(a)-(c) Formation energies for point defects in alumina (a) calculated using the PBE XC functional under Al-rich conditions, and calculated using the HSE06 hybrid functional under (b) Al-rich and (c) O-rich conditions, respectively. The line slopes correspond to the defect charge states according to Eq. (1) . (d)-(e) Binding energies of He interstitials to vacancies for (d) oxygen and (e) aluminum.
Formation energies for point defects in titania calculated using (a) the PBE XC functional under Ti-rich conditions as well as the HSE06 hybrid functional under, (b) Ti-rich, and (c) O-rich conditions.
Formation energies for point defects in yttria calculated using the PBE XC functional under (a) Y-rich and (b) O-rich conditions. (c) Binding energies of He interstitials to vacancies.
Formation energies of He interstitials in various oxides as a function of the free volume at the interstitial site. Filled symbols indicate values at the respective equilibrium volumina.
Migration barriers for He interstitials in several oxides as a function of the relative change in He–nearest neighbor separation. The inset shows the same data as a function of the relative change in the Voronoi volume of the He site. The light gray stripe is intended as a guide for the eye.
Binding energy of He interstitial clusters in several oxides as well as iron (Fe data from Ref. 5 ) as a function of (a) the number of He atoms in the clusters and (b) the density of He interstitial sites in the host material.
(a) and (b) Illustration of the Baker-Nutting orientation relationship. Projection of (a) rocksalt MgO and (b) body-centered cubic Fe along . The latter is rotated by 45° about the  axis such that the and directions are parallel to each other. (c) Variation of interface energy as a function of the lateral displacement of the two crystals with respect to each other.
Plane-averaged charge density and out-of-plane strain for the (a),(b) Fe—MgO and (c),(d) Fe—FeO—MgO interfaces as a function of position perpendicular to the interface plane. The colored spheres in the bottom panel indicate the atomic positions.
Helium interstitial formation energies and their respective Voronoi volumina for the (a) Fe—MgO and (b) Fe—FeO—MgO interfaces as a function of position perpendicular to the interface. Helium atoms placed in the range indicated by the horizontal bars relax into the interface.
Schematic energy landscape for He interstitial migration in an ODS steel. In the Fe matrix formation energies are high but migration barriers are low, while the opposite applies for the oxide particles. The smallest formation energies and thus the highest solubilities are predicted in the interface region. Strain fields can lead to gradients near the interface that depending on the sign of the strain field can either increase or decrease toward the interface.
Overview of computational parameters used in calculations of properties of the ideal bulk systems as well as point defects. Migration barrier calculations for Y2O3 and the rocksalt structured oxides were carried using the parameters given in brackets.
Migration barriers in eV for He interstitials in several oxides. Note that jump directions are approximate. : change in Voronoi volume of He site between initial state and saddle point normalized by volume of initial state; : change in He–nearest neighbor distance between initial state and saddle point normalized by initial neighbor distance.
Structural and electronic properties of Al2O3, TiO2, and Y2O3 from experiment and calculations. a, c: lattice parameters (Å), : rhombohedral angle (deg), , u: internal structural parameters, : band gap(eV), : electronic contribution to dielectric constant, : ionic contribution to dielectric constant, : static dielectric constant (sum of and ). The subscripts and indicate the dielectric constant perpendicular and parallel to the rhombohedral  axis (equivalent to the  axis in the hexagonal setting), respectively. Note that for alumina the thermal band gap is reduced with respect to the optical gap (given here) due to polaronic effects (Ref. 49 ). Experimental data for alumina from Refs. 49 and 50 , for titania from Refs. 51–53 , and for yttria from Refs. 54 and 55 .
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