## 3 Results
### Defect Reactions versus Individual Defects  
In the preceding discussion, we have focused on the properties of isolated point defects. However, in reality, charged defects are never created in isolation, but by means of a series of reactions involving multiple defects; this is typically described via Kroger-Vink notation [71]. These reactions obey strict conservation rules of mass, charge, and site (maintenance of constant ratio of cationic to anionic sites in the structure) balance. There have been several theoretical studies [9; 27; 28; 72] that develop a point defect reaction model for UO\({}_{2}\) in order to predict defect concentration, formation energies, lattice constant, and transport properties under thermodynamic conditions prevailing in experiments [26]. Figure 8: Relative change (\(\Delta d\)/\(d_{0}\)) in nearest neighbor distances (solid circles) induced by the uranium vacancy in the neutral (blue) and charged (4\(-\)) (red) state in: (**a**) \(2\times 2\times 3\) and (**b**) \(3\times 3\times 3\) DFT supercells. The dashed lines mark the % fraction volume change of the entire supercell, relative to a defect-free neutral system, to show the overall expansion or contraction of the lattice in the respective cases. In this section, we employ a similar approach based on a point defect model to compute defect reaction volumes of the net reaction, as opposed to the individual constituents. We illustrate this by using the example of the oxygen interstitial and the uranium vacancy, which are more likely to form in UO\({}_{2}\) under oxidizing/transmutation conditions. The reaction for incorporating oxygen at an interstitial site is given as  
\[\frac{1}{2}\text{U}_{3}\text{O}_{8}(s)+V_{i}^{\times}\rightleftharpoons\text{O} _{i}^{\prime\prime}+2h.+\frac{3}{2}\text{U}\text{O}_{2}(s) \tag{12}\]  
where \(V_{i}^{\times}\) is a vacant octahedral interstitial site, in the fluorite structure, O\({}_{i}^{\prime\prime}\) is the oxygen ion at the interstitial site, and \(h.\) represents an U\({}^{5+}\) ion (hole) not bounded to the oxygen interstitial O\({}_{i}^{\prime\prime}\). The volume change, \(\Delta\Omega^{\prime\prime\prime\prime}\), for the reaction (Equation (12)) is then given as the sum of the individual relaxation volumes (as listed in Table 2, without any corrections to volumes), \(\Delta\Omega^{\prime}\), obtained from separate DFT supercell calculations of the charged (2-) oxygen interstitial and the U\({}^{5+}\) ion,  
\[\Delta\Omega_{\text{O}_{i}}^{\prime\prime\prime\prime}=\Delta\Omega^{\prime} \big{(}\text{O}_{i}^{\prime\prime}\big{)}+2\Delta\Omega^{\prime}(h.)+\frac{3 }{2}\Omega_{\text{U}\text{O}_{2}}-\frac{1}{2}\Omega_{\text{U}\text{3}\text{O} _{8}} \tag{13}\]  
The reaction volume \(\Delta\Omega_{\text{O}_{i}}^{\prime\prime\prime\prime}\) for the oxygen interstitial reaction is computed to be \(-25.8\) A\({}^{3}\), implying contraction of the lattice due to formation of this overall neutral collection of defects (note that, in this calculation, the interstitial and the holes are assumed to be infinitely far apart). The sum of the relaxation volumes of charged oxygen interstitial and holes is \(-3.2\) A\({}^{3}\), similar to the relaxation volume, \(\Delta\Omega^{\prime}\big{(}\text{O}_{i}^{\times}\big{)}\) of a neutral oxygen interstitial (\(-3.38\) A\({}^{3}\)) calculated within a DFT supercell, which is not surprising if the neutral O interstitial can be understood as a charged oxygen interstitial with two bound holes. Similarly, a defect reaction for the formation of a uranium vacancy, is given as:  
\[\text{U}_{\text{U}}^{\times}+\text{U}_{3}\text{O}_{8}(s)\rightleftharpoons V_{\text{U}}^{\prime\prime\prime\prime}+4h.+4(\text{U}\text{O}_{2})_{SRG} \tag{14}\]  
where point defects \(V_{\text{U}}^{\prime\prime\prime\prime}\) and \(h.\) are the fully charged uranium and U\({}^{5+}\) ion (hole), respectively, and \(\big{(}\text{U}\text{O}_{2}\big{)}_{SRG}\) represents the corresponding lattice atoms present at a site of repeatable growth (SRG) such as a surface, dislocation or grain boundary.