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Trapping and diffusion of fission products in ThO2 and CeO2
Journal of Nuclear Materials 414 (2011) 464–470
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Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat
Trapping and diffusion of fission products in ThO2 and CeO2 H.Y. Xiao a,⇑, Y. Zhang b,a, W.J. Weber a,b a b
Department of Materials Science & Engineering, University of Tennessee, Knoxville, TN 37996, USA Materials Science & Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
a r t i c l e
i n f o
Article history: Received 16 March 2011 Accepted 16 May 2011 Available online 23 May 2011
a b s t r a c t The trapping and diffusion of Br, Rb, Cs and Xe in ThO2 and CeO2 have been studied using an Ab Initio total energy method in the local-density approximation of density functional theory. Fission products incorporated in cation mono-vacancy, cation–anion di-vacancy and Schottky defect sites are found to be stable, with the cation mono-vacancy being the preferred site in most cases. In both oxides, Rb and Cs are the most likely to be trapped, and Xe is more difficult to incorporate than other fission products. The energy barriers for migration of each species in ThO2 and CeO2 are also calculated. Alkali metals are relatively more mobile than other fission products, and bromine is the least mobile. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Thoria (ThO2) exhibits excellent behavior under irradiation and is an important nuclear fuel material for advanced reactors, such as high-temperature gas cooled and light water reactors [1]. Compared to UO2-based fuel, thoria has better thermo-physical properties and chemical stability, which ensures better in-pile performance and a more stable waste form, and the thoria-based fuel cycle generates lower amounts of plutonium and long-lived minor actinides [1–5]. Due to growing global energy demand and increasing concern with the proliferation of nuclear weapons and the radioactivity of the waste, there has been renewed interest in thorium-based fuels, which could provide an abundant, safe and clean source of energy and a means to burn current radioactive waste [6–8]. With the burn-up of fissile isotopes in nuclear reactor fuels, gaseous and volatile fission products, such as Kr, Xe, Br, Rb, Cs and I, are produced. The progressive production and release of the fission gases results in fuel pin swelling, development of microcracks and voids, and deterioration of thermal conduction across the fuel-clad gap [9]. The volatile fission products are produced in smaller quantities than gaseous products; however, they have detrimental chemical interactions with the fuel matrix because of their highly corrosive nature, which eventually leads to the degradation of mechanical properties, brittle fracture and stress corrosion cracking of the clad [9]. Understanding the incorporation and migration of the fission products in nuclear fuel materials is thus of significant importance for predicting the structural evolution, properties and performance of fuels. ⇑ Corresponding author. E-mail address: [email protected] (H.Y. Xiao). 0022-3115/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jnucmat.2011.05.037
Due to the importance of thoria in the nuclear industry, a number of investigations on the release and transport behavior of fission gases and volatiles in thoria and thoria-based fuels have been reported; [4,10–16] however, significant discrepancies exist in the migration energy and diffusion mechanism [10]. For example, Shiba [17] reported a high energy barrier of 4.95 eV for xenon release from thoria-6% urania (UO2) and suggested that Xe transport occurred through a tetra-vacancy mechanism. Naik [18] reported an activation energy of 2.48 eV for Xe migration in thoria-1% urania, while they suggested the migration occurred through interstitial motion. Recently, Shirsat et al. [11] studied Xe transport in a solid solution of thoria-2% urania and obtained a migration energy of 1.9 eV. They also suggested that the increase in vacancy concentration decreases the migration energy, indicative of a vacancy-assisted mechanism. In the case of volatile products, Naik et al. [19] showed that iodine diffuses differently from Xe, and it undergoes transport via the anionic sublattice, which is in contrast with earlier work [20] that the diffusion rates of iodine and xenon were comparable. It was also found that, at low irradiation dose, the alkali metal fission products Rb and Cs behave similarly to Xe [10,21]. Under comparable operating conditions, the fission product migration and release rates have been reported to be lower for thoria than for urania fuels [1]. Thus far, the incorporation and migration of fission products in urania have been extensively studied theoretically and experimentally, [22–29] whereas only a few theoretical studies of thoria [30] have been reported. Ceria (CeO2) has the same cubic fluorite-type crystal structure as UO2 and PuO2 and exhibits similar defect behavior under irradiation [31–33]. As a result, CeO2 has been used as a non-radioactive surrogate for UO2 and PuO2 in many laboratories. In the present work, a systematic study of trapping and diffusion of fission products (FP = Br, Rb, Cs and I) in ThO2 and CeO2 has been carried out
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465
based on a density functional theory method plus Hubbard U correction (DFT + U). Fission product incorporation, solution and migration energies, as well as the effects of the electronic structure on the behavior of the FP, have been determined. These results provide significant information for understanding the transport and release behavior of fission products in nuclear fuels.
energy of 450 eV for the plane wave basis set. Structural optimizations were carried out at variable volume in the limit of infinitely dilute concentrations of fission products, and under the condition that all residual forces should be smaller than 0.01 eV/Å.
2. Computational details
3.1. Incorporation of fission products in ThO2 and CeO2
All the calculations were performed with the Vienna Ab Initio Simulation Package (VASP) using the projector augmented wave method [34]. The exchange–correlation effects were treated using the local-density approximation (LDA) in the Ceperley-Alder parametrization with spin-polarized effects considered. The strong on-site Coulomb repulsion was modified by considering Hubbard U correction proposed by Dudarev et al., [35] in which only the difference between U and J is significant, and U and J can be treated as a single parameter Ueff = U–J. In this study, we used a Ueff = 6.0 eV for Ce, as developed and validated in our previous work [36]. ThO2 contains no occupied 5f states; whereas for partially reduced ThO2, the thorium 5f states are partially occupied [36]. Since our work involves defective ThO2, a Hubbard U correction is also applied to thorium-containing systems in order to improve electron localization, as shown in our previous study [36]. For an effective U value of 4 eV for Th, the calculated lattice constant and band gap (O 2p-Th 6d) are 5.60 Å and 4.75 eV, respectively, comparable to the experimental lattice constant of 5.60 Å and band gap of 5.75 eV [37–39]. The LDA + U results are shown to be in better agreement with experiments than the conventional LDA results of 5.53 Å for lattice constant and 4.32 eV for band gap. Computations were based on a 2 2 2 supercell (96 atoms) with a 2 2 2 k-point sampling in reciprocal space and a cutoff
To predict the behavior of fission products in thoria and ceria, we first fully relax the structures and calculate the incorporation energies of Br, Rb, Cs and Xe trapped in an octahedral interstitial site (Octa.) and in the following defects: (1) cation mono-vacancy (VM, M = Th or Ce); (2) oxygen mono-vacancy (VO); (3) cation– anion di-vacancy (VMO, M = Th or Ce); (4) tri-vacancy or Schottky defect (Vtrio). The di-vacancy is formed by removing one cation and its nearest-neighboring anion. For the Schottky defects, three configurations [22,40] have been considered, as shown in Fig. 1b–d, in which the
3. Results and discussion
Fig. 1. Schematic view of bulk ThO2 and three types of Schottky defects. The blue and red spheres represent the thorium and oxygen atoms, respectively. The oxygen vacancy is indicated by the yellow sphere. (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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Table 1 Incorporation energies (eV) of fission products in ThO2 and CeO2.VM (M = Th or Ce): a cation vacancy. VO: an oxygen vacancy. VMO: a cation–anion di-vacancy. Vtrio: a Schottky defect. Octa.: an octahedral interstitial. The values in bold characters indicate the most stable trapping site. VM
VO
ThO2 (lattice constant: 5.6 Å) Br 4.69 0.38 Rb 5.93 7.89 Cs 5.65 9.11 Xe 1.37 9.45 CeO2 (lattice constant: 5.4 Å) Br 5.18 0.35 Rb 5.34 2.20 Cs 5.15 3.94 Xe 1.14 7.19
VMO
V trio
V trio
V trio
3.11 5.77 5.19 0.08
2.45 2.64 1.88 0.20
1.70 2.10 1.44 0.88
0.50 2.26 1.59 1.68
3.56 5.72 8.14 9.53
3.16 5.42 4.78 0.49
1.83 3.88 3.57 0.20
1.38 3.13 2.44 1.46
0.11 2.92 2.01 2.27
4.18 4.24 6.77 10.05
ð1Þ
ð2Þ
ð3Þ
Octa.
Fig. 2. Incorporation energies of fission products in different sites of ThO2 and CeO2.
are less likely to be trapped in these sites than in other vacancytype defects since a certain amount of energy is needed to
accommodate a fission product. For Br, Rb and Cs in ThO2, the lowest incorporation energy corresponds to the VTh site, followed by VThO and Vtrio sites. This is consistent with the incorporation of Cs, [27,28,41] Sr, [28] and Mo [41] in UO2. Since the incorporation energies become larger with increasing defect size, steric effects are suggested to be involved in the incorporation [22]. In the case of Xe, the most stable trapping site is the cation mono-vacancy site, whereas the second most stable site is the Schottky defect. Of the ð1Þ three types of Schottky defects investigated, the V trio site is the most stable for all the fission products in ThO2. We also note that the incorporation energies of fission products in ThO2 follow a trend: Rb < Cs < Br < Xe, suggesting that alkali metal products are the most likely to be trapped and that xenon is not readily incorporated. In all cases, structural modification induced by the fission product incorporation is trivial. Fig. 3 illustrates the optimized configurations for Rb occupying the VTh, VThO and Vtrio sites. In this case, the atomic relaxation is small, and the trapped Rb atom is always located at the cation sublattice site. Similar to the case of ThO2, the oxygen vacancy and octahedral interstitial sites are also not favorable for all the fission products in CeO2. VCe is the only site at which the incorporation energy for Xe is negative. This site exhibits the lowest incorporation energy for Br and Cs as well, whereas for Rb incorporation the di-vacancy site is 0.08 eV lower in energy than the VCe site. Of all the fission products investigated, Rb is more readily incorporated than other fission products, and xenon is the least likely to be trapped. Comparing the trapping behavior of fission products in ThO2 and CeO2, we find that Br is more readily incorporated in CeO2, whereas other fission products prefer to be trapped in ThO2. The magnetic moments of ThO2 and CeO2 before and after incorporation of fission products are listed in Table 2. Monovacancy or di-vacancy induced magnetic moments are found for both ThO2 and CeO2 with the exception of the oxygen vacancy in ThO2. Our recent work [36] on oxygen vacancy formation in ThO2 and CeO2 has shown that the ground states are antiferromagnetic (S = 0 lB) and ferromagnetic (S = 2 lB) for reduced ThO2 and CeO2, respectively, and the results for reduced CeO2 compare well with experiments [42]. In ThO2, Br incorporation in an oxygen vacancy site has no effect on the magnetism of the system, whereas a magnetic moment of 0.82–1.83 lB is induced by the incorporation of other fission products. For CeO2 with Br trapped in an oxygen vacancy site, the magnetic moments are reduced, and alkali metals incorporation increases the magnetic moments, as compared with the value of 1.88 lB for the reduced CeO2. Bader charge analysis of the fission products, as presented in Table 3, also shows that, at the oxygen vacancy site, Br exhibits different character from other fission products. In both ThO2 and CeO2 the Br trapped in the VO site gains some charge from its neighboring
VO Rb
(a) Rb_VTh
Rb
(b) Rb_VThO
VO Rb
(c) Rb_Vtrio
Fig. 3. Schematic view of the optimized configurations for Rb occupying Th vacancy (a), Th–O di-vacancy (b) and Schottky(1) defect (c). The blue, red and purple spheres represent thorium, oxygen and rubidium atoms, respectively. The oxygen vacancy is indicated by the yellow sphere. (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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H.Y. Xiao et al. / Journal of Nuclear Materials 414 (2011) 464–470 Table 2 Magnetic moments (lB) of ThO2 and CeO2 before and after incorporation of fission products.
Br Rb Cs Xe
VMO
V trio
ð1Þ
V trio
ð2Þ
V trio
ð3Þ
Octa.
EV trio Fapp ¼ EV trio BV trio ;
3.37 0.83 2.52 2.58 1.88
0 0 1.82 1.83 0.82
1.67 0.82 0.32 0.85 1.80
0 0.81 0.35 0.40 0
0 0.81 0.38 0.39 0
0 0.79 0.36 0.35 0
0 0.88 0.58 0.66 0
where EFDO and EV trio are the oxygen Frenkel pair and Schottky defect formation energies, respectively, and BV MO and BV trio are the binding energies of di-vacancy and tri-vacancy, respectively. The formation energies of oxygen Frenkel pair and Schottky defects are calculated by [22,30]
3.39 0.84 2.55 2.60 1.90
1.88 0.98 2.88 2.88 1.92
1.69 0.82 0.73 0.86 1.82
0 0.81 0.94 0.94 0
0 0.82 0.95 0.92 0
0 0.80 0.91 0.90 0
0 0.89 0.97 0.96 0
VM
VO
VMO
V trio
V trio
V trio
Octa.
+1.10 +0.82 +1.15 +1.24
0.50 +0.17 +0.27 +0.26
+0.50 +0.79 +0.86 +0.64
0.39 +0.67 +0.66 +0.05
0.28 +0.66 +0.64 +0.04
0.19 +0.68 +0.61 +0.04
0.22 +0.70 +0.46 +0.33
+1.24 +0.88 +1.19 +1.40
0.36 +0.70 +0.81 +0.33
+0.61 +0.76 +0.87 +0.74
0.24 +0.77 +0.77 +0.12
0.10 +0.80 +0.80 +0.10
0.09 +0.79 +0.74 +0.08
0.06 +0.68 +0.86 +0.45
ð1Þ
ð2Þ
ð3Þ
atoms, while other fission product incorporation on this site results in charge loss. For incorporation on the cation mono-vacancy and cation–anion di-vacancy sites, all the fission products lose electrons to their neighboring atoms, as shown in Table 3. The magnetic moments of ThO2 and CeO2 are decreased significantly by bromine and alkali metals incorporation. In the case of Xe, the magnetic moments for the VM site trapping also decrease considerably, while incorporation in the di-vacancy site increases the magnetic moments slightly. Incorporation in Schottky defects or an octahedral interstitial site results in induced magnetic moments for all the fission products in both ThO2 and CeO2, and charge gain and loss are found for bromine and other fission products, respectively. The valance configurations considered in the present work are: Br 4s2 4p5, Rb 4s2 4p6 5s1, Cs 5s2 5p6 6s1 and Xe 5s2 5p6. The electronic configuration and the size of the fission products, and the trapping site are suggested to have significant effects on the charge transfer between the fission products and its neighboring atoms, as well as the magnetic properties of the incorporation system.
ð4Þ
Nþ1 EFDO ¼ EN1 2 EN ; V O þ EI O
N1 EV trio ¼ EN1 V M þ 2 EV O 3
ð5Þ N1 N E ; N
ð6Þ
where EN1 V M is the total energy of the supercell containing one M vacancy, ENþ1 is the total energy of the supercell containing one oxyIO gen interstitial, and EN is the total energy of the supercell without any defect. For ThO2, the oxygen Frenkel pair and Schottky defect formation energies are calculated to be 9.4 and 21.3 eV, respectively, agreeing well with the results of 9.8 and 20.6 eV reported by Yun et al. [30]. The EFDO of 6.96 eV and EV trio of 15.3 eV in CeO2 are shown to be smaller than the results for ThO2. The existence of two oxidation states (Ce3+ and Ce4+) may facilitate the formation of point defects, resulting in lower defect formation energies in CeO2. The binding energies of the di-vacancy are 7.68 and 5.05 eV for ThO2 and CeO2, respectively. The Schottky defect binding energies of 14.75–15.34 eV in ThO2 are also larger than the values of 9.62–10.24 eV in CeO2. Table 4 lists the apparent defect formation energies calculated for ThO2 and CeO2. The solution energies at 0 K obtained from the apparent defect formation energies and incorporation energies are shown in Table 5 and are plotted in Fig. 4. The solution energies are positive in all cases, suggesting that being in solution is an endothermic process. In ThO2, the preferential solution site for all fission products is the di-vacancy site. Gupta et al. [27] also found this site is more preferable for Cs in UO2. The solubility of fission products in ThO2 follows a trend of Rb > Cs > Br > Xe. Rubidium exhibits the lowest solution energy of 0.43 eV, and the energy of Xe is as high as 6.12 eV. The solution energy of 1.01 eV for Cs is larger than the value of 0.2 eV [27,41] for Cs in UO2 obtained by the generalized gradient approximation method plus Hubbard U correction (GGA + U). For CeO2, the alkali metals are also preferentially inserted in the di-vacancy site, while bromine and xenon prefer the VCe and Schottky sites, respectively. Comparing the behavior of fission products in ThO2 and CeO2, it is found that the alkali metals are more soluble in ThO2, and bromine and xenon show comparable solubility in both oxides. However, the large solution energies of Xe in both oxides indicate that it is more favorable to precipitate into bubbles or clusters, consistent with the behavior of Xe in UO2 [46]. 3.3. Migration of fission products in ThO2 and CeO2
3.2. Solubility of fission products in ThO2 and CeO2 The solubility of an FP atom in a trap site X can be assessed by its solution energy, which can be obtained by its incorporation enXInc XFapp ergy and the apparent defect formation energy: EXSol . FP ¼ EFP þ E In the framework of the point defect model introduced by Matzke [43] and Lidiard [44], EXFapp = kT ln ([X]), and assuming defect concentrations are dominated by oxygen Frenkel pairs, the apparent formation energies of mono-, di- and tri-vacancy at 0 K can be expressed as follows [41,45]:
EFDO ¼ ; 2
ð1Þ
EV M Fapp ¼ EV trio EFDO ;
ð2Þ
EV O Fapp
ð3Þ
VO
Table 3 Bader charge (|e|) of fission products in ThO2 and CeO2. The positive and negative signs denote charge loss and charge gain, respectively.
ThO2 Br Rb Cs Xe CeO2 Br Rb Cs Xe
EFDO BV MO ; 2
VM ThO2 Br Rb Cs Xe CeO2
EV MO Fapp ¼ EV trio
The energy barrier for fission product migration in ThO2 and CeO2 is investigated by the climbing-image nudged elastic band (CI-NEB) method, [47] and the activation energy is determined by the energy difference between the system with the migration
Table 4 Apparent defect formation (EFapp) and binding (B) energies (eV) in ThO2 and CeO2.
ThO2 CeO2
Fapp
E B EFapp B
VM
VO
VMO
V trio
V trio
V trio
11.88 – 8.35 –
4.7 – 3.5 –
6.20 7.68 6.78 5.05
6.53 14.75 5.69 9.62
5.94 15.34 5.07 10.24
6.07 15.21 5.15 10.16
ð1Þ
ð2Þ
ð3Þ
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Table 5 Solution energies (eV) of fission products in ThO2 and CeO2. The values in bold characters indicate the most stable solution sites.
ThO2 Br Rb Cs Xe CeO2 Br Rb Cs Xe
VM
VO
VMO
V trio
ð1Þ
V trio
ð2Þ
V trio
Octa.
7.18 5.95 6.23 10.51
4.32 12.59 13.81 14.15
3.09 0.43 1.01 6.12
4.08 3.89 4.65 6.33
4.24 3.84 4.50 6.82
5.57 3.81 4.48 7.75
3.56 5.72 8.14 9.53
3.17 3.01 3.20 7.21
3.83 5.68 7.42 10.67
3.62 1.36 2.00 7.27
3.86 1.81 2.12 5.89
3.70 1.94 2.63 6.53
5.26 2.23 3.14 7.42
4.18 4.24 6.77 10.05
ð3Þ
Fig. 5. Pathway of (a) VM (M = Th or Ce) and (b) VMO migration in ThO2 and CeO2.
Table 6 Vacancy migration energies (eV) in ThO2 and CeO2.
Cation mono-vacancy Di-vacancy
Fig. 4. Solution energies of fission products in different sites of ThO2 and CeO2.
atom located at the saddle point and that at the trap site. Since the octahedral interstitial sites are less favorable than the vacancy or vacancy cluster for fission product incorporation, we only investigate fission product migration via the vacancy-assisted mechanism [24,30] without considering the octahedral interstitial mechanism. We first investigate the energies of vacancy migration without the incorporation of any fission product atoms. The mechanisms for cation mono-vacancy and cation–anion di-vacancy migration are illustrated in Fig. 5, and the corresponding energies are given in Table 6. The migration energies of 6.67 eV for Th vacancy and 5.02 eV for di-vacancy are larger than the results reported by Yun et al., [30] but compare well with the work of Colbourn and Mackrodt [48]. Yun et al. [30] calculated the migration energies of the Th vacancy and di-vacancy between their nearest lattice sites employing the GGA method. They found that the oxygen vacancy does not affect the movement of the Th vacancy because the migration energy of the Th vacancy in the effective migration of the di-vacancy remains unchanged [30]. However, our calculations and the work performed by Colbourn and Mackrodt [48] show that the oxygen vacancy decreases the migration energy of the thorium vacancy. In CeO2 the migration energy of the di-vacancy is 0.8 eV smaller than that of the Ce vacancy, and the migration energies
ThO2
Ref. [48]
Ref. [30]
CeO2
6.67 5.02
7.04 5.36
4.47 4.47
5.08 4.28
of the cation vacancy and di-vacancy are found to be lower than those in ThO2. The high migration energies in both oxides indicate that the mobility of these defects is low [30]. Fig. 6 illustrates the optimized configurations of the saddle-point states for Th vacancy and di-vacancy migration in ThO2. In each case, the atomic relaxation is shown to be significant. For Th vacancy migration, the saddle-point state is situated roughly midway between the starting and final configurations along the h1 1 0i direction, and the two nearest-neighboring oxygen atoms of the migrating atom shift 0.68 Å away from their sublattice sites along the h0 0 1i and direction, respectively. For di-vacancy migration, the h0 0 1i migrating Th atom moves 0.42 Å away from the center of the migration pathway along the h0 0 1i direction. Also, two neighboring oxygen atoms of the oxygen vacancy relax inward by 0.6 Å 0 0i direction, respectively. along the h1 0 0i and h1 Table 7 lists the energy barriers for Br, Rb, Cs and Xe migration via a cation vacancy- or a di-vacancy-assisted mechanism in ThO2 and CeO2. The absolute values for each mechanism are shown to be lower than the corresponding energies for self-diffusion of the cation mono-vacancy and cation–anion di-vacancy. For Br and Xe migration, the energies via the Th vacancy-assisted mechanism are lower than those via the di-vacancy. The optimized configurations of the saddle-point states for Br and Xe migrating via the Th vacancy are illustrated in Fig. 7a and d, respectively. The structural modification is shown to be very small for Br migration. In the case of Xe, two neighboring oxygen atoms relax outward by 1.56 Å 0 0i direction, respectively, exhibiting a along the h1 0 0i and h1 Schottky defect structure. This suggests that for Xe diffusion in ThO2, the Schottky defect-assisted mechanism is more favorable than either the Th vacancy or the di-vacancy mechanisms. Yun et al. [30] have calculated the binding energy of a single VO separated by Xe-vacancy complexes in ThO2 using the GGA method and suggested Xe diffusion by the vacancy-assisted mechanism is unfavorable. Recently, the work performed by Shirsat et al. [11] showed that the vacancies play an important role in Xe migration in ThO2, and the increase of vacancy concentration decreases the energy barrier, indicative of a vacancy-assisted mechanism. Also, the migration energy of 1.96 eV reported in their work is in excellent agreement with our result of 1.9 eV. Naik [18] and Shiba [17] have reported energy barriers of 2.48 eV for Xe transport in
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VO VTh
VTh
VTh
VTh Thmig
Thmig
(b) VThO migration
(a) VTh migration
Fig. 6. Schematic view of the optimized configurations of the saddle-point states for Th vacancy (a) and Th–O di-vacancy (b) migration in ThO2. The blue and red spheres represent the thorium and oxygen atoms, respectively. The larger and smaller yellow spheres represent the VTh and VO, respectively. The migrating Th is indicated by the dark red sphere. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Table 7 Migration energies (eV) of Br, Rb, Cs and Xe via a cation vacancy(VM, M = Th or Ce) or a di-vacancy-assisted mechanism in ThO2 and CeO2.
Br Rb Cs Xe
VTh
VThO
VCe
VCeO
3.42 1.35 1.08 1.90
4.67 0.81 0.66 2.73
4.14 2.31 0.89 3.51
4.40 0.85 0.77 2.66
thoria-1% urania and 4.95 eV for xenon release from thoria-6% urania, respectively. For Rb and Cs migration via the Th vacancy, the migration energies determined from our calculations are negative.
The negative sign indicates that the so-called ‘‘saddle-point’’ position is more stable than the starting position and that the role of two sites should be exchanged, meaning that the interstitial migration will occur in the reverse direction [49]. Similar to the case of Xe migration, the optimized configurations also include Schottky defect structures. In the presence of an oxygen vacancy, the migration energies become positive and the absolute values become smaller. Fig. 7b and c shows the optimized configuration for Rb and Cs migrating via the di-vacancy, respectively. In both oxides, one oxygen atom shifts 0.7 Å away from its sublattice site, suggesting that there is a competition between the di-vacancy and the Schottky defect mechanism. The migration energy of 0.66 eV for Cs in pure thoria, however, is significantly smaller than the
VO VTh
VTh
VTh
VTh
Rb
Br
(a) Br migration via VTh
(b) Rb migration via VThO
VO VTh
VTh Cs
(c) Cs migration via VThO
VTh
VTh Xe
(d) Xe migration via VTh
Fig. 7. Schematic view of the optimized configurations of the saddle-point states for Br migrating via the VTh-assisted mechanism (a); Rb migrating via the VThO-assisted mechanism (b); Cs migrating via the VThO-assisted mechanism (c); Xe migrating via the VTh-assisted mechanism (d).
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activation energy of 4.8–6.9 eV for Cs migration in thoria-6% urania reported by Akabori and Fukuda [15]. In CeO2, the alkali metals are also more mobile than other fission products, and the high energy barrier of 4.14 eV for Br migration suggests that its mobility is relatively low. The presence of an oxygen vacancy decreases the migration energy of Rb, Cs and Xe, and increases the energy barrier for Br migration. It is interesting to find that the migration energies for Rb and Cs via the VCeO mechanism are still negative, suggesting that the interstitial mechanism is favorable for alkali metals migration in CeO2. Dorado et al. [22] have suggested that in UO2 the DFT + U method yields an increased number of energy minima, i.e., metastable states. In our work, we have made convergence tests of the total energies for all the systems to make sure they are completely converged. Unfortunately, no experimental work about fission product behavior in CeO2 are currently available for direct comparison. It is expected that the present calculations will be useful for further experimental and theoretical investigations of the transport and release behavior of fission products in nuclear materials. 4. Conclusions The incorporation and migration behavior of fission products in ThO2 and CeO2 have been investigated by a DFT + U method. Incorporation at oxygen vacancy and interstitial sites are found to be less stable than cation mono-vacancy, cation–anion di-vacancy and Schottky defect sites, and the cation mono-vacancy sites are the more favorable sites in most cases. In ThO2, the lowest incorporation energies are 4.69, 5.93, 5.65 and 1.37 eV for Br, Rb, Cs and Xe, respectively, which are generally lower than those in CeO2. Of all the fission products, Rb is the most likely to be trapped and xenon is less readily incorporated than other fission products. Charge gain or loss was determined to result from the incorporation of fission products on different sites. The predicted solution energies are shown to be positive in all cases, and the lowest energy is 0.43 eV for Rb in ThO2. The energy barriers for Th vacancy, Th–O di-vacancy, Ce vacancy and Ce–O di-vacancy migration have been calculated to be 6.67, 5.02, 5.08 and 4.28 eV, respectively, indicative of immobility of these point defects in both oxides. The energies for fission product migration via the cation mono-vacancy and cation–anion di-vacancy mechanisms are shown to be lower than the corresponding energies for self-diffusion of cation mono-vacancy and cation–anion di-vacancy. In both ThO2 and CeO2, Rb and Cs are more mobile than Br and Xe, and the mobility of Xe is the lowest. The vacancy-assisted mechanism is shown to be favorable for the migration of all fission products migration in ThO2. The migration mechanism and migration energy for Xe are shown to be in good agreement with recent experiments. In CeO2, it is found that the vacancy-assisted mechanism is preferred for Br and Xe, while Rb and Cs prefer the interstitial mechanism. Acknowledgements This work was supported as part of the Materials Science of Actinides, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences. The theoretical calculations were performed using the supercomputer resources at the Environmental Molecular Sciences
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Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat
Trapping and diffusion of fission products in ThO2 and CeO2 H.Y. Xiao a,⇑, Y. Zhang b,a, W.J. Weber a,b a b
Department of Materials Science & Engineering, University of Tennessee, Knoxville, TN 37996, USA Materials Science & Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
a r t i c l e
i n f o
Article history: Received 16 March 2011 Accepted 16 May 2011 Available online 23 May 2011
a b s t r a c t The trapping and diffusion of Br, Rb, Cs and Xe in ThO2 and CeO2 have been studied using an Ab Initio total energy method in the local-density approximation of density functional theory. Fission products incorporated in cation mono-vacancy, cation–anion di-vacancy and Schottky defect sites are found to be stable, with the cation mono-vacancy being the preferred site in most cases. In both oxides, Rb and Cs are the most likely to be trapped, and Xe is more difficult to incorporate than other fission products. The energy barriers for migration of each species in ThO2 and CeO2 are also calculated. Alkali metals are relatively more mobile than other fission products, and bromine is the least mobile. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Thoria (ThO2) exhibits excellent behavior under irradiation and is an important nuclear fuel material for advanced reactors, such as high-temperature gas cooled and light water reactors [1]. Compared to UO2-based fuel, thoria has better thermo-physical properties and chemical stability, which ensures better in-pile performance and a more stable waste form, and the thoria-based fuel cycle generates lower amounts of plutonium and long-lived minor actinides [1–5]. Due to growing global energy demand and increasing concern with the proliferation of nuclear weapons and the radioactivity of the waste, there has been renewed interest in thorium-based fuels, which could provide an abundant, safe and clean source of energy and a means to burn current radioactive waste [6–8]. With the burn-up of fissile isotopes in nuclear reactor fuels, gaseous and volatile fission products, such as Kr, Xe, Br, Rb, Cs and I, are produced. The progressive production and release of the fission gases results in fuel pin swelling, development of microcracks and voids, and deterioration of thermal conduction across the fuel-clad gap [9]. The volatile fission products are produced in smaller quantities than gaseous products; however, they have detrimental chemical interactions with the fuel matrix because of their highly corrosive nature, which eventually leads to the degradation of mechanical properties, brittle fracture and stress corrosion cracking of the clad [9]. Understanding the incorporation and migration of the fission products in nuclear fuel materials is thus of significant importance for predicting the structural evolution, properties and performance of fuels. ⇑ Corresponding author. E-mail address: [email protected] (H.Y. Xiao). 0022-3115/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jnucmat.2011.05.037
Due to the importance of thoria in the nuclear industry, a number of investigations on the release and transport behavior of fission gases and volatiles in thoria and thoria-based fuels have been reported; [4,10–16] however, significant discrepancies exist in the migration energy and diffusion mechanism [10]. For example, Shiba [17] reported a high energy barrier of 4.95 eV for xenon release from thoria-6% urania (UO2) and suggested that Xe transport occurred through a tetra-vacancy mechanism. Naik [18] reported an activation energy of 2.48 eV for Xe migration in thoria-1% urania, while they suggested the migration occurred through interstitial motion. Recently, Shirsat et al. [11] studied Xe transport in a solid solution of thoria-2% urania and obtained a migration energy of 1.9 eV. They also suggested that the increase in vacancy concentration decreases the migration energy, indicative of a vacancy-assisted mechanism. In the case of volatile products, Naik et al. [19] showed that iodine diffuses differently from Xe, and it undergoes transport via the anionic sublattice, which is in contrast with earlier work [20] that the diffusion rates of iodine and xenon were comparable. It was also found that, at low irradiation dose, the alkali metal fission products Rb and Cs behave similarly to Xe [10,21]. Under comparable operating conditions, the fission product migration and release rates have been reported to be lower for thoria than for urania fuels [1]. Thus far, the incorporation and migration of fission products in urania have been extensively studied theoretically and experimentally, [22–29] whereas only a few theoretical studies of thoria [30] have been reported. Ceria (CeO2) has the same cubic fluorite-type crystal structure as UO2 and PuO2 and exhibits similar defect behavior under irradiation [31–33]. As a result, CeO2 has been used as a non-radioactive surrogate for UO2 and PuO2 in many laboratories. In the present work, a systematic study of trapping and diffusion of fission products (FP = Br, Rb, Cs and I) in ThO2 and CeO2 has been carried out
H.Y. Xiao et al. / Journal of Nuclear Materials 414 (2011) 464–470
465
based on a density functional theory method plus Hubbard U correction (DFT + U). Fission product incorporation, solution and migration energies, as well as the effects of the electronic structure on the behavior of the FP, have been determined. These results provide significant information for understanding the transport and release behavior of fission products in nuclear fuels.
energy of 450 eV for the plane wave basis set. Structural optimizations were carried out at variable volume in the limit of infinitely dilute concentrations of fission products, and under the condition that all residual forces should be smaller than 0.01 eV/Å.
2. Computational details
3.1. Incorporation of fission products in ThO2 and CeO2
All the calculations were performed with the Vienna Ab Initio Simulation Package (VASP) using the projector augmented wave method [34]. The exchange–correlation effects were treated using the local-density approximation (LDA) in the Ceperley-Alder parametrization with spin-polarized effects considered. The strong on-site Coulomb repulsion was modified by considering Hubbard U correction proposed by Dudarev et al., [35] in which only the difference between U and J is significant, and U and J can be treated as a single parameter Ueff = U–J. In this study, we used a Ueff = 6.0 eV for Ce, as developed and validated in our previous work [36]. ThO2 contains no occupied 5f states; whereas for partially reduced ThO2, the thorium 5f states are partially occupied [36]. Since our work involves defective ThO2, a Hubbard U correction is also applied to thorium-containing systems in order to improve electron localization, as shown in our previous study [36]. For an effective U value of 4 eV for Th, the calculated lattice constant and band gap (O 2p-Th 6d) are 5.60 Å and 4.75 eV, respectively, comparable to the experimental lattice constant of 5.60 Å and band gap of 5.75 eV [37–39]. The LDA + U results are shown to be in better agreement with experiments than the conventional LDA results of 5.53 Å for lattice constant and 4.32 eV for band gap. Computations were based on a 2 2 2 supercell (96 atoms) with a 2 2 2 k-point sampling in reciprocal space and a cutoff
To predict the behavior of fission products in thoria and ceria, we first fully relax the structures and calculate the incorporation energies of Br, Rb, Cs and Xe trapped in an octahedral interstitial site (Octa.) and in the following defects: (1) cation mono-vacancy (VM, M = Th or Ce); (2) oxygen mono-vacancy (VO); (3) cation– anion di-vacancy (VMO, M = Th or Ce); (4) tri-vacancy or Schottky defect (Vtrio). The di-vacancy is formed by removing one cation and its nearest-neighboring anion. For the Schottky defects, three configurations [22,40] have been considered, as shown in Fig. 1b–d, in which the
3. Results and discussion
Fig. 1. Schematic view of bulk ThO2 and three types of Schottky defects. The blue and red spheres represent the thorium and oxygen atoms, respectively. The oxygen vacancy is indicated by the yellow sphere. (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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H.Y. Xiao et al. / Journal of Nuclear Materials 414 (2011) 464–470
Table 1 Incorporation energies (eV) of fission products in ThO2 and CeO2.VM (M = Th or Ce): a cation vacancy. VO: an oxygen vacancy. VMO: a cation–anion di-vacancy. Vtrio: a Schottky defect. Octa.: an octahedral interstitial. The values in bold characters indicate the most stable trapping site. VM
VO
ThO2 (lattice constant: 5.6 Å) Br 4.69 0.38 Rb 5.93 7.89 Cs 5.65 9.11 Xe 1.37 9.45 CeO2 (lattice constant: 5.4 Å) Br 5.18 0.35 Rb 5.34 2.20 Cs 5.15 3.94 Xe 1.14 7.19
VMO
V trio
V trio
V trio
3.11 5.77 5.19 0.08
2.45 2.64 1.88 0.20
1.70 2.10 1.44 0.88
0.50 2.26 1.59 1.68
3.56 5.72 8.14 9.53
3.16 5.42 4.78 0.49
1.83 3.88 3.57 0.20
1.38 3.13 2.44 1.46
0.11 2.92 2.01 2.27
4.18 4.24 6.77 10.05
ð1Þ
ð2Þ
ð3Þ
Octa.
Fig. 2. Incorporation energies of fission products in different sites of ThO2 and CeO2.
are less likely to be trapped in these sites than in other vacancytype defects since a certain amount of energy is needed to
accommodate a fission product. For Br, Rb and Cs in ThO2, the lowest incorporation energy corresponds to the VTh site, followed by VThO and Vtrio sites. This is consistent with the incorporation of Cs, [27,28,41] Sr, [28] and Mo [41] in UO2. Since the incorporation energies become larger with increasing defect size, steric effects are suggested to be involved in the incorporation [22]. In the case of Xe, the most stable trapping site is the cation mono-vacancy site, whereas the second most stable site is the Schottky defect. Of the ð1Þ three types of Schottky defects investigated, the V trio site is the most stable for all the fission products in ThO2. We also note that the incorporation energies of fission products in ThO2 follow a trend: Rb < Cs < Br < Xe, suggesting that alkali metal products are the most likely to be trapped and that xenon is not readily incorporated. In all cases, structural modification induced by the fission product incorporation is trivial. Fig. 3 illustrates the optimized configurations for Rb occupying the VTh, VThO and Vtrio sites. In this case, the atomic relaxation is small, and the trapped Rb atom is always located at the cation sublattice site. Similar to the case of ThO2, the oxygen vacancy and octahedral interstitial sites are also not favorable for all the fission products in CeO2. VCe is the only site at which the incorporation energy for Xe is negative. This site exhibits the lowest incorporation energy for Br and Cs as well, whereas for Rb incorporation the di-vacancy site is 0.08 eV lower in energy than the VCe site. Of all the fission products investigated, Rb is more readily incorporated than other fission products, and xenon is the least likely to be trapped. Comparing the trapping behavior of fission products in ThO2 and CeO2, we find that Br is more readily incorporated in CeO2, whereas other fission products prefer to be trapped in ThO2. The magnetic moments of ThO2 and CeO2 before and after incorporation of fission products are listed in Table 2. Monovacancy or di-vacancy induced magnetic moments are found for both ThO2 and CeO2 with the exception of the oxygen vacancy in ThO2. Our recent work [36] on oxygen vacancy formation in ThO2 and CeO2 has shown that the ground states are antiferromagnetic (S = 0 lB) and ferromagnetic (S = 2 lB) for reduced ThO2 and CeO2, respectively, and the results for reduced CeO2 compare well with experiments [42]. In ThO2, Br incorporation in an oxygen vacancy site has no effect on the magnetism of the system, whereas a magnetic moment of 0.82–1.83 lB is induced by the incorporation of other fission products. For CeO2 with Br trapped in an oxygen vacancy site, the magnetic moments are reduced, and alkali metals incorporation increases the magnetic moments, as compared with the value of 1.88 lB for the reduced CeO2. Bader charge analysis of the fission products, as presented in Table 3, also shows that, at the oxygen vacancy site, Br exhibits different character from other fission products. In both ThO2 and CeO2 the Br trapped in the VO site gains some charge from its neighboring
VO Rb
(a) Rb_VTh
Rb
(b) Rb_VThO
VO Rb
(c) Rb_Vtrio
Fig. 3. Schematic view of the optimized configurations for Rb occupying Th vacancy (a), Th–O di-vacancy (b) and Schottky(1) defect (c). The blue, red and purple spheres represent thorium, oxygen and rubidium atoms, respectively. The oxygen vacancy is indicated by the yellow sphere. (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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H.Y. Xiao et al. / Journal of Nuclear Materials 414 (2011) 464–470 Table 2 Magnetic moments (lB) of ThO2 and CeO2 before and after incorporation of fission products.
Br Rb Cs Xe
VMO
V trio
ð1Þ
V trio
ð2Þ
V trio
ð3Þ
Octa.
EV trio Fapp ¼ EV trio BV trio ;
3.37 0.83 2.52 2.58 1.88
0 0 1.82 1.83 0.82
1.67 0.82 0.32 0.85 1.80
0 0.81 0.35 0.40 0
0 0.81 0.38 0.39 0
0 0.79 0.36 0.35 0
0 0.88 0.58 0.66 0
where EFDO and EV trio are the oxygen Frenkel pair and Schottky defect formation energies, respectively, and BV MO and BV trio are the binding energies of di-vacancy and tri-vacancy, respectively. The formation energies of oxygen Frenkel pair and Schottky defects are calculated by [22,30]
3.39 0.84 2.55 2.60 1.90
1.88 0.98 2.88 2.88 1.92
1.69 0.82 0.73 0.86 1.82
0 0.81 0.94 0.94 0
0 0.82 0.95 0.92 0
0 0.80 0.91 0.90 0
0 0.89 0.97 0.96 0
VM
VO
VMO
V trio
V trio
V trio
Octa.
+1.10 +0.82 +1.15 +1.24
0.50 +0.17 +0.27 +0.26
+0.50 +0.79 +0.86 +0.64
0.39 +0.67 +0.66 +0.05
0.28 +0.66 +0.64 +0.04
0.19 +0.68 +0.61 +0.04
0.22 +0.70 +0.46 +0.33
+1.24 +0.88 +1.19 +1.40
0.36 +0.70 +0.81 +0.33
+0.61 +0.76 +0.87 +0.74
0.24 +0.77 +0.77 +0.12
0.10 +0.80 +0.80 +0.10
0.09 +0.79 +0.74 +0.08
0.06 +0.68 +0.86 +0.45
ð1Þ
ð2Þ
ð3Þ
atoms, while other fission product incorporation on this site results in charge loss. For incorporation on the cation mono-vacancy and cation–anion di-vacancy sites, all the fission products lose electrons to their neighboring atoms, as shown in Table 3. The magnetic moments of ThO2 and CeO2 are decreased significantly by bromine and alkali metals incorporation. In the case of Xe, the magnetic moments for the VM site trapping also decrease considerably, while incorporation in the di-vacancy site increases the magnetic moments slightly. Incorporation in Schottky defects or an octahedral interstitial site results in induced magnetic moments for all the fission products in both ThO2 and CeO2, and charge gain and loss are found for bromine and other fission products, respectively. The valance configurations considered in the present work are: Br 4s2 4p5, Rb 4s2 4p6 5s1, Cs 5s2 5p6 6s1 and Xe 5s2 5p6. The electronic configuration and the size of the fission products, and the trapping site are suggested to have significant effects on the charge transfer between the fission products and its neighboring atoms, as well as the magnetic properties of the incorporation system.
ð4Þ
Nþ1 EFDO ¼ EN1 2 EN ; V O þ EI O
N1 EV trio ¼ EN1 V M þ 2 EV O 3
ð5Þ N1 N E ; N
ð6Þ
where EN1 V M is the total energy of the supercell containing one M vacancy, ENþ1 is the total energy of the supercell containing one oxyIO gen interstitial, and EN is the total energy of the supercell without any defect. For ThO2, the oxygen Frenkel pair and Schottky defect formation energies are calculated to be 9.4 and 21.3 eV, respectively, agreeing well with the results of 9.8 and 20.6 eV reported by Yun et al. [30]. The EFDO of 6.96 eV and EV trio of 15.3 eV in CeO2 are shown to be smaller than the results for ThO2. The existence of two oxidation states (Ce3+ and Ce4+) may facilitate the formation of point defects, resulting in lower defect formation energies in CeO2. The binding energies of the di-vacancy are 7.68 and 5.05 eV for ThO2 and CeO2, respectively. The Schottky defect binding energies of 14.75–15.34 eV in ThO2 are also larger than the values of 9.62–10.24 eV in CeO2. Table 4 lists the apparent defect formation energies calculated for ThO2 and CeO2. The solution energies at 0 K obtained from the apparent defect formation energies and incorporation energies are shown in Table 5 and are plotted in Fig. 4. The solution energies are positive in all cases, suggesting that being in solution is an endothermic process. In ThO2, the preferential solution site for all fission products is the di-vacancy site. Gupta et al. [27] also found this site is more preferable for Cs in UO2. The solubility of fission products in ThO2 follows a trend of Rb > Cs > Br > Xe. Rubidium exhibits the lowest solution energy of 0.43 eV, and the energy of Xe is as high as 6.12 eV. The solution energy of 1.01 eV for Cs is larger than the value of 0.2 eV [27,41] for Cs in UO2 obtained by the generalized gradient approximation method plus Hubbard U correction (GGA + U). For CeO2, the alkali metals are also preferentially inserted in the di-vacancy site, while bromine and xenon prefer the VCe and Schottky sites, respectively. Comparing the behavior of fission products in ThO2 and CeO2, it is found that the alkali metals are more soluble in ThO2, and bromine and xenon show comparable solubility in both oxides. However, the large solution energies of Xe in both oxides indicate that it is more favorable to precipitate into bubbles or clusters, consistent with the behavior of Xe in UO2 [46]. 3.3. Migration of fission products in ThO2 and CeO2
3.2. Solubility of fission products in ThO2 and CeO2 The solubility of an FP atom in a trap site X can be assessed by its solution energy, which can be obtained by its incorporation enXInc XFapp ergy and the apparent defect formation energy: EXSol . FP ¼ EFP þ E In the framework of the point defect model introduced by Matzke [43] and Lidiard [44], EXFapp = kT ln ([X]), and assuming defect concentrations are dominated by oxygen Frenkel pairs, the apparent formation energies of mono-, di- and tri-vacancy at 0 K can be expressed as follows [41,45]:
EFDO ¼ ; 2
ð1Þ
EV M Fapp ¼ EV trio EFDO ;
ð2Þ
EV O Fapp
ð3Þ
VO
Table 3 Bader charge (|e|) of fission products in ThO2 and CeO2. The positive and negative signs denote charge loss and charge gain, respectively.
ThO2 Br Rb Cs Xe CeO2 Br Rb Cs Xe
EFDO BV MO ; 2
VM ThO2 Br Rb Cs Xe CeO2
EV MO Fapp ¼ EV trio
The energy barrier for fission product migration in ThO2 and CeO2 is investigated by the climbing-image nudged elastic band (CI-NEB) method, [47] and the activation energy is determined by the energy difference between the system with the migration
Table 4 Apparent defect formation (EFapp) and binding (B) energies (eV) in ThO2 and CeO2.
ThO2 CeO2
Fapp
E B EFapp B
VM
VO
VMO
V trio
V trio
V trio
11.88 – 8.35 –
4.7 – 3.5 –
6.20 7.68 6.78 5.05
6.53 14.75 5.69 9.62
5.94 15.34 5.07 10.24
6.07 15.21 5.15 10.16
ð1Þ
ð2Þ
ð3Þ
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H.Y. Xiao et al. / Journal of Nuclear Materials 414 (2011) 464–470
Table 5 Solution energies (eV) of fission products in ThO2 and CeO2. The values in bold characters indicate the most stable solution sites.
ThO2 Br Rb Cs Xe CeO2 Br Rb Cs Xe
VM
VO
VMO
V trio
ð1Þ
V trio
ð2Þ
V trio
Octa.
7.18 5.95 6.23 10.51
4.32 12.59 13.81 14.15
3.09 0.43 1.01 6.12
4.08 3.89 4.65 6.33
4.24 3.84 4.50 6.82
5.57 3.81 4.48 7.75
3.56 5.72 8.14 9.53
3.17 3.01 3.20 7.21
3.83 5.68 7.42 10.67
3.62 1.36 2.00 7.27
3.86 1.81 2.12 5.89
3.70 1.94 2.63 6.53
5.26 2.23 3.14 7.42
4.18 4.24 6.77 10.05
ð3Þ
Fig. 5. Pathway of (a) VM (M = Th or Ce) and (b) VMO migration in ThO2 and CeO2.
Table 6 Vacancy migration energies (eV) in ThO2 and CeO2.
Cation mono-vacancy Di-vacancy
Fig. 4. Solution energies of fission products in different sites of ThO2 and CeO2.
atom located at the saddle point and that at the trap site. Since the octahedral interstitial sites are less favorable than the vacancy or vacancy cluster for fission product incorporation, we only investigate fission product migration via the vacancy-assisted mechanism [24,30] without considering the octahedral interstitial mechanism. We first investigate the energies of vacancy migration without the incorporation of any fission product atoms. The mechanisms for cation mono-vacancy and cation–anion di-vacancy migration are illustrated in Fig. 5, and the corresponding energies are given in Table 6. The migration energies of 6.67 eV for Th vacancy and 5.02 eV for di-vacancy are larger than the results reported by Yun et al., [30] but compare well with the work of Colbourn and Mackrodt [48]. Yun et al. [30] calculated the migration energies of the Th vacancy and di-vacancy between their nearest lattice sites employing the GGA method. They found that the oxygen vacancy does not affect the movement of the Th vacancy because the migration energy of the Th vacancy in the effective migration of the di-vacancy remains unchanged [30]. However, our calculations and the work performed by Colbourn and Mackrodt [48] show that the oxygen vacancy decreases the migration energy of the thorium vacancy. In CeO2 the migration energy of the di-vacancy is 0.8 eV smaller than that of the Ce vacancy, and the migration energies
ThO2
Ref. [48]
Ref. [30]
CeO2
6.67 5.02
7.04 5.36
4.47 4.47
5.08 4.28
of the cation vacancy and di-vacancy are found to be lower than those in ThO2. The high migration energies in both oxides indicate that the mobility of these defects is low [30]. Fig. 6 illustrates the optimized configurations of the saddle-point states for Th vacancy and di-vacancy migration in ThO2. In each case, the atomic relaxation is shown to be significant. For Th vacancy migration, the saddle-point state is situated roughly midway between the starting and final configurations along the h1 1 0i direction, and the two nearest-neighboring oxygen atoms of the migrating atom shift 0.68 Å away from their sublattice sites along the h0 0 1i and direction, respectively. For di-vacancy migration, the h0 0 1i migrating Th atom moves 0.42 Å away from the center of the migration pathway along the h0 0 1i direction. Also, two neighboring oxygen atoms of the oxygen vacancy relax inward by 0.6 Å 0 0i direction, respectively. along the h1 0 0i and h1 Table 7 lists the energy barriers for Br, Rb, Cs and Xe migration via a cation vacancy- or a di-vacancy-assisted mechanism in ThO2 and CeO2. The absolute values for each mechanism are shown to be lower than the corresponding energies for self-diffusion of the cation mono-vacancy and cation–anion di-vacancy. For Br and Xe migration, the energies via the Th vacancy-assisted mechanism are lower than those via the di-vacancy. The optimized configurations of the saddle-point states for Br and Xe migrating via the Th vacancy are illustrated in Fig. 7a and d, respectively. The structural modification is shown to be very small for Br migration. In the case of Xe, two neighboring oxygen atoms relax outward by 1.56 Å 0 0i direction, respectively, exhibiting a along the h1 0 0i and h1 Schottky defect structure. This suggests that for Xe diffusion in ThO2, the Schottky defect-assisted mechanism is more favorable than either the Th vacancy or the di-vacancy mechanisms. Yun et al. [30] have calculated the binding energy of a single VO separated by Xe-vacancy complexes in ThO2 using the GGA method and suggested Xe diffusion by the vacancy-assisted mechanism is unfavorable. Recently, the work performed by Shirsat et al. [11] showed that the vacancies play an important role in Xe migration in ThO2, and the increase of vacancy concentration decreases the energy barrier, indicative of a vacancy-assisted mechanism. Also, the migration energy of 1.96 eV reported in their work is in excellent agreement with our result of 1.9 eV. Naik [18] and Shiba [17] have reported energy barriers of 2.48 eV for Xe transport in
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VO VTh
VTh
VTh
VTh Thmig
Thmig
(b) VThO migration
(a) VTh migration
Fig. 6. Schematic view of the optimized configurations of the saddle-point states for Th vacancy (a) and Th–O di-vacancy (b) migration in ThO2. The blue and red spheres represent the thorium and oxygen atoms, respectively. The larger and smaller yellow spheres represent the VTh and VO, respectively. The migrating Th is indicated by the dark red sphere. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Table 7 Migration energies (eV) of Br, Rb, Cs and Xe via a cation vacancy(VM, M = Th or Ce) or a di-vacancy-assisted mechanism in ThO2 and CeO2.
Br Rb Cs Xe
VTh
VThO
VCe
VCeO
3.42 1.35 1.08 1.90
4.67 0.81 0.66 2.73
4.14 2.31 0.89 3.51
4.40 0.85 0.77 2.66
thoria-1% urania and 4.95 eV for xenon release from thoria-6% urania, respectively. For Rb and Cs migration via the Th vacancy, the migration energies determined from our calculations are negative.
The negative sign indicates that the so-called ‘‘saddle-point’’ position is more stable than the starting position and that the role of two sites should be exchanged, meaning that the interstitial migration will occur in the reverse direction [49]. Similar to the case of Xe migration, the optimized configurations also include Schottky defect structures. In the presence of an oxygen vacancy, the migration energies become positive and the absolute values become smaller. Fig. 7b and c shows the optimized configuration for Rb and Cs migrating via the di-vacancy, respectively. In both oxides, one oxygen atom shifts 0.7 Å away from its sublattice site, suggesting that there is a competition between the di-vacancy and the Schottky defect mechanism. The migration energy of 0.66 eV for Cs in pure thoria, however, is significantly smaller than the
VO VTh
VTh
VTh
VTh
Rb
Br
(a) Br migration via VTh
(b) Rb migration via VThO
VO VTh
VTh Cs
(c) Cs migration via VThO
VTh
VTh Xe
(d) Xe migration via VTh
Fig. 7. Schematic view of the optimized configurations of the saddle-point states for Br migrating via the VTh-assisted mechanism (a); Rb migrating via the VThO-assisted mechanism (b); Cs migrating via the VThO-assisted mechanism (c); Xe migrating via the VTh-assisted mechanism (d).
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activation energy of 4.8–6.9 eV for Cs migration in thoria-6% urania reported by Akabori and Fukuda [15]. In CeO2, the alkali metals are also more mobile than other fission products, and the high energy barrier of 4.14 eV for Br migration suggests that its mobility is relatively low. The presence of an oxygen vacancy decreases the migration energy of Rb, Cs and Xe, and increases the energy barrier for Br migration. It is interesting to find that the migration energies for Rb and Cs via the VCeO mechanism are still negative, suggesting that the interstitial mechanism is favorable for alkali metals migration in CeO2. Dorado et al. [22] have suggested that in UO2 the DFT + U method yields an increased number of energy minima, i.e., metastable states. In our work, we have made convergence tests of the total energies for all the systems to make sure they are completely converged. Unfortunately, no experimental work about fission product behavior in CeO2 are currently available for direct comparison. It is expected that the present calculations will be useful for further experimental and theoretical investigations of the transport and release behavior of fission products in nuclear materials. 4. Conclusions The incorporation and migration behavior of fission products in ThO2 and CeO2 have been investigated by a DFT + U method. Incorporation at oxygen vacancy and interstitial sites are found to be less stable than cation mono-vacancy, cation–anion di-vacancy and Schottky defect sites, and the cation mono-vacancy sites are the more favorable sites in most cases. In ThO2, the lowest incorporation energies are 4.69, 5.93, 5.65 and 1.37 eV for Br, Rb, Cs and Xe, respectively, which are generally lower than those in CeO2. Of all the fission products, Rb is the most likely to be trapped and xenon is less readily incorporated than other fission products. Charge gain or loss was determined to result from the incorporation of fission products on different sites. The predicted solution energies are shown to be positive in all cases, and the lowest energy is 0.43 eV for Rb in ThO2. The energy barriers for Th vacancy, Th–O di-vacancy, Ce vacancy and Ce–O di-vacancy migration have been calculated to be 6.67, 5.02, 5.08 and 4.28 eV, respectively, indicative of immobility of these point defects in both oxides. The energies for fission product migration via the cation mono-vacancy and cation–anion di-vacancy mechanisms are shown to be lower than the corresponding energies for self-diffusion of cation mono-vacancy and cation–anion di-vacancy. In both ThO2 and CeO2, Rb and Cs are more mobile than Br and Xe, and the mobility of Xe is the lowest. The vacancy-assisted mechanism is shown to be favorable for the migration of all fission products migration in ThO2. The migration mechanism and migration energy for Xe are shown to be in good agreement with recent experiments. In CeO2, it is found that the vacancy-assisted mechanism is preferred for Br and Xe, while Rb and Cs prefer the interstitial mechanism. Acknowledgements This work was supported as part of the Materials Science of Actinides, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences. The theoretical calculations were performed using the supercomputer resources at the Environmental Molecular Sciences
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