DENSITY FUNCTIONAL THEORY CALCULATIONS OF DEFECT FORMATION ENERGIES IN ThO 2, CeO 2 and (Th, Ce)O 2 MIXED OXIDES

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1 Transactions, SMiRT-23 X, Paper ID 871 DENSITY FUNCTIONAL THEORY CALCULATIONS OF DEFECT FORMATION ENERGIES IN ThO 2, CeO 2 and (Th, Ce)O 2 MIXED OXIDES N. Kuganathan 1, P. S. Ghosh 2, R. W Grimes 1, C.O. T. Galvin 1, A. Arya 2, B.K. Dutta 3 and G. K. Dey 2 1 Department of Materials, Faculty of Engineering, Imperial College, London, SW7 2AZ, UK 2 Material Science Division, Bhabha Atomic Research Centre, Trombay, Mumbai , India 3 Human Resource Development Division, Bhabha Atomic Research Centre, Trombay, Mumbai , India ABSTRACT Fission gasses, such as Xe, formed during normal reactor operation are accommodated initially at defect sites in the fuel lattice and are known to have a deleterious effect on fuel performance, particularly at high levels of burnup. Using first-principles density functional theory, we calculate the three most stable structures of Scottky defects (SD) in ThO 2, CeO 2 and Th 0.94 Ce 0.06 O 2 and accommodate a single Xe atom in the most favourable of these to investigate how Xe atom interacts with defects. Our simulations reproduce the experimental ThO 2 and CeO 2 bulk crystal structures well, establishing the quality of the pseudopotentials and basis sets used. Most favourable Schottky defect in all three structures is SD2. In larger supercells, SD3 is dominant as observed in the classical effective potential simulation. A single Xe atom is most stable when accommodated within a SD1 and is highly unlikely to occupy the interstitial site. This is because the SD provides a larger volume to accommodate a large Xe atom. Incorporation can be further facilitated by increasing the number of Schottky defects. Substitution of Ce for Th in ThO 2 does not affect the overall trend observed in the pure (end-member) lattices due to the relatively small lattice distortions. INTRODUCTION Over the last three decades there has been increased emphasis in finding an alternative to urania (UO 2 ), which remains the main fuel component in fissile nuclear reactors. Materials that can replace UO 2 should have low cost, high abundance, high proliferation resistance and be able to accommodate high burn-up (IAEA, 2005). Thoria (ThO 2 ) has been identified as a possible alternative to urania partly because thorium is three times more abundant in the earth s crust. Thoria is a highly stable oxide and there are additional physical advantages such as higher thermal conductivity, higher melting temperature, higher corrosion resistance and lower thermal expansion compared to uranium based fuels (Chen, 2014). The use of thorium in a mixed oxide (MOX) fuels is of particular interest because they couple surplus fissile plutonium with a robust, fertile ceramic matrix (ThO 2 ) to give a viable fuel form that can be used in current and new-build water-cooled reactors (Mann, 2010). Ceria (CeO 2 ) has been used as a surrogate material to simulate PuO 2 or the other actinides with +4 valency (Mathews, 2001; Mathews, 2000; Lee, 1999). ThO 2 and CeO 2 form almost an ideal solid solution across the complete homogeneity and composition range. CeO 2 and PuO 2 have quite similar physicochemical properties, namely, ionic sizes in octahedral and cubic coordination, melting points, standard enthalpy of formation and specific heat (Tyagi, 2002). The plutonium chemistry can be effectively simulated using CeO 2 in place of highly active PuO 2. The thermo-physical properties, the bulk and lattice thermal expansion behaviour of thorium based systems with PuO 2 and the oxides of the some fission products have been investigated experimentally (Mathews, 2001; Tyagi, 2002; Barrier, 2006; Yildiz, 2007; Bukaemskiy, 2009; Mathews, 2000). Theoretical calculations based on DFT have also been

2 performed to investigate the energetic, electronic, elastic and oxygen vacancy formation and migration in Th 1-x Ce x O 2 (Sevik, 2009; Kanchana, 2006; Xiao, 2011). Fission product rare gases (Xe and Kr), produced during the fission process in nuclear fuels, affect in different ways fuel performance due to their virtual insolubility in the fuel matrix. When fission gases are released, the temperature in the system increases due to a decrease in thermal conductivity across the fuel clad gap, resulting in an increase in fuel pressure. In order to improve the fuel performance, it is necessary to understand the behaviour of these gasses. Rare gases can be accommodated at defect sites in the fuel matrix. The smallest, charge neutral group of vacancies in ThO 2 is a Schottky defect cluster (SD) which plays a significant role in the accommodation of gaseous fission products. The SD has a greater volume than an interstitial for large xenon atoms. Over the past decade, computational modelling based on DFT has been used extensively to investigate defect properties of nuclear materials. Here we employ DFT to calculate the most stable structures of SDs, Xe@SDs and the incorporation energetics of Xe in ThO 2, CeO 2 and Th 0.94 Ce 0.06 O 2. COMPUTATIONAL MODELLING The calculations were carried out using the spin-polarized mode of density functional theory as implemented in the VASP (Kresse, 1996) package for ab initio simulations. The exchange-correlation term was modelled using the generalized gradient approximation (GGA) parameterized by Perdew, Burke, and Ernzerhof (PBE) (Perdew, 1996) The standard projected augmented wave (PAW) potentials and a plane-wave basis set with the cut-off value of 500 ev were employed in the calculations. For the bulk structures, we used Monkhorst-Pack k-points for the Brillouin zone integration. For all defect structures a supercell was employed with a k-point mesh. The total energy was minimized with respect to the atomic coordinates and the lattice constants until the forces acting on the atoms were smaller than ev/å. In order to describe the behaviour of the localized Th and Ce f states, we included the orbital-dependent, Coulomb potential (Hubbard U) and the exchange parameter J within the DFT+U calculations. In this work, we used a Hubbard parameter, U = 4.5, and an exchange parameter, J = 0.5, to describe the self-interaction of 4f and 5f electron of Ce and Th respectively. Dispersion was included by using the pair-wise force field as implemented by Grimme (Grimme, 2010). RESULTS & DISCUSSION Bulk ThO 2 and CeO 2 Crystal structures of ThO 2 and CeO 2 are isomorphic, exhibiting a face-cantered cubic fluorite type lattice with experimentally determined lattice parameters: a = 5.60 (Leigh, 2006) and 5.41 Å (Duclos, 1988) respectively. Figure 1 shows this structure and the chemical environments of thorium (forming a cube with eight O atoms) and oxygen (forming a tetrahedron with four Th atoms).

3 Figure 1. Conventional cubic fluorite unit cell of ThO 2 and CeO 2 (a) Th atom surrounded by eight O atoms forming a cube (b) and O atom forming tetrahedron with four adjacent Th atoms (c). Energy minimization calculations were performed on cubic fluorite structures of ThO 2 and CeO 2 to obtain the equilibrium lattice constants and bond distances, thereby enabling an assessment (through comparison with experiment) of the quality of the pseudopotentials and basis set used for Th, Ce and O. The calculated equilibrium lattice constants, bond distances and band gap (tabulated in Table 1) are in good agreement with the experiment. Table 1. Calculated and experimental lattice constants, bond distances and band gap of bulk ThO 2 and CeO 2. Parameter ThO 2 CeO 2 calc expt calc expt a = b = c (Å) α = β = γ ( ) Th (Ce) O (Å) E gap The calculated total density of states of fluorite ThO 2 and CeO 2 are shown in Figure 2. ThO 2 is a typical insulator. The present calculated energy band gap for ThO 2 is 4.70 ev. Our calculation underestimates the band gap as reported in the other theoretical calculations (Murphy, 2014, Szpunar, 2013; Xiao, 2011) compared to the experimental value of 5.75 ev (Rodine, 1971). The underestimation of band gap is due to the drawback of exchange-correlation approximation.

4 Figure 2. Density of states calculated for (a) pure ThO 2 and (b) pure CeO 2 [Blue dot lines corresponds to the Fermi energy level]. In this work the band gap of O 2p -Ce 4f in CeO 2 is found to be 2.50 ev. Hay et al. (Hay, 2006) have calculated this band gap using different functionals and found that all of the gaps are significantly smaller than the 3.0 ev value (Wuilloud, 1984) observed experimentally. The gap with the HSE functional, however, is 3.3 ev and agrees well with experiment (Hay, 2006). Thorium/cerium mixed oxides In the literature, thoria-ceria mixed oxides with different compositions have been synthesised and characterised (Mathews, 2001; Tyagi, 2002; Barrier, 2006; Yildiz, 2007; Bukaemskiy, 2009; Mathews, 2000; Yildiz, 2007; Chen, 2014). Santos et al. have used a co-precipitation technique to synthesise sintered pellets of thoria-ceria material with the mean composition of (Th Ce 0.06 )O 2 (Santos, 1990). Our current simulation focusses on the experimental composition reported by Santos et al. In order to consider the defects in Th/Ce mixed oxides, 6% of Th (2 Th atoms out of 32) was replaced by Ce in a supercell of ThO 2. We have considered three different configurations of (Th Ce 0.06 )O 2 to find the lowest energy structure; the relaxed structures together with the relative energies are shown in Figure 3. Figure 3. Possible configurations considered for the mixing of Ce in ThO 2 supercell with relative energies. The red spheres represent oxygen, green thorium and yellow cerium.

5 Calculations show that configuration 2 is the lowest in energy although the other two configurations are energetically very close. For subsequent defect simulations, we considered the lowest energy configuration (configuration 2). In pure CeO 2, the Ce O bond distance is 2.36 Å and this value increases to 2.41 Å in the mixed oxide. In pure ThO 2, the Th O bond distance is 2.44 Å and this value stays the same in the mixed oxide. As the ionic sizes of Th and Ce are quite similar, in all relaxed composites, Ce perfectly occupies the Th positions with +4 charge showing that a homogenous solid solution can be formed with this composition. Scottky trio defects We calculated the charge neutral Schottky trio defects. There are three different Schottky trio defects possible in fluorite type structures and they are shown in Figure 4. Figure 4. Three different possible configurations of Schottky trio defects in ThO 2 or CeO 2. Our calculations predict that the Schottky 2 (SD2) defect exhibits the lowest in energy in the supercell for ThO 2, CeO 2 and (Th,Ce)O 2. Using classical effective potential simulation, Cooper et al. found that the Schottky 3 defect is the most stable for all supercell sizes (Cooper, 2014). We have considered all three defects in a supercell of ThO 2 and found that the energy difference between SD2 and SD3 defects is very small. Thus DFT calculation indicates that for actinide oxides supercell size influences the Schottky trio defects and in larger supercells the Schottky 3 defect will become dominant. Table 2 shows the relative energies calculated for the trio defects in supercells of ThO 2, CeO 2 and (Th 0.94, Ce 0.06 )O 2. Table 2. Relative energies for the Schottky trio defects calculated in the supercells of ThO 2, CeO 2 and (Th 0.94, Ce 0.06 )O 2 Schottky defect ThO 2 Relative energy (ev) CeO 2 (Th 0.94, Ce 0.06 )O 2 SD SD SD Murphy et al. have recently calculated Schottky trio defects in pure ThO 2 using DFT+GGA method and found the same trend that we identify in our calculations for ThO 2 (Murphy, 2014). Thompson et al. have also seen the similar trend in their Schottky trio defect calculation using DFT+GGA+U method in a fluorite type UO 2 supercell (Thompson, 2000).

6 trio defects We have also investigated Xe incorporation in all three Schottky defects. We find that a single Xe incorporates more favorably in SD1. Figure 5 shows DFT optimized structures for a Xe atom incorporated in three different Schottky trio defects. Figure 5. DFT optimised structures of Xe atom incorporated in (a) SD1, (b) SD2 and (c) SD3 defects in ThO 2. A Xe atom interacts more with two oxygen vacancies in SD1 than in the other two Schottky defects. In the SD1, the two oxygen vacancies are close to the interstitial Xe atom, which is roughly located at the thorium site and thus offer the roughly spherical Xe atoms a more effective volume. In the other two Schottky defects, the oxygen vacancies are further apart and the Schottky trio is elongated. The degree to which the sites can accommodate Xe is reflected in the relative energies of the defect clusters shown in the table 3. Table 3. Relative energies for a single Xe incorporated in Schottky trio defects calculated in the supercells of ThO 2, CeO 2 and (Th 0.94, Ce 0.06 )O 2. Relative energy (ev) Xe@Schottky defect ThO 2 CeO 2 (Th 0.94, Ce 0.06 )O 2 Xe@SD Xe@SD Xe@SD In all three structures, Xe is the best accommodated within a SD1 cluster and the relative energy trend is the same. This indicates that mixing a small proportion of Ce within ThO 2 does not affect the formation of the Xe defect clusters. Incorporation energy The incorporation energy (E inc ) is defined to be the energy required to incorporate one Xe atom at a preexisting defect cluster or at an interstitial site. E inc can be expressed as follows: E inc = E tot E ThO2 E Xe, (1) Where E tot is the energy of Xe@SD1 in a ThO 2 supercell or the energy of the ThO 2 supercell with an incorporated Xe, E ThO2 is the energy of the ThO 2 supercell with or without SD1, and E Xe the energy of an isolated Xe atom. The calculated incorporation energies are tabulated in Table 4.

7 Table 4. Incorporation energies for the Xe in Schottky trio defects calculated in supercells of ThO 2, CeO 2 and (Th 0.94, Ce 0.06 )O 2. Composites E inc (ev) Xe@ThO Xe@CeO Xe@(Th 0.94, Ce 0.06 )O Xe@ThO 2 _SD Xe@CeO 2 _SD Xe@(Th 0.94, Ce 0.06 )O 2 _SD We found that in all cases Xe incorporation is highly unlikely in the perfect lattices as there is not much enough volume to accommodate a large Xe atom. Creation of a SD in all three lattices facilitates the incorporation of Xe as a SD offers a greater volume. In the present study, all calculations were performed at 0 K. CONCLUSION Spin-polarized GGA+U calculations have performed to investigate the most stable structures of perfect and Schottky trio defects with and without a single Xe atom in ThO 2, CeO 2 and (Th 0.94, Ce 0.06 )O 2. Our simulations show good reproduction of experimentally observed face-cantered cubic fluorite type lattice of ThO 2 and CeO 2. The most favourable Schottky trio defect in all three structures is SD2. In larger supercells, SD3 is dominant as observed in the classical effective potential simulation. A single Xe atom is energetically most favourably accommodated within a SD1 and is highly unlikely to occupy the interstitial position of all three lattices. Incorporation becomes more favourable with pre-existing SD1 as this cluster has a more effective volume to accommodate a large spherical Xe atom. Incorporation may be more facilitated by increasing the number of Schottky defects. Substitution of small concentration of Ce in ThO 2 does not affect the overall trend observed in the pure lattices due to the relatively small associated lattice distortion. ACKNOWLEDGEMENTS We thank RCUK for financial support under the INDO-UK collaborative program and High Performance Computing (HPC) facility at Imperial College London for computational facilities. REFERENCES Barrier, D.; Bukaemskiy, A. A.; Modolo, G. (2006). Thoria, a quasi-inert matrix for actinides dispositions. Journal of Nuclear Materials, 352 (1 3), Bukaemskiy, A. A.; Barrier, D.; Modolo, G. (2009) Thermal and crystallization behaviour of ThO 2 CeO 2 system. Journal of Alloys and Compounds, 485 (1 2), Chen, C.-F.; Kelly, J.; Asphjell, Ø.; Papin, P. A.; Forsyth, R. T.; Guidry, D. R.; Safarik, D. J.; Llobet, A.(2014). Processing of ThO 2 /CeO 2 Ceramic Fuel. Journal of the American Ceramic Society, 97 (10), Cooper, M. W. D.; Rushton, M. J. D.; Grimes, R. W. (2014). A many-body potential approach to modelling the Thermos mechanical properties of actinide oxides. Journal of Physics: Condensed Matter, 26 (10), Duclos, S. J.; Vohra, Y. K.; Ruoff, A. L.; Jayaraman, A.; Espinosa, G. P.(1998). High-pressure x-ray diffraction study of CeO 2 to 70 GPa and pressure-induced phase transformation from the fluorite structure.physical Review B, 38 (11),

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