairs as a function of the interatomic distance.
narrow region of 2.02e2.26 Å. Again, the binding energy gradually converges to zero as the distance exceeds 4.0 Å. As representatives, we further analyze the charge density difference of He-Ne/ H-Ne pairs in a-Fe solid. In Fig. 8, lower electron-density regions can be seen around the Ne (He) atom for the six configurations, indicating that Ne atom can act as a trapping site for H and He impurities. It can be seen that there is charge accumulation at the H surrounding region and the <110> and <111> configurations are more visible than the <100> one. This well explains the lower formation energies of H/He occupying in the nearby region of Ne. These attraction strengths provide a preliminary explanation for the difference of irradiation effects on Fe lattice between He and
(1) From the energetic point of view, the most stable site for a He/Ne/Ar atom in perfect bcc Fe lattice is the substitutional site, and the tetrahedral site is more stable than the octahedral one by 0.2 eV. The formation energies of interstitial Ne and Ar atoms are two and three times that for He. With the 128-atom supercell, the volume expansion of 0.61% for HeTsite increases to 1.60% for NeT-site and 1.97% for ArT-site. Thus, higher formation energies of Ne and Ar are originated from larger lattice expansion relative to He. The insertion of noble gas atom hardens Fe crystal by increasing the bulk modulus. (2) Two interstitial noble gas atoms can form stable pair with the <110> configurations, which is more stable than the <111> and <100> configurations and the separated situation. The formation energy of noble gas pair considerably increase from He to Ar. The strength of hybridization between noble gas atom and Fe atoms and the atomic size should be
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Fig. 8. Charge density differences of He-Ne and H-Ne pairs with <100>, <110> and <111> configurations inside Fe lattice, respectively. Blue contour denotes regions of charge depletion, and yellow contour denotes regions of charge accumulation (isovalue is 0.004 e/Å3). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
responsible for the trend of formation energy and volume expansion. (3) According to binding energy calculations, the strengths of Ne-Ne (1.91 eV) and Ar-Ar (1.40 eV) attractions are significantly stronger than He-He (0.37 eV). Such strong attraction means that clustering of He/Ne/Ar atoms can occur by self-trapping, which can be explained by the lower electron density induced by interstitial noble gas atoms and its strong repulsion with the surrounding Fe atoms. This trend is similar with the behavior of noble gas atoms in metal W. (4) We have demonstrated the appreciable capability of noble gas atom Ne/Ar to bind H or He. A single H/He atom energetically prefers to locate at the interstitial sites near Ne/Ar rather than the other interstitial positions. The binding energies of He-Ne (1.01 eV) and He-Ar (0.85 eV) pairs are larger than those of H-He (0.03 eV), H-Ne (0.22 eV) and H-Ar (0.10) in bulk Fe lattice. The attractive distance remains in the order of 1.7e3.0 Å. Thus, interstitial noble gas atoms like Ne or Ar act as a trapping site for H/He impurities and can form stable clusters even in absence of the other defects like vacancies. The above results not only provide a preliminary explanation for the difference of irradiation effects on Fe crystal between He, Ne/Ar, and HeþH/NeþH/ArþH, but also improve our understanding about the behavior of noble gas atoms and gas bubble formation in iron under irradiation.
Acknowledgements This work was supported by the National Magnetic Confinement Fusion Energy Research Project of China (2015GB118001), the China Postdoctoral Science Foundation (2015M581325), and the Fundamental Research Funds for the Central Universities of China (3132017063).
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First-principles study of noble gas atoms in bcc Fe Pengbo Zhang a, b, *, Jianhua Ding a, Dan Sun a, Jijun Zhao a, ** a b
Key Laboratory of Materials Modification by Laser, Ion and Electron Beams (Dalian University of Technology), Ministry of Education, Dalian 116024, China Department of Physics, Dalian Maritime University, Dalian 116026, China
h i g h l i g h t s Interstitial noble gas atoms tend to stay together with each other by self-trapping. H/He prefers to locate interstitial sites nearby Ne atom than other interstitials. Noble gas atoms like Ne can act as a trapping site for H/He impurities in bcc Fe.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 January 2017 Received in revised form 17 May 2017 Accepted 17 May 2017 Available online 19 May 2017
We investigate the energetics and clustering trend of noble gas atoms (He, Ne, and Ar) in bcc Fe, and their interactions with vacancy or H/He impurities using first-principles calculations. We determine the formation energy of single and double noble gas atoms inside Fe host lattice as well as the resulted volume changes. The Ne/Ar formation energy is two and three times that of He. The attraction between Ne/Ar and vacancy is stronger than He-vacancy, indicating higher dissolution energy of Ne/Ar. The interstitial Ne-Ne/Ar-Ar pairs have stronger attractions (1.91 eV/1.40 eV) than He-He (0.37 eV), forming stable <110> configurations. Such strong attraction means that He/Ne/Ar tend to aggregate, which can be well explained by the lower electron density induced by interstitial noble gas atoms and its strong repulsion with Fe atoms. Moreover, H/He energetically prefers to occupy the tetrahedral sites nearby Ne/Ar atom. The attraction energies of He-Ne/He-Ar pairs (1.01 eV/-0.85 eV) are much stronger than those of H-Ne/ H-Ar (0.22 eV/0.10 eV) and their charge density differences are discussed. The distinct attraction strengths by various noble gas atoms provide a preliminary explanation for the difference in irradiation effects on Fe solid by He, Ne, Ar, and HeþH/NeþHe. These findings improve our understanding about the behavior of noble gas atoms and gas bubble formation in iron under irradiation. © 2017 Elsevier B.V. All rights reserved.
Keywords: Noble gas Fe Interaction Irradiation First principles
1. Introduction High-strength Fe-based alloys of bcc phase have been widely used in nuclear reactors [1e3]. Large amounts of He and H impurities are produced in the neutron-irradiated Fe solid and alloys via (n, a) transmutation reactions, which play an important role in microstructure evolution and degradation of mechanical properties, known as the irradiation damage effect [4e7]. Thus, understanding the effect of He atoms in metals and alloys is crucial for
* Corresponding author. Key Laboratory of Materials Modification by Laser, Ion and Electron Beams (Dalian University of Technology), Ministry of Education, Dalian 116024, China. ** Corresponding author. E-mail addresses: [email protected] (P. Zhang), [email protected] (J. Zhao). http://dx.doi.org/10.1016/j.jnucmat.2017.05.022 0022-3115/© 2017 Elsevier B.V. All rights reserved.
development of materials resistant to irradiation under fusion and fission environments. Experimentally, multi-beam irradiations by He/Ne/Ar ions have been intensively carried out to investigate He embrittlement and irradiation damage in metals and alloys [8e15]. Previous TEM experiments [8,11,13] observed formation of noble gas bubbles of He, Ne, or Ar in pure Ni, W metals and Fe alloys. He atoms can be deeply trapped in vacancies and grain boundaries, leading to bubble formation [16,17]. The volume change associated with void swelling is quite large (over 10%) [18]. Until now, experimental data on defect energies of the noble gas atoms in bcc Fe solid and its alloys, especially the formation/binding energy of noble gas atoms and their binding energies with vacancy/H/He, are generally scarce. Density functional theory (DFT) calculations have been extensively performed to investigate the defect formation energy, diffusion and clustering of He atoms in bcc Fe [19e28]. Fu and
P. Zhang et al. / Journal of Nuclear Materials 492 (2017) 134e141
Willaime examined the stability of He and He-vacancy clusters, their results showed that He migration barrier is very small (about 0.06 eV) and the strength of He-He attraction in a-Fe lattice is 0.43 eV [19]. Seletskaia reported that magnetic interactions influence the behavior of He in bcc Fe [20] and revealed that strong hybridization between He and the surrounding Fe atoms results in high defect formation energy. Sakuraya et al. investigated the effects of H, He, C, and N impurities in a-Fe [24] and found the magnetic moment of Fe slightly increases after He insertion. Our recent studies demonstrated that interstitial He greatly reduces the vacancy formation energy in bcc Fe [22] and interstitial He-He has strong attraction in bcc Fe, Cr, Mo and W metals, suggesting that He clustering may occur by self-trapping [29]. Kong revealed He clustering in W host lattice by self-trapping [30]. However, the effects of other noble gas atoms and their impact on H/He behaviors in bcc Fe have received much less attention in terms of atomic simulations. Owing to the accumulation of He and other noble gas atoms in pure Fe, steels, and other metals, accurate determination of defect formation energies and their interaction strengths are important for developing predictive models for the quantification of irradiation damage. In this paper, we investigate the energetics and clustering trend of noble gas atoms (He, Ne, and Ar), as well as their interactions with vacancy and H/He impurities in bcc Fe solid using first-principles calculations. Firstly, we determine the occupying site, volume changes and formation energy of single and dual noble gas atoms in Section 3.1. We then calculate binding energy between a noble gas atom and a vacancy in Section 3.2. We further calculate binding energy between two noble gas atoms (Ne/Ar and H/He) and discuss their electronic structures in Section 3.3 and 3.4. We find that interstitial noble gas atoms tend to stay together with each other and are able to attract He/H impurities. The present results provide vital insights into the behavior of noble gas atoms and gas bubble formation in iron under irradiation. 2. Computational methods and models All calculations were performed using spin-polarized density functional theory and the plane-wave pseudopotential approach [31,32], as implemented in the Vienna Ab initio Simulation Package (VASP 5.3.5) [33,34]. We adopted the generalized gradient approximation (GGA) with the Perdew and Wang (PW91) functional [35] for the exchange-correlation interaction and the projector-augmented wave (PAW) potentials [36,37] for the ionelectron interaction. A 128-atom bcc supercell (4 4 4 unit cells) of Fe was used and the cutoff energy of plane-wave basis was set as 350 eV. For accurately describing the interaction between two noble atoms and the defect formation energy, single-pointenergy calculations were carried out with a higher energy cutoff of 500 eV. During relaxations, the Brillouin zone integration was achieved using a Methfessel-Pazton smearing width of 0.1 eV. The Brillouin zones were sampled with 3 3 3 k points by Monkhorst-Pack scheme [38]. The atomic positions were fully relaxed at constant volume until the force on each atom is less than 0.005 eV/Å. The climbing image nudged elastic-band (CI-NEB) method [39,40] was used to determine the diffusion barriers of noble gas atom in bcc Fe solid. Five images between the initial and final configurations were considered and all images were relaxed until the force on each atom is less than 0.01 eV/Å. Formation energy of an interstitial and substitutional impurity X (X ¼ H, He, Ne, Ar) in Fe host solid can be defined by Refs. [19,41]: f
E ðXinterstitial Þ ¼ Eð1X; NFeÞ E½NFe EðXisolated Þ;
Ef ðXsubstitutional Þ ¼ Eð1X; NFeÞ ðN 1ÞE½NFe=N EðXisolated Þ; (2) respectively. Here E(1X, NFe) and E[NFe] are the energies of supercell with and without a X atom, respectively; E(Xisolated) is the energy of an isolated noble X atom in vacuum, while E(Hisolated) is half of the energy of a H2 molecules in vacuum (3.40 eV from our calculations). By definition, a positive energy denotes endothermic process, while a negative energy denotes exothermic. Binding energy between two defects, i.e., A and B, can be defined by:
Eb ðA; BÞ ¼ Ef ðA þ BÞ Ef ðfar A þ BÞ:
Table 1 The lattice constants (a), bulk modulus (B), magnetic moment mB, and formation energies (eV) of intrinsic defects (mono-/di-vacancy and self-interstitial) and H defects in bcc Fe in comparison with the available experimental and theoretical results [19,42e48,50,52,53]. Type
This work
Theor.
Exp.
a (Å)
2.83 177 2.16 2.16 4.12 4.10 4.96 3.85 4.53 0.30 0.46 0.16 0.62
2.83a, 2.88b 174a, 180b, 160d 2.20a, 2.32d 2.14e, 2.12f,2.17g 4.13f, 4.08h 4.04f, 4.01h 4.97f, 4.64b 3.93f, 3.64b 4.58f, 4.34b 0.27i 0.45i 0.13g 0.58g
2.87c 168c 2.22c 2.0 ± 0.2c
B (GPa)
mB
Ef (Vac) Ef1nn (Vac2) Ef2nn (Vac2) Ef<100>(SIA) Ef<110>(SIA) Ef<111>(SIA) Ef(HT-site) Ef(HO-site) Ed(HO-T) Eb(H-vacancy) a b c d e
g h
and
(3)
Here Ef(AþB) is the formation of energy for coexistence of A and B defects, Ef(far AþB) is the formation of energy for A and B defects far away from each other. Obviously, a negative binding energy means attractive interaction between two defects, while a positive one means repulsive interaction. Using the present computational scheme, we first determine the bulk properties of bcc Fe and the intrinsic defect formation energies of monovacancy, divacancy and self-interstitials (SIA) in Fe host lattice in comparison with the available experimental data and theoretical results, as summarized in Table 1. The calculated lattice constants (a), bulk modulus (B) and magnetic moment mB of bulk Fe solid are 2.83 Å, 177 GPa and 2.16 mB, respectively, in good agreement with the experimental values (2.87 Å, 168 GPa and 2.22 mB) [42] and previous DFT results (2.83 Å, 174 GPa and 2.20 mB [43]; 2.88 Å 180 GPa and 2.31 mB [44]), respectively. The calculated monovacancy formation energy of 2.16 eV coincides with the experimental data of 2.0 ± 0.2 eV [42,45] and previous DFT values of 2.14 eV [46] and 2.12 eV [47]. The divacancy formation energy (Vac2) as first (1nn) and second (2nn) nearest neighbor are 4.12 eV and 4.10 eV, respectively, in line with previous theoretical results (4.13 eV and 4.04 eV [47], 4.08 eV and 4.01 eV [19,44]). For selfinterstitials, the <110> configuration is most stable with the formation energy of 3.85 eV relative to both <100> and <111> configurations, which accords with other DFT results of 3.93 eV [47]
f
(1)
135
i
DFT calculations (PAW-GGA-VASP) from Ref. [43]. DFT calculations (PBE-GGA-SIESTA) from Refs. [19,44]. The experimental values from Refs. [42,45]. DFT calculations (PW91-GGA-VASP) from Ref. [53]. DFT calculations (PW91-GGA-VASP) from Ref. [46]. DFT calculations (PBE-GGA-VASP) from Ref. [47]. DFT calculations (PAW-GGA-VASP) from Ref. [48]. DFT calculations (PBE-GGA-PLATO) from Refs. [50,52]. DFT calculations (PBE-GGA-GPAW) from Ref. [24].
136
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and 3.64 eV [19,44]. The difference in calculated H formation energies between tetrahedral (0.30 eV) and octahedral (0.46 eV) interstitial sites is 0.16 eV, also consistent with 0.18 eV from Sakuraya's DFT calculations [24] and 0.13 eV from Ohsawa's DFT calculations [48]. Overall speaking, for a given type of defect, the calculated energy differences between different configurations also coincide well with the previous DFT values. The small differences between ours and previous results may originate from the computational details. 3. Results and discussion 3.1. Formation energy of single/double noble gas atoms Noble gas atoms with closed-shell electronic configurations are typically insoluble in most metals. To find the most stable site for each kind of atom, we first consider three possible occupation sites in a perfect bcc Fe lattice: tetrahedral (T-site), octahedral (O-site), and substitutional (S-site). The formation energies and volume changes of He, Ne, and Ar at these three sites are calculated and summarized in Table 2. Energetically, the interstitial T-site for He, Ne and Ar atom is more stable than the O-site, corresponding to the formation energies of 4.67 eV, 9.03 eV, and 11.57 eV, respectively. For the three elements, the energy difference between O-site and Tsite is nearly identical, i.e., about 0.20 eV. The substitutional site for a He/Ne/Ar atom is the lowest-energy configuration with the formation energies of 4.38 eV, 5.76 eV, and 8.03 eV, respectively. Overall, the formation energy increases rapidly with increase of the atomic number, while the values for Ne and Ar are two and three times that for He. Upon optimization of the simulation supercell, the interstitial noble gas atom induces strong relaxation in the Fe lattices (see Table 2). The volume expansion of 0.61% for HeT-site increases to 1.60% for NeT-site and 1.97% for ArT-site, while the substitutional defect causes less lattice distortion by about 0.01e0.80%. We also analyze local lattice expansion (stress field) [24]. For He insertion, the local lattice expansion of T-site, O-site and S-site configurations is about 11%, 17% and 0%, respectively. The local expansion for Ne insertion increases to about 28%, 40% and 2% for the T-site, O-site and S-site configurations, respectively. For Ar insertion, the expansion increases to about 31%, 45%, and 4% for the three configurations, respectively. After comparison, we find that Ne and Ar insertions cause larger lattice expansion. As a result, larger lattice expansion should be reasonable for the higher formation energy of Ne/Ar in Fe with regard to He. The calculated migration barriers of He, Ne and Ar are 0.06 eV, 0.10 eV and 0.16 eV, respectively, which is identical to the previous DFT data by Fu and
Willaime for He (Em¼0.06 eV) [19]. Besides, after insertion of noble gas atom, the bulk modulus (B) of Fe lattice increases with the atomic number of noble gas. For Fe crystal with a He/Ne/Ar atom at the T-site of 128-atom supercell, the bulk modulus (B) is predicted as 184 GPa, 188 GPa, and 197 GPa, respectively, which is obvious higher than that of perfect Fe crystal (177 GPa). This indicates that incorporation of noble gas atoms make iron solid harder (higher bulk modulus). However, it is noteworthy that such effect would be much weaker in the realistic situation, since the concentration 1:127 of noble gas atoms to iron in our supercell model is too high. To understand the clustering trend and stability of noble gas atoms related to possible bubble formation, we first determine the formation energies for various Ne-Ne and Ar-Ar configurations in bcc Fe lattice, and the representative results are displayed in Fig. 1. Overall, the interstitial noble gas atoms of Ne/Ar always prefer to stay together rather than being separated in Fe host lattice, similar to the behavior of He atoms [19,29]. Energetically, the <110> configurations are most stable for Ne-Ne and Ar-Ar pairs, and followed by the <111>, <100> and far (separated) configurations. This trend is the same as previously reported for He-He [29,49] and selfinterstitial configurations [44,50]. The formation energy of Ar-Ar pair (21.71e23.05 eV) is markedly higher than that of Ne-Ne pair (16.38e18.29 eV) by over 5.0 eV. Upon optimization, the Ne-Ne distance (1.92e1.94 Å) is always smaller than the Ar-Ar distance (2.53e2.66 Å) by about 0.6 Å, compared with the even shorter HeHe distance of 1.65 Å [29]. Meanwhile, there are large volume expansions of about 3.0% for Ne-Ne configurations and 4.0% for Ar-Ar configurations, compared to 1.2% for He-He configurations. Clearly, larger lattice expansion caused by noble gas atom corresponds to higher defect formation energy. To explain the physical origin of the higher defect formation energies and relative stability to He, we calculate the local densities of states (LDOS) on selected Ne/Ar and Fe atoms for the Ne-Ne/ArAr configurations. Fig. 2 displays the d-projected DOS of Fe atom and the p-projected DOS of noble gas atom. Apparently, the overall similarity in the shapes of Fe-d and Ne-p/Ar-p states indicates the strong hybridization between them. As for a closed-shell atom in metal, it is well known that stronger hybridization and larger distortion on the LDOS of the surrounding lattice atoms corresponds to the higher formation energy [41]. The Ar-Fe hybridization is much stronger than Ne-Fe, which means higher formation energy of Ar-Ar pair and stronger repulsion between Fe and Ar. For Ne-Ne configurations, Ne as <110> pair causes relatively less distortion on the LDOS of the nearby Fe atoms than the <111> and <100> pairs (see Fig. 2a). At the same time, the height of Ne p states as <110> configuration is substantially lower than the latter two
Table 2 The defect volume changes and formation energies of He, Ne, and Ar at the T-site, Osite and S-site of bcc Fe in comparison with pervious DFT results [19,20,24,44,46]. The last two columns list migration energy of noble gas atom and bulk modulus of Fe within a tetrahedral noble gas atom.
DV(%) HeT-site HeO-site HeS-site NeT-site NeO-site NeS-site ArT-site ArO-site ArS-site a b c d
DFT DFT DFT DFT
0.61 0.64 0.01 1.60 1.63 0.41 1.97 2.22 0.80
calculations calculations calculations calculations
Ef (eV) 4.67 4.87 4.38 9.03 9.22 5.76 11.57 11.76 8.03
Ref. a
b
c
d
4.56 , 4.39 , 4.91 , 4.60 4.75a, 4.57b, 5.11c, 4.37d 4.34a, 4.22b, 4.08d e e e e e e
(PAW-GGA) from Ref. [46]. (PBE-GGA) from Refs. [19,44]. (PBE-GGA from Ref. [24]. (PAW-GGA) from Ref. [20].
Em (eV)
B (GPa)
0.06 (0.06b) e e 0.10 e e 0.16 e e
184 e e 188 e e 197 e e
Fig. 1. Formation energies of Ne-Ne and Ar-Ar pairs for the <100>, <110>, <111> and far configurations in bcc Fe lattice, respectively.
P. Zhang et al. / Journal of Nuclear Materials 492 (2017) 134e141
137
Fig. 2. Local DOS for (a,c) Fe atom, (b) Ne and (d) Ar atom with three Ne-Ne/Ar-Ar pairs. The black lines denote the DOS of pure Fe. The blue, pink and green lines denote the DOS of Fe with Ne-Ne/Ar-Ar as the <100>, <110> and <111> configurations, respectively. The Fermi energy of the supercell with the Ne/Ar defect pair is 0.00 eV. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
configurations at the Fermi level. Similar results are also observed in the three Ar-Ar configurations. Ar as the <111> and <100> pairs cause stronger distortion in DOS than <100> pair (see Fig. 2c and d). Therefore, stronger hybridization between noble gas atom and lattice Fe atoms and larger DOS distortion should be responsible for higher formation energy and larger volume expansion. 3.2. Binding energy between a noble gas atom and a vacancy Vacancies as trapping sites play a significant role in bubble formation. Here we consider a noble gas atom trapped by a vacancy (Vac) and obtain the binding energies of 2.45 eV, 5.41 eV, 5.83 eV for He-Vac, Ne-Vac, and Ar-Vac configurations, respectively. In the case of He, the dissolution energy via interstitial diffusion is the summation of He-Vac binding energy and He migration energy: 2.45 þ 0.06 ¼ 2.51 eV, in reasonable agreement with pervious DFT calculation (2.36 eV) [19]. Apparently, the binding energy considerably increases from He-Vac to Ne-Vac and Ar-Vac, indicating much stronger attraction of Ne or Ar atom to a vacancy defect. The calculated migration barriers for Ne and Ar atom are 0.10 eV and 0.16 eV, respectively; thus Ne and Ar have much higher dissolution energies from vacancies (5.51 eV and 5.99 eV) than He. Since the atomic radius and covalent radius of Ne and Ar are larger than He, the trend of binding energies of these noble gas atoms in Fe host lattice might be understood by their atomic radius and covalent radius. Fig. 3 shows the local densities of states (LDOS) on selected He/Ne/Ar and Fe atoms surrounding vacancy. Ne or Ar atom at vacancy causes larger DOS distortion of Fe than He, and the hybridization between the Fe-d and Ne-p/Ar-p states is stronger than Fe-He hybridization. The Ne/Ar atom occupying vacancy center reduces more in DOS distortion of Fe and hybridization between Fe and Ne/Ar (repulsion energy with Fe) with respect to He because of the stronger repulsion of Ne-Fe/Ar-Fe, thus resulting in stronger attraction of vacancy to Ne/Ar. 3.3. Binding energy between two noble gas atoms To understand the clustering behavior of noble gas atoms inside
Fig. 3. Local DOS for Fe atom, He, Ne and Ar atom at vacancy. The black lines denote the DOS of pure Fe. The blue, pink and green lines denote the DOS of Fe nearby the noble atom and the DOS of He/Ne/Ar, respectively. The Fermi energy of the supercell with the He/Ne/Ar defect at vacancy is 0.00 eV. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fe host lattice, we calculate the binding energy between two noble gas atoms as a function of the interatomic distance, as shown in Fig. 4. General speaking, the binding energies of Ne-Ne (1.91 eV) and Ar-Ar (1.40 eV) pairs are stronger than that of He-He (0.37 eV from our calculation, 0.43 eV [19] and 0.37 eV [29] from previous studies), which correspond to larger equilibrium distances of 1.92 Å (Ne-Ne) and 2.58 Å (Ar-Ar) than 1.57 Å for He-
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pair is higher than that of Ne-Ne pair. By comparison, we observe that the binding nature of Ne-Ne and Ar-Ar pairs is similar to that of He-He in Fe host lattice [29], while their interaction strengths are different. Combining with the volume changes and DOS analysis above (Section 3.1), the stability and clustering behavior can be nicely explained by the attractive interaction originated from the reduced valence-electron density by interstitial noble gas atoms and the repulsive interaction between the gas atom and Fe atom. 3.4. Binding energy between noble gas atoms and H/He
Fig. 4. Binding energy of He-He/Ne-Ne/Ar-Ar pairs as a function of the interatomic distance.
He. Within the distances of 1.90e4.00 Å, all binding energies of NeNe and Ar-Ar pairs are negative, meaning that the interaction between two noble gas atoms is attractive and the Ne-Ne attraction is stronger. Beyond 4.00 Å, the binding energy gradually converges to zero with increasing the interatomic distance. Such strong attraction suggests that noble gas atoms can aggregate in Fe host lattice by self-trapping under irradiations [30], while the effects of Ne and Ar atoms are more pronounced. We turn to discuss the nature of attractive interaction between two Ne/Ar atoms. The charge density difference of Ne-Ne/Ar-Ar pair in Fe host lattice is shown in Fig. 5. The presence of interstitial NeNe/Ar-Ar pair in Fe lattice induces strong perturbation of local charge density distributions, forming the low electron density area surrounding the noble gas atoms. Lower electron density corresponds to lower defect formation energy [51]; thus this formed low density region can act as a preferable site for noble gas atoms. Since Ne and Ar atoms have a closed-shell electronic configuration, stronger polarization of charge densities would cost more extra energy. This can explain well why the formation energy of Ar-Ar
To gain an insight into the effect of noble gas impurities on H/He retention properties in Fe, we determine the formation energies and stable configurations of single H/He atom near and far away from the interstitial Ne or Ar atom, as shown in Fig. 6. Energetically, H and He impurities prefer to occupy interstitial sites in the region of interstitial Ne/Ar atom. The <111> configurations are most stable for both H-Ne and He-Ne pairs with the formation energies of 0.10 eV and 3.76 eV, respectively, which are obviously lower than the far T-sites (0.32 eV for H and 4.77 eV for He). Similar results are observed for H and He nearby Ar atom, corresponding to the lower formation energies of 0.13 eV and 3.80 eV than the far T-sites, respectively. By comparison, the equilibrium distance of H-Ar/HeAr is generally larger than H-Ne/He-Ne by about 0.2e0.8 Å. To obtain their attraction strength, we calculate the binding energy of H-He/H-Ne/H-Ar/He-Ne/He-Ar pairs as a function of the interatomic distance. As shown in Fig. 7, the binding energies of He-Ne (1.01 eV) and He-Ar (0.85 eV) pairs are larger than those of H-He (0.03 eV), H-Ne (0.22 eV) and H-Ar (0.10 eV) in bulk lattice of Fe, indicating that the former pairs have much stronger attractive interaction than the latter ones. Within the distances of 1.7e3.0 Å, we observe a strong attraction region for He-Ne and HeAr pairs, corresponding to the binding energy of 1.01 ~ 0.25 eV. Such strong attraction indicates that He and Ne/Ar atoms can trap with each other in absence of other defects. By contrast, a weak attraction of H-He/H-Ne/H-Ar pairs is limited in a relatively
Fig. 5. Charge density difference of Ne-Ne pairs and Ar-Ar pairs with the <100>, <110> and <111> configurations in Fe host lattice, respectively. Blue contour denotes region of charge depletion, and yellow contour denotes region of charge accumulation (isovalue is 0.005 e/Å3). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 6. Formation energies and atomic configurations of H and He near and far away from an interstitial Ne or Ar atom, respectively.
Ne/Ar as well HeþH/NeþH irradiations [10,12]. 4. Conclusions First-principle calculations have been carried out to investigate the energetic and aggregation tendency of noble gas atoms (He, Ne, Ar), and their interactions with vacancy, H and He defects inside bcc Fe lattice. Interstitial noble gas atoms tend to stay together with each other and can attract He/H impurities. The electronic structure factor and atomic sizes play major roles in the defect formation energy and the clustering behavior of noble gas atoms. Our main conclusions are summarized as follows: Fig. 7. Binding energy of H-He/H-Ne/H-Ar and He-Ne/He-Ar pairs as a function of the interatomic distance.
narrow region of 2.02e2.26 Å. Again, the binding energy gradually converges to zero as the distance exceeds 4.0 Å. As representatives, we further analyze the charge density difference of He-Ne/ H-Ne pairs in a-Fe solid. In Fig. 8, lower electron-density regions can be seen around the Ne (He) atom for the six configurations, indicating that Ne atom can act as a trapping site for H and He impurities. It can be seen that there is charge accumulation at the H surrounding region and the <110> and <111> configurations are more visible than the <100> one. This well explains the lower formation energies of H/He occupying in the nearby region of Ne. These attraction strengths provide a preliminary explanation for the difference of irradiation effects on Fe lattice between He and
(1) From the energetic point of view, the most stable site for a He/Ne/Ar atom in perfect bcc Fe lattice is the substitutional site, and the tetrahedral site is more stable than the octahedral one by 0.2 eV. The formation energies of interstitial Ne and Ar atoms are two and three times that for He. With the 128-atom supercell, the volume expansion of 0.61% for HeTsite increases to 1.60% for NeT-site and 1.97% for ArT-site. Thus, higher formation energies of Ne and Ar are originated from larger lattice expansion relative to He. The insertion of noble gas atom hardens Fe crystal by increasing the bulk modulus. (2) Two interstitial noble gas atoms can form stable pair with the <110> configurations, which is more stable than the <111> and <100> configurations and the separated situation. The formation energy of noble gas pair considerably increase from He to Ar. The strength of hybridization between noble gas atom and Fe atoms and the atomic size should be
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Fig. 8. Charge density differences of He-Ne and H-Ne pairs with <100>, <110> and <111> configurations inside Fe lattice, respectively. Blue contour denotes regions of charge depletion, and yellow contour denotes regions of charge accumulation (isovalue is 0.004 e/Å3). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
responsible for the trend of formation energy and volume expansion. (3) According to binding energy calculations, the strengths of Ne-Ne (1.91 eV) and Ar-Ar (1.40 eV) attractions are significantly stronger than He-He (0.37 eV). Such strong attraction means that clustering of He/Ne/Ar atoms can occur by self-trapping, which can be explained by the lower electron density induced by interstitial noble gas atoms and its strong repulsion with the surrounding Fe atoms. This trend is similar with the behavior of noble gas atoms in metal W. (4) We have demonstrated the appreciable capability of noble gas atom Ne/Ar to bind H or He. A single H/He atom energetically prefers to locate at the interstitial sites near Ne/Ar rather than the other interstitial positions. The binding energies of He-Ne (1.01 eV) and He-Ar (0.85 eV) pairs are larger than those of H-He (0.03 eV), H-Ne (0.22 eV) and H-Ar (0.10) in bulk Fe lattice. The attractive distance remains in the order of 1.7e3.0 Å. Thus, interstitial noble gas atoms like Ne or Ar act as a trapping site for H/He impurities and can form stable clusters even in absence of the other defects like vacancies. The above results not only provide a preliminary explanation for the difference of irradiation effects on Fe crystal between He, Ne/Ar, and HeþH/NeþH/ArþH, but also improve our understanding about the behavior of noble gas atoms and gas bubble formation in iron under irradiation.
Acknowledgements This work was supported by the National Magnetic Confinement Fusion Energy Research Project of China (2015GB118001), the China Postdoctoral Science Foundation (2015M581325), and the Fundamental Research Funds for the Central Universities of China (3132017063).
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