APPLIED PHYSICS LETTERS 93, 201903 共2008兲

Ab initio investigation on oxygen defect clusters in UO2+x Hua Y. Geng,1,a兲 Ying Chen,1 Yasunori Kaneta,1 and Motoyasu Kinoshita2,3 1

Department of Systems Innovation, The University of Tokyo, Hongo 7-3-1, Tokyo 113-8656, Japan Nuclear Technology Research Laboratory, Central Research Institute of Electric Power Industry, Tokyo 201-8511, Japan 3 Japan Atomic Energy Agency, Ibaraki 319-1195, Japan 2

共Received 1 September 2008; accepted 5 November 2008; published online 19 November 2008兲 Oxygen defect clustering in uranium dioxide had been indicated in powder neutron diffraction measurements, and an empirical clustering mechanism had been proposed to explain the data. However, using first-principles LSDA+ U calculations, we find that this empirical model, in fact, cannot work. A more physically reasonable model is proposed based on a thermodynamical competition between point defects and cuboctahedral clusters. This mechanism interprets the puzzled origin of the observed asymmetric interstitial O⬘ and O⬙ naturally. It also gives a good and consistent agreement with all available experimental data, except the high occupation of the O⬙ site. © 2008 American Institute of Physics. 关DOI: 10.1063/1.3035846兴 Uranium dioxide 共UO2兲 is in fluorite structure under ambient pressure. This structure is quite simple and commonly appears in rare earth and actinide oxides. However, its anion defects are not trivial. Its complicated behavior in UO2 has drawn particular interests; partly due to as the nowadays widely adopted nuclear fuel, the most important applications of UO2 are highly related to defects. The current model of defective UO2 was empirically derived from powder neutron diffraction measurements.1–4 These experiments indicated that oxygen interstitials cannot occupy the octahedral site 共Oi兲 of the cation FCC lattice, which is a commonly expected position for interstitial by the consideration of symmetry and available space. Instead, all oxygen interstitials had been detected at positions displaced from this site along 具110典 and 具111典 directions about 1 Å 共site O⬘ and O⬙兲, respectively. To explain the puzzled origin of these asymmetric interstitials, Willis suggested a clustering model empirically—to form the so-called Willis clusters.2,3 Willis cluster is an extensive defect formed by a linear combination of interstitial O⬘, O⬙ and vacancy Ov, and is labeled according to the composition O⬘ : Ov : O⬙. This structure was proposed after a hypothesis, which states that a stable asymmetric interstitial is possible if O⬘ pushes its nearest lattice oxygen out along the 具111典 direction to an O⬙ site and forms a split-interstitial-like structure.2,3 A typical Willis cluster of 2:2:2 is shown in Fig. 1共a兲. All Willis clusters have a similar geometry, different only in the number of O⬘ interstitials and oxygen vacancies that were inserted inbetween the two end O⬙’s.5 This clustering interpretation of defective UO2 invalidated the simple point defect model and its associating diffusion mechanism, both of which are widely employed to describe fuel behaviors.6–8 It also led to confusion because it was difficult to investigate the Willis clusters thoroughly. Meanwhile people were unsure how large the clustering effects should be.9–11 Furthermore, since the explicit atomic coordinates were unavailable from the experiments, the empirical picture contains ambiguities and is therefore uncera兲

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tain. In some cases the Willis clusters even cannot account for the neutron diffraction patterns. For example a quite different cluster, the cuboctahedral cluster 共COT兲, was required in order to describe U4O9 and U3O7 phases, despite their close relationship with the fluorite structure.12–15 To get rid of this difficulty, one has to a resort to a theoretical method that deals with atomic coordinates explicitly. Modern quantum mechanical approach based on the density functional theory is one of the best choice.16 It provides an accurate description for most material behaviors. For example the bulk stoichiometric UO2 was investigated by this method with the local density approximation plus Hubbard correction 共LSDA+ U兲,17 and the electronic structure and defect induced charge transfers were well described.8,18 Application of this method to nonstoichiometric UO2+x also gave amazing agreement with available experiments. The following observations have been well reproduced: 共i兲 the negative volume change induced by oxygen interstitials, 共ii兲 predominance of oxygen defects when x ⬎ 0, and 共iii兲 exclusive COT clusters at the high x region 共in U4O9 / U3O7 phase兲.1–4,6–8,12–14,19 However, a systematic investigation on Willis clusters is still absent. In this work we will apply this method to study the stability of Willis clusters, including 1:2:2, 2:2:2, and 3:3:2 共with two inequivalent configurations, respectively兲, as well

FIG. 1. 共Color online兲 Relaxation process of a Willis 2:2:2 cluster 共a兲 to a V-3O⬙ split interstitial 共b兲. Each cube indicates an oxygen cage and the arrows point to the atomic relaxation directions. In 共a兲, the top interstitial moves to annihilate the nearest vacancy 共Ov兲, and the two O⬘ interstitials relax downwards to their nearby O⬙ sites.

93, 201903-1

© 2008 American Institute of Physics

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TABLE I. First-principles results for oxygen defect clusters in UO2+x: ⌬V is the defect induced volume change per fluorite cubic cell; E f , Eef , and E pf are the overall defect formation energy, formation energy per excess oxygen, and formation energy per oxygen Frenkel pair, respectively.

V − 3O⬙ V − 4O⬙

x

⌬V 共Å3兲

E f 共eV兲

Eef 共eV兲

E pf 共eV兲

1 / 16 3 / 32

−0.21 −0.42

−3.88 −6.74

−1.94 −2.25

3.72 3.96

as 4:3:2 and 2:3:2 共with three inequivalent configurations兲 clusters. All clusters were modeled in a defective cubic supercell with otherwise 96 atoms 共U32O64兲. Larger supercells were also tried to check the size effect that arisen from the periodic boundaries. The defective structures were then optimized fully until all forces and stress were less than 0.01 eV/ Å. Other computational parameters, such as the kinetic energy cutoff and sampling k-points, are the same as in Ref. 8. Calculated results showed that all inequivalent configurations of the same kind of the Willis cluster always relaxed to the same final structure. For example, all 1:2:2 clusters relaxed to a point oxygen interstitial that occupies the octahedral site 共Oi兲; all 2:2:2 and 3:3:2 clusters relaxed to a V-3O⬙ cluster, a split interstitial with three O⬙’s surrounding one oxygen vacancy 共Ov兲; the 4:3:2 cluster relaxed to a V-4O⬙ cluster, a split interstitial with four O⬙’s surrounding one Ov; and all 2:3:2 clusters decayed to a point Oi. The relaxation process of a 2:2:2 cluster is shown in Fig. 1共a兲, in which the arrows indicate the atomic movement directions. Moreover, the final structure, a V-3O⬙ cluster, is given in Fig. 1共b兲. In detail, the top O⬙ in Fig. 1共a兲 relaxed to annihilate the nearest Ov eventually, while the two O⬘’s moved downward to their nearest O⬙ sites to form a V-3O⬙ cluster with the bottom O⬙ interstitial. Similarly, in clusters 1:2:2 and 2:3:2, the O⬘ interstitials were not capable of supporting the structure and all interstitials annihilated the nearby vacancies, with the remaining interstitial moving to an Oi site; the four O⬘ interstitials in the 4:3:2 cluster also could not prevent the O⬙ interstitials from annihilating, and they relaxed to the O⬙ positions that surrounding the middle vacancy to form a V-4O⬙ cluster. These depict a general picture of the Willis cluster’s failure. Different from the case in COT cluster where the nestlike structure provides an integral support to stabilize the asymmetric interstitials,19 here the open structure of the Willis cluster makes the O⬘ interstitials energetically unfavorable, which cannot push their nearest lattice oxygens out along the 具111典 direction. Instead, they prefer to relax to O⬙ sites to form a split interstitial 共V-3O⬙ / V-4O⬙兲 or to annihilate nearby vacancies directly. For clusters bigger than 4:3:2, we have tried several primary calculations, but failed to find any evidence that they favored. Also, no energy trap has been detected in all structure optimizations, which indicates that the Willis cluster is not even metastable. As discussed above, all Willis clusters will relax to point interstitial Oi, cluster V-3O⬙, V-4O⬙, or their combinations. It becomes necessary to clarify the role of these clusters on the material behavior. The calculated static properties of clusters V-3O⬙ and V-4O⬙ are listed in Table I, where the energetics information can be used to evaluate the defect concentrations at finite temperature and composition x with the independent cluster approximation.8,19 This method is exact at low tem-

FIG. 2. Defect concentrations of point oxygen interstitial, uranium vacancy, COT-o, V-3O⬙ and V-4O⬙ clusters at 1500 K. All other components are negligible.

peratures and low defect concentrations where atomic vibrational effects and correlation among clusters are negligible. The formation energy E f and E pf in Table I were calculated according to Eqs. 共1兲 and 共14兲 of Ref. 8, respectively, except the latter that also needs to be compensated with the corresponding point oxygen vacancies to form Frenkel pairs.19 For a thermodynamically equilibrium system consisting of point vacancies 共on uranium and oxygen sublattices兲, point interstitials 共uranium and oxygen兲, and oxygen clusters 共V-3O⬙, V-4O⬙, COT-o, and COT-v兲, the composition x must satisfy x = 2共关VU兴 − 关IU兴兲 + 关IO兴 + 兺ini␳i + 2兺 jn j␳ j − 2关VO兴. Here i runs over COT-v and COT-o clusters 共see Ref. 19 for details兲, and j over V-3O⬙ and V-4O⬙ clusters, and n is the excess number of oxygen within each cluster. The coefficient 2 before the second summation arose from the fact that these clusters are defined on the oxygen sublattice. Quantities in square brackets denote the point defect concentrations, and ␳ denote the cluster concentrations. Using the similar equilibrium equations as in Refs. 8 and 19, we computed the defect concentrations in the closed regime where no particle exchange with the exterior occurs. The results at 1500 K are shown in Fig. 2. This temperature was selected to ensure that Oi has enough concentration to compete with other defects, whose temperature dependency can be found in Fig. 3 of Ref. 19. All unshown components are small enough to be ignored. We see that the clusters V-3O⬙ and V-4O⬙ compete with each other throughout the whole x region. However, their concentrations are always one order smaller than the uranium vacancy and two orders smaller than that of Oi and COT-o clusters. At lower temperatures their concentrations reduce further and can be neglected completely. This indicates a general physical picture about oxygen defects in UO2+x, i.e., the nontrivial oxygen interstitials in the hyperstoichiometric regime are only Oi and COT-o clusters. Oxygen vacancy and all asymmetric O⬙ and O⬘ interstitials come from COT-o clusters. This picture describes very well the high x region where the U4O9 / U3O7 phase dominates.19 It also explains the puzzled origin of the observed asymmetric interstitials and the non-negligible oxygen vacancy at T ⬃ 300 K and x ⬃ 0.1 satisfactorily. However, a quantitative discrepancy remains: the experimental occupation numbers of O⬘ and O⬙ sites are 0.08:0.16 共Ref. 2兲

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FIG. 3. Pseudophase diagram of oxygen interstitials in UO2+x, where the hatched area marks the region in which the neutron diffraction measurements had been performed.

关which had been corrected to 0.13:0.12 later 共Ref. 3兲兴, 0.14:0.12, and 0.33:0.10 共Ref. 4兲 when x is at around 0.110.13, which evidently asks for a low O⬘ : O⬙ ratio. However, COT-o cluster can provide only a fixed ratio of 12:1.19 This disagreement unlikely can be solved in the current model. It has already been confirmed experimentally and theoretically that in U4O9 / U3O7 phase, the dominated cluster is COT-o only. That is to say, the O⬘ : O⬙ ratio should approach 12:1 at the high x limit. Besides, the current model always predicts a monotonic variation in concentration along composition x. It seems impossible that the V-3O⬙ / V-4O⬙ cluster 共contribution to O⬙兲 would have a considerable concentration at x ⯝ 0.11 but would disappear later. A possible solution to this problem is to take the decomposition/ reassembly process of COT clusters into account explicitly. However, this complicates the situation and is beyond the scope of this study. On the experimental side, there is also some ambiguity that existed in the postdata analysis. For example the correction of the O⬘ : O⬙ ratio from 0.08:0.16 to 0.13:0.12 共Refs. 2 and 3兲 and the vagueness in U4O9 / U3O7 phase about the position of the center oxygen in COT clusters12–14,19 demonstrate the existence of such uncertainty. This is because it is very difficult to extract the explicit atomic geometry from powder neutron diffraction measurements, especially for UO2+x, which is not an ordered phase, and some peaks have been broadened drastically. Another experimental cause that might affect the O⬙ occupation is the oxygen partial pressure. If experiments were performed in an open atmosphere and grains of the sample were fine enough, then the above model needs to be revised to take oxygen exchange with the exterior into account. Thus the defect concentration becomes proportional to10 ␳i ⬃ exp共−Eif / ␬BT兲 关PO2 f共T兲兴ni/2, where the oxygen partial pressure PO2 could modify the concentration of V-3O⬙ / V-4O⬙ cluster greatly. Anyway, it requires further

endeavor from both theorists and experimentalists to remove this discrepancy. In summary we investigated the defect clustering structure in UO2+x by first principles calculations. The Willis clusters were shown to generally fail. A more reasonable picture about oxygen defective UO2+x was established based on a thermodynamical competition between Oi and COT-o clusters, which explained most available experiments very well. The last confusion is about the role of the point interstitial Oi in UO2+x. Willis once argued that oxygen should not occupy this site based on his observations.1,2 The above discussion indicated that this argument might be inappropriate. In order to understand the Willis observation that Oi is negligible, we should take the effects of temperature and composition into account. Using the defect concentrations calculated above, one can plot a pseudophase diagram for oxygen defects in UO2+x, as shown in Fig. 3. It clearly illustrates that the predominant region of Oi is at low x and high temperatures. 关Note that each COT-o cluster contributes 13 single oxygen interstitials and the figure was plotted according to which kind of interstitial is the majority 共from Oi or from COT-o cluster兲兴. Considering that all experiments available so far were conducted in the hatched region,2–4 which is far from the Oi territory, it is then understandable why experiments failed to detect Oi. Support from the Budget for Nuclear Research of the Ministry of Education, Culture, Sports, Science and Technology of Japan based on the screening and counseling by the Atomic Energy Commission is acknowledged. B. T. M. Willis, J. Phys. 共France兲 25, 431 共1964兲. B. T. M. Willis, Proc. Br. Ceram. Soc. 1, 9 共1964兲. 3 B. T. M. Willis, Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. A34, 88 共1978兲. 4 A. D. Murray and B. T. M. Willis, J. Solid State Chem. 84, 52 共1990兲. 5 C. R. A. Catlow, Proc. R. Soc. London, Ser. A 353, 533 共1977兲. 6 Hj. Matzke, J. Chem. Soc., Faraday Trans. 2 83, 1121 共1987兲. 7 A. B. Lidiard, J. Nucl. Mater. 19, 106 共1966兲. 8 H. Y. Geng, Y. Chen, Y. Kaneta, M. Iwasawa, T. Ohnuma, and M. Kinoshita, Phys. Rev. B 77, 104120 共2008兲. 9 K. Govers, S. Lemehov, M. Hou, and M. Verwerft, J. Nucl. Mater. 366, 161 共2007兲. 10 J. P. Crocombette, F. Jollet, L. T. Nga, and T. Petit, Phys. Rev. B 64, 104107 共2001兲. 11 M. Freyss, T. Petit, and J. P. Crocombette, J. Nucl. Mater. 347, 44 共2005兲. 12 D. J. M. Bevan, I. E. Grey, and B. T. M. Willis, J. Solid State Chem. 61, 1 共1986兲. 13 R. I. Cooper and B. T. M. Willis, Acta Crystallogr., Sect. A: Found. Crystallogr. A60, 322 共2004兲. 14 F. Garrido, R. M. Ibberson, L. Nowicki, and B. T. M. Willis, J. Nucl. Mater. 322, 87 共2003兲. 15 L. Nowicki, F. Garrido, A. Turos, and L. Thome, J. Phys. Chem. Solids 61, 1789 共2000兲. 16 G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 共1996兲. 17 V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B 44, 943 共1991兲. 18 H. Y. Geng, Y. Chen, Y. Kaneta, and M. Kinoshita, Phys. Rev. B 75, 054111 共2007兲, and references therein. 19 H. Y. Geng, Y. Chen, Y. Kaneta, and M. Kinoshita, Phys. Rev. B 77, 180101共R兲 共2008兲. 1 2

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