First-Principles Calculation of Point Defects in Uranium Dioxide

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1 Materials Transactions, Vol. 47, No. 11 (26) pp to 2657 Special Issue on Advances in Computational Materials Science and Engineering IV #26 The Japan Institute of Metals First-Principles Calculation of Point Defects in Uranium Dioxide Misako Iwasawa 1, Ying Chen 2, Yasunori Kaneta 2, Toshiharu Ohnuma 1, Hua-Yun Geng 2 and Motoyasu Kinoshita 3;4 1 Materials Science Research Laboratory, Central Research Institute of Electric Power Industry, Tokyo , Japan 2 Department of Quantum Engineering and Systems Science, School of Engineering, The University of Tokyo, Tokyo , Japan 3 Nuclear Technology Research Laboratory, Central Research Institute of Electric Power Industry, Tokyo , Japan 4 Japan Atomic Energy Agency, Ibaraki , Japan A first-principles calculation for uranium dioxide (UO 2 ) in an antiferromagnetic structure with four types of point defects, uranium vacancy, oxygen vacancy, uranium interstitial, and oxygen interstitial, has been performed by the projector-augmented-wave method with generalized gradient approximation combined with the Hubbard U correction. Defect formation energies are estimated under lattice relaxation for supercells containing 1, 2, and 8 unit cells of UO 2. The electronic structure, the atomic displacement and the stability of defected systems are obtained, and the effects of cell sizes on these properties are discussed. The results form a self-consistent dataset of formation energies and atomic distance variations of various point defects in UO 2 with relatively high precision. We show that a supercell with 8 UO 2 unit cells or larger is necessary to investigate the defect behavior with reliable precision, since point defects have a wide-ranging effect, not only on the first nearest neighbor atoms of the defect, but on the second neighbors and on more distant atoms. [doi:1.232/matertrans ] (Received August 3, 26; Accepted October 6, 26; Published November, 26) Keywords: first-principles method, density functional theory, generalized gradient approximation, projector-augmented-wave method, Hubbard U correction, electronic structure, uranium dioxide, point defect, formation energy, lattice relaxation 1. Introduction Uranium dioxide, UO 2, is widely used as fuel in nuclear power generation. In the high burn-up of the fuel, the radial periphery of UO 2 pellets forms a characteristic fine grain, which is known as a rim structure. 1) To clarify the microscopic mechanism of this structuring behavior, a basic understanding of the thermodynamic, structural, and kinetic properties of UO 2 is very important for both the practical operation of nuclear reactors and the theoretical interest in this metal oxide with strongly correlated electrons. In particular, the rim structure is believed to be formed by accumulated irradiation damage and the effects of high energy electronic excitation. Since it is difficult to analyze these complicated dynamical processes using current theoretical tools, the first step of this research is to provide by information on the point defect behavior in UO 2, which is the most fundamental information on the elemental processes in the formation of the complex defect structure. Petit et al. have carried out a series of electronic structure calculations on UO 2 with point defects by the linear muffintin orbital method with the atomic-sphere approximation (LMTO-ASA) and by the plane-wave pseudopotential method based on density functional theory (DFT) with the local density approximation (LDA) or the generalized gradient approximation (GGA). 2 5) They investigated the electronic structure of the defect systems, and estimated the formation energies of several types of point defects in UO 2. Their results qualitatively provided the correct trend of defect formation in UO 2. However, their studies have two limitations. One is the relatively small supercell (U 8 O 16 ) used in each calculation, which may lead to an overestimation of the defect formation energies. The other is the LDA or GGA scheme adopted in all calculations, which gave a metallic ground state. UO 2 is well known to be a conventional Mott insulator, 6) but in the framework of the LDA or GGA, UO 2 is predicted to be a metal. 2 5) This failure of the conventional LDA or GGA has been attributed to the absence of strong correlations for the 5f electrons in the uranium atom in UO 2. The LDA+U 7,8) method has been developed to describe such a correlation by introducing a strong intra-atomic interaction. This approach has been used for uranium 5f electrons, 6,9,1) and has yielded the electronic structure as a Mott insulator for UO 2 in the ground state. In the present study, we perform a series of comprehensive calculations on UO 2 with four types of point defects to obtain point defect properties with high accuracy. We have focused on the following two aspects: to construct a suitable theoretical framework for UO 2 that reflects the correct antiferromagnetic insulator ground state of UO 2 crystal by incorporating the effect of strongly correlated 5f electrons in uranium atoms through the Hubbard U correction, and to calculate point defect properties such as formation energies and lattice deformations by taking the currently plausible maximal supercell (2 2 2 unit cells of UO 2,U 32 O 64 )as the model system. In the next section, the calculation method and the calculation systems are briefly described. In Section 3, results obtained for various cases are presented and discussed. The results are then summarized in Section Calculation Method The present work focuses on four types of point defects in UO 2 : uranium vacancy (), oxygen vacancy (), uranium interstitial () and oxygen interstitial (). The ideal UO 2 crystal has a CaF 2 - type structure (cf12, Fm3m), and its unit cell is shown in Fig. 1(a). Systems containing point defects are modeled using supercell technique. A specific supercell has been

2 2652 M. Iwasawa et al. (a) (c) (e) a c B C U2 U1 A b U-atom O-atom (b) (d) (f) Fig. 1 Calculation systems. (a) Unit cell of ideal UO 2 crystal, and relaxed defect systems in U 32 O 64 supercell: (b) Ideal, (c), (d) O- vacancy, (e), and (f). constructed to introduce each point defect. Examples of an U- vacancy and an are illustrated as site A and site B, respectively, in Fig. 1(a). For the interstitial defects, only the octahedral site (the center of the oxygen cube) is taken into account for both U- and s, as marked by C in Fig. 1(a). To examine the effect of the cell size, calculations are performed for three types of supercells with containing 1, 2, and 8 unit cells of UO 2. We use the following notation to show the number of atoms in each unit cell, U 4 O 8 (1 1 1), U 8 O 16 (2 1 1) and U 32 O 64 (2 2 2) for each type of defect. Figures 1(b) (f) show five U 32 O 64 (2 2 2) supercells corresponding to an ideal UO 2 crystal and the four types of point defects. Note that these defected structures show the situations after relaxation, but the initial regular structures can be imaged easily. To discuss the ground state properties of UO 2 with defects, we assume an antiferromagnetic structure for electron spins in our calculations in accordance with experimental results. 11) The magnetic moments of uranium ions lie in the (1) plane ferromagnetically, and this plane is stacked along the [1] direction antiferromagnetically, as illustrated in Fig. 1(a). An antiferromagnetic configuration remains when defects are introduced. All lattice constants and cell shapes are relaxed by minimizing the stress tensor, and all atomic positions are relaxed by minimizing the Hellman-Feynman forces using the RMM-DIIS algorithm. 12) In the paramagnetic state, UO 2 crystal belongs to the CaF 2 -type structure with cubic symmetry. However, the antiferromagnetic structure causes UO 2 to have a tetragonal lattice. The initial atomic positions and lattice constants are set corresponding to the cubic lattice, while during calculation, the lattice symmetries turn to tetragonal for U 4 O 8 and U 32 O 64, and orthorhombic for U 8 O 16 as a natural result of introducing antiferromagnetism. Calculations in the present work are carried out by the projector-augmented-wave (PAW) method, 13,14) using PBE- GGA exchange-correlation functional, ) as implemented in the Vienna ab initio Simulation Package (VASP). 16) The spin polarization calculations are performed, whereas the spinorbit interactions are neglected as a simplification of the complex magnetism in UO 2. The Hubbard U correction is introduced to describe strongly correlated uranium 5f electrons in accordance with a simplified version of GGA+U proposed by Dudarev et al. 8) They determined the LSDA+U parameters to be U ¼ 4:5 ev and J ¼ :51 ev by a careful estimation using LMTO method. 9) It may be reasonable to select these values when we use the simplified version of GGA+U for Hubbard U correction. We undertook electronic structure calculation for U 4 O 8 by the PAW-GGA+U method using these values in the antiferromagnetic configuration whose lattice constant was experimental value,.547 nm. The electronic structure of U 4 O 8 obtained was that of a Mott insulator, so we decided to adopt these values in our PAW- GGA+U calculations. Regarding to other calculation parameters, the cut-off energy of plane-waves is set to 4 ev. The Wigner-Seitz radii of uranium and oxygen atoms are.88 and.82 nm, respectively. The Monkhorst-Pack 17) (4,4,4) set is used for k-space sampling in U 4 O 8, whereas a (3,3,3) set is used in large cells to reduce the amount of calculation. The tetrahedron method with Blöchl s corrections 18) is used in k-mesh integration. The bulk modulus is estimated by fitting to the Birch-Murnaghan equation of state 19) based on the total energy vs unit cell volume plot. 3. Results and Discussion 3.1 Structural properties of pure UO 2 First, the calculation for an ideal UO 2 crystal was carried out to test the framework with a combination of several theoretical treatments so as to obtain an optimized calculation condition. The calculated equilibrium properties are listed in Table 1, as well as the available values from experiments 2) and other calculations 21,22) for comparison. A good agreement among these data can be found. The tetragonal symmetry (a ¼ b 6¼ c) in the present result is due to the c-axis-direction antiferromagnetic structure. Figure 2 shows the total density of states (DOS) of ideal UO 2 and the site projected density of states (PDOS) for uranium (U) and oxygen (O) atoms. Since U 7s, 6p and 6d components are very small, only the U 5f components are illustrated for the U atom. Because of the antiferromagnetic configuration, there are two types of U sites: U1 indicates the U site in the c ¼ plane with up spin, and U2, the U site in the c ¼ 1=2 plane with down spin, as shown in Fig. 1. It can be seen that the strong correlation for 5f electrons of U

3 First-Principles Calculation of Point Defects in Uranium Dioxide 2653 Table 1 Calculated equilibrium properties of relaxed structure with point defects in UO 2 : lattice constants (nm) and bulk moduli (GPa). Experimental and theoretical values available are also listed for comparison. Method Lattice constant, Bulk modulus, l/nm k/gpa experimental 2Þ Petit et al. 21Þ LMTO-LDA LMTO-GGA Dudarev et al. 9Þ LMTO-LSDA LMTO-LSDA+U Crocombette et al. 4Þ DFT-LDA Freyss et al. 5Þ DFT-GGA Yun et al. 22Þ PAW-GGA+U this work (U 4 O 8 ) DOS, n(e)/states/ev - - Total up spin U-5f up spin PAW-GGA+U down spin down spin O-2p up spin down spin U1 U2 a ¼ b ¼ :552, c ¼ :547 atoms results in the removal of the degeneracy of 5f bands near the Fermi level. The 5f bands split into two energy regions, one is mainly the top of the valence states and the Energy, E/eV Fig. 2 Total DOS and PDOS of ideal UO 2 crystal with antiferromagnetic configuration. U1 and U2 are uranium atoms at the c ¼ and c ¼ :5 planes with different spin directions, respectively. (Vertical bar indicates Fermi level.) other is the bottom of the conduction states, which correctly reproduces an insulator characteristic. It is also found that the 2p orbitals of O predominate the lower valence states and partially hybridize with 5f orbitals of U at the top of the valence band. These calculated results are consistent with the experimental and theoretical values reported previously, 6) which confirms that the method we employed can be taken as a suitable platform for further investigation of various defect structures. 3.2 Electronic structures of defect systems Figure 3 shows the total DOS of ideal and four defect structures in U 32 O 64, as well the PDOS of defect-related sites in various types of defects. The components of the first nearest neighbor ( n.n. hereafter), 2nd n.n. and 3rd n.n. U atoms are plotted in red, green and blue, respectively. The reference sites to which the neighboring atoms are referred in different situations are, in Fig. 3(a), the U atom site; in Figs. 3(b) and (c), the U vacancy and O vacancy sites, respectively; and in Figs. 3(d) and (e), the U interstitial and O interstitial sites, respectively. For the ideal crystal, only the contributions of the 1st n.n. and 2nd n.n. atoms are drawn, because, with respect to one U atom, the 3rd n.n. U atoms are fully geometrically equivalent to the 1st n.n. U atoms but only at a different distance from the U atom, their contributions are the same as those of the 1st n.n. U atoms (red). Comparing the DOS of defect structures with the ideal situation, the effect of introducing point defects and of the introduced defects themselves on the electronic structures can be explained as following. The electronic structure of the system with the, U 31 O 64 (Fig. 3(b)), shows p-type semiconductor behavior with an acceptor level at.5 ev higher than the top of the valence band. The missing U atom results in an O 2p hole state; at same time, the antiferromagnetism is broken and a non zero magnetic moment in the U 5f bands occurs. It is interesting to see the effect of the 1st n.n. U atoms on the vacancy (red) most appears at the top of the valence band, whereas the acceptor level that newly appeared in the energy gap is mainly created by the 2nd n.n. U atoms (green). It is also observed that the PDOS distribution of the 3rd n.n. U atoms (blue) is significantly different from that of 1st n.n. U atoms (red) although they are the same in the ideal crystal. Such a redistribution of the band structure is due to the fact that the interatomic interactions among atoms are amended according to the vacancy introduced. The situation for the, U 32 O 63 (Fig. 3(c)), seems to be that of an n-type semiconductor with a donor level at 1.7 ev higher than the top of the valence band, near the bottom of the conduction band. This impurity state is mainly formed from the 5f states of U (which almost cannot be seen in the figure due to its small value) hybridized with planewave-like states that are attributed to an interstitial component of about 45% electrons in that state. It is intriguing to note the deviation of the PDOS of the 1st and 3rd n.n. U atoms, and the remaining antiferromagnetism with the existing O vacancy. The electronic structure of the, U 33 O 64 (Fig. 3(d)), is as that of an n-type semiconductor. The component of the U interstitial atom steadily crosses the

4 2654 M. Iwasawa et al. 1 (a) ideal U 32 O 64 (a) U-atom O-atom -1 1 (b) U 31 O 64 (b) (c) DOS, n(e)/states/ev (c) U 32 O 63 (d) U 33 O 64 (d) (e) -1 1 (e) U 32 O 65 Fig. 4 Partial charge density distributions projected onto (11) plane corresponding to U 5f band at the top of valence band spanned from Fermi level down to lower energy states by 2: ev. (a) Ideal UO 2 crystal, and relaxed defect systems in U 32 O 64 supercell: (b), (c), (d) and (e). (Blue and red indicate low and high values of charge, respectively.) Energy, E/eV Fig. 3 Total DOS of ideal UO 2 crystal and four types of point defects in U 32 O 64 supercell (thin line), and PDOS of defect relevant sites. (a) Ideal UO 2 crystal: total DOS and PDOS of 1st n.n. (red), 2nd n.n. (green) and 3rd n.n. (blue) U atom sites with respect to U atom. (b) case: total DOS and PDOS of 1st n.n. (red), 2nd n.n. (green) and 3rd n.n. (blue) U atom sites with respect to U vacancy site. (c) case: total DOS and PDOS of 1st n.n. (red), 2nd n.n. (green) and 3rd n.n. (blue) U atom sites with respect to O vacancy site. (d) case: total DOS and PDOS of 1st n.n. (red) U atom sites with respect to interstitial U site and U interstitial site (bold). (e) case; total DOS and PDOS of 1st n.n. (red) U atom sites with respect to interstitial O site and O interstitial site (bold). (Vertical bar indicates Fermi level.) range from the bottom of the valence band to the top of the conduction band, although the portion of the single atom is too small to be seen in the figure. It can be seen that the donor level is formed mainly from 5f electrons of the 1st n.n. U atoms (red) and is located at about 1.3 ev higher than the top of the valence band. The defect state is similar to the U 32 O 63 case, but the extra U 5f electrons impose a larger magnetic effect to form an obvious asymmetric distribution of majority and minority spins. The system, U 32 O 65 (Fig. 3(e)), acts as a p- type semiconductor; the acceptor level, which consists of mainly 1st n.n. U 5f (red) states slightly hybridized by the 2p of the interstitial O atom (which almost cannot be seen), spans the energy gap while showing complicated features. The effect of the single O interstitial atom (bold black line) can be observed at the joint place of two parts of the valence band. The defect state is similar to the U 31 O 64 case, but since some of the occupied 5f electrons on the U site are absorbed into the interstitial O site, the unoccupied 5f states appear above the Fermi level. This also leads to a large magneticstate redistribution in the DOS that creates a remarkable shift of the majority and minority spin states in the energy space. Figure 4 shows the partial charge density distributions projected onto the (11) plane in the lattice corresponding to the 5f band at the top of the valence band spanned from the Fermi level down to the lower energy states by 2: ev, for ideal and defect systems in U 32 O 64 supercells. The localized

5 First-Principles Calculation of Point Defects in Uranium Dioxide 2655 Table 2 Calculated lattice parameters and volume variations of relaxed structures with point defects in UO 2 with respect to the ideal crystal in three types of supercells: U 4 O 8,U 8 O 16 and U 32 O 64. The volume variation is calculated as the difference from the ideal crystal in each system. (Values in parentheses are converted to one unit cell of UO 2 :U 4 O 8.) Lattice constant, l/nm Equilibrium Volume a b c volume, V/nm 3 variation, (%) Ideal U 4 O 8 cell Ideal 1.133(.5517) (.5376) U 8 O 16 cell 1.996(.5498) (.5591) (.5467) Ideal 1.133(.5516) 1.133(.5516) 1.936(.5468) (.5494) 1.988(.5494) 1.944(.5472) U 32 O 64 cell 1.19(.555) 1.19(.555) 1.1(.55) (.5547) 1.193(.5547) 1.142(.5521) (.557) 1.113(.557) 1.937(.5468) f-orbital feature is clearly displayed. Note that a high spherical charge density exists in UO 2, particularly around O atoms. The non spherical charge density of occupied U 5f electrons, atomic displacement and change in bonding state in various defect structures can be observed clearly, particularly in the case. 3.3 Lattice relaxations surrounding defects A summary of the lattice parameters and the volume variations of relaxed structures in the ideal crystal and the four types of point defects in the UO 2 system are shown in Table 2. These four types of point defects in the U 32 O 64 supercell after relaxation are displayed in Figs. 1(c) (f), and the ideal U 32 O 64 is shown in Fig. 1(b). We found that the lattice parameter c is less than a (or b)in all cases. This can be assigned to the asymmetry of charge densities in the ða; bþ plane and along the c-axis, which results from the magnetic interaction among partially occupied 5f electrons. The volume variations with respect to different sizes of supercells listed in Table 2 are depicted in Fig. 5. One can see that the volume variations are large in the case of the U defects; the causes a large swelling, whereas the induces a decrease in volume shrink. On the other hand, O defects do not cause significant variations in volume. Figure 5 also shows the effective ranges of various point defects. We found rather large volume variations in the U 8 O 16 cell; even in our largest cell, U 32 O 64, the volume variations still do not reach zero. This clearly indicates the insufficiency of the U 8 O 16 cell as a non interacting point defect model. It is of high interest to analyze the details of changes in atomic distances among the defect site and its neighboring atoms in our largest supercell U 32 O 64 (Fig. 6). It is noteworthy that all distances of the 1st n.n. atoms with respect to Volume Variation (%) Number of unit cells of UO 2 Fig. 5 Volume variations of relaxed defect structures in UO 2 (in %). Distance Variation, d/nm U 32 O 64 supercell Distance to Defect, d/nm Fig. 6 Bond length variations in relaxed defect structures in UO 2 (in %).

6 2656 M. Iwasawa et al. Table 3 Formation energies (ev) of point defects in UO 2 in three types of supercells: U 4 O 8,U 8 O 16 and U 32 O 64. Other calculated results available are also listed for comparison. U 4 O 8 U 8 O 16 U 32 O 64 AF AF NM Freyss Crocombette AF Þ 3.3 2Þ Þ 6.7 2Þ Þ 7.3 2Þ Þ 2.9 2Þ.44 1) Ref. 5) by GGA, 2) Ref. 4) by LDA NM 1Þ NM 2Þ defect sites in the four defect structures are increased, whereas for further distant atoms, the corresponding distance varies complicatedly according to the types of defect. In Fig. 6, several pairs of points at about.27,.39 and.46 nm from the defect site are the same neighboring atoms with slight distance differences due to the tetragonal symmetry of the lattice. In the case of the (solid line in Fig. 6), the eight 1st n.n. O atoms move outward from the vacancy site to form a large vacancy hole, although the total volume change is negative, as shown in Table 2. This large vacancy hole manifests the feature of the ionic crystal, and is caused by effective Coulomb repulsion between the 1st n.n. O atoms due to the missing cation. The further n.n. atoms adjust themselves in an oscillatory manner, i.e., the twelve 2nd n.n. U atoms turn to be closer to the vacancy due to the removal of the Coulomb interactions among the cations; then the next nearest atoms adjust themselves slightly further from the defect site, and so on. In the case of the (dashed line in Fig. 6), the displacement of n.n. atoms is similar to the behavior in the previous case of the ; a vacancy hole is created, originating from repulsive forces among the four 1st n.n. U atoms. However, the displacements of the six 2nd n.n. O atoms depend on their direction with respect to the vacancy site. Though the two O atoms along the [1] direction in the upper or lower (1) plane from the vacancy site approach the vacancy site very slightly, the four O atoms in the same (1) plane as the vacancy site become rather closer to the vacancy site. These complex phenomena result from the asymmetric occupied 5f electrons at the U site, which are released from the removed O. The O removal corresponds to an additional two electrons in the crystal, and then these electrons mainly occupy the unoccupied U 5f orbitals. This situation can be seen in Fig. 4(c), as well as the asymmetry of charge density. The two additional electrons spread to U sites from the vacancy site, and the charge density around the U atoms near the vacancy is modified markedly from the cubic symmetry. In the case of interstitials at the octahedral site, the displacements of the 1st n.n. atoms may be attributed to the atomic size effect of interstitial atoms. However, displacements of further n.n. atoms also affect ionic interactions among atoms. In fact, a larger variation in 2nd n.n. than in 1st n.n. atoms is observed in both and defects; this cannot be explained simply by the size effect. Figure 6 also gives information on the convergence of the supercell size to the model of noninteracting defects. It can be seen that the changes in the volumes and bond lengths of the four defect structures become stable towards an asymptotic limit with increasing of the supercell size to U 32 O 64. The most important point manifested in Fig. 6 is that the long-range interactions exist among the defect and the surrounding atoms. In the U 32 O 64 supercell, there are still large distance variations of 2% at about.4 nm from the defect site, which is beyond the range of the U 8 O 16 cell. This suggests that a large U 32 O 64 supercell is necessary to estimate the precise defect properties. This will be discussed again in the next section. 3.4 Defect formation energies The vacancy formation energy can be defined by EV F X ¼ EV N 1 X E N þ E X ; ð1þ where EV F X is the vacancy formation energy of atom X (X ¼ U, O), EV N 1 X is the calculated free energy of a cell with defect X, E N is the free energy of an ideal crystal without a defect, and E X is the internal energy of the pure substance formed from an X-atom in the reference state. -Uranium (oc4, Cmcm) is used as the reference state of U, and an oxygen molecule in a 1. nm cubic cell is taken as the reference state of O. For -uranium, the free energy was also calculated by the GGA+U method with the same U and J values as used by Dudarev 9) in order to obtain a consistent reference state for all UO 2 calculations in the present work. The interstitial formation energy is defined in a similar way to eq. (1): EI F X ¼ EX Nþ1 E N E X ; ð2þ where EI F X is the interstitial formation energy of atom X, and EX Nþ1 is the calculated free energy of the cell when an X atom is placed at the interstitial site. The calculated formation energies of the four types of defects, U- and O-vacancies, and U- and s, in the three kinds of supercells are shown in Table 3 and plotted in Fig. 7. A gradual decrease in the values of formation energies with increasing size of the supercell can be observed. It can be seen that the has a large positive formation energy, whereas the and have roughly half the formation energy of the defect. The most noteworthy result is the negative formation energy of the in the octahedral site, which implies that the has lower energy than the O 2 molecule, and that UO 2 becomes oxidized in the presence of O, which is in agreement with the well-known fact that UO 2 is inclined to be hyperstoichiometric.

7 First-Principles Calculation of Point Defects in Uranium Dioxide 2657 Formation Energy, E/eV introduced by point defects can be explained in the picture of the ionic interactions among atoms. The important effect of the magnetism in the UO 2 system is revealed through the present calculations. We believe that our results for the supercell of UO 2 unit cells (U 32 O 64 ) provide reliable formation energies and volume relaxation values and also suggest that a larger supercell is necessary for investigating the impurity behavior with higher precision, since point defects have a wide-ranging effect, not only on the first nearest neighbor atoms of the defect but on the second nearest neighbor and on more distant atoms. Acknowledgement Table 3 also gives the values for U 8 O 16 obtained by Freyss et al. 5) and Crocombette et al. 4) as a comparison. Note that there is a large discrepancy in the values of formation energy between the two calculations, particularly for the ; our value is about double that of Freyss et al. s value, and moreover, Freyss et al. s formation energy of the O- interstitial is more than 1 times our result. In addition to the difference in values arising from the different methods, we expect that the difference in magnetic states also resulted in this discrepancy, since our calculations treated all systems as antiferromagnetic (AF) structures, whereas Freyss et al. used nonmagnetic (NM) ground states. To clarify this point, we performed the NM calculation on U 8 O 16. The results are also listed in Table 3 under NM next to the column of our AF values for U 8 O 16. One can see that our NM calculation gives very similar formation energies to Freyss et al. s values, and we can confirm the necessity of using AF configuration to obtain accurate values of formation energy. Although it is difficult to see the full convergence from Fig. 7 even in the 8 unit cells of the U 32 O 64 supercell, we believe that the defect formation energies in a large supercell would be more accurate than in small cells. 4. Conclusion Number of unit cells of UO 2 Fig. 7 Formation energies (ev) of point defects in UO 2 in three types of supercells: U 4 O 8,U 8 O 16 and U 32 O 64. We have performed first-principles electronic structure calculations of point defects in UO 2, in which we included antiferromagnetism, strong correlations, and the relaxation of lattice and atomic positions. The results form for the first time, a self-consistent dataset of formation energies and atomic distance variations of various point defects in UO 2 with relatively high precision, which provides important information on the elementary processes of complex structure formation, and provides a basis for further investigations of the thermodynamic, stability and kinetic properties of UO 2. The change in the electronic structures in UO 2 This study was financially supported by the Budget for Nuclear Research of the Ministry of Education, Culture, Sports, Science and Technology, based on the screening and counseling by the Atomic Energy Commission. REFERENCES 1) T. Sonoda, M. Kinoshita, I. L. F. Ray, T. Wissb, H. Thiele, D. Pellottiero, V. V. Rondinella and Hj. Matzke: Nucl. Instr. and Meth. B 191 (22) ) T. Petit, C. Lemaignan, F. Jollet, B. Bigot and A. Pasturel: Philos. Mag. B 77 (1998) ) T. Petit, G. Jomard, C. Lemaignan, B. Bigot and A. Pasturel: J. Nucl. Mater. 275 (1999) ) J. P. Crocombette, F. Jollet, L. Thien Nga and T. Petit: Phys. Rev. B 64 (21) ) M. Freyss, T. Petit and J. P. Crocombette: J. Nucl. Mater. 347 (25) ) F. Jollet, T. Petit, S. Gota, N. Thromat, M. Gautier-Soyer and A. Pasturel: J. Phys.: Condens. Matter 9 (1997) ) A. I. Liechtenstein, V. I. Anisimov and J. Zaanen: Phys. Rev. B 52 (1995) R5467 R547. 8) S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys and A. P. Sutton: Phys. Rev. B 57 (1998) ) S. L. Dudarev, G. A. Botton, S. Y. Savrasov, Z. Szotek, W. M. Temmerman and A. P. Sutton: phs. stat. sol. (a) 166 (1998) ) R. Laskowski, G. K. H. Madsen, P. Blaha and K. Schwarz: Phys. Rev. B 69 (24) 1448(R). 11) B. C. Frazer, G. Shirane, D. E. Cox and C. E. Olsen: Phys. Rev. 14 (1965) A1448 A ) P. Pulay: Chem. Phys. Lett. 73 (198) ) P. E. Blöchl: Phys. Rev. B 5 (1994) ) G. Kresse and J. Joubert: Phys. Rev. B 59 (1999) ) J. P. Perdew, K. Burke and M. Ernzerhof: Phys. Rev. Lett. 77 (1996) ) G. Kresse and J. Furthmüller: Phys. Rev. B 54 (1996) ) H. J. Monkhorst and J. D. Pack: Phys. Rev. B 13 (1976) ) P. E. Blöchl, O. Jepsen and O. K. Andersen: Phys. Rev. B 49 (1994) ) F. D. Murnaghan: Proc. Natl. Acad. Sci. U.S.A. 3 (1944) ) M. Idiri, T. Le Bihan, S. Heathman and J. Rebizant: Phys. Rev. B 7 (24) ) T. Petit, B. Morel, C. Lemaignan, A. Pasturel and B. Bigot: Philos. Mag. B 73 (1996) ) Y. Yun, H. Kim, H. Kim and K. Park: Int. J. Korean Nucl. Soc. Nucl. Eng. and Tech. 37 (25)