2.2.2. Diffusion of Xe/Kr Cluster in UO2

Generally, diffusion in solids occurs with point defects [18]. The point defect concentration is thermally activated, and it increases as the temperature increases. Migration is also a thermally activated process, accelerated by an increasing temperatures. Hence, the diffusion coefficients and diffusion energy barriers of small Xe and Kr clusters (number of atoms < 6) in UO2 are calculated by the mean square displacement (MSD) method, which can intuitively reflect the strength of the self-diffusion ability of particles. A box of 10 *a*<sup>0</sup> × 10 *a*<sup>0</sup> × 10 *a*<sup>0</sup> (*a*<sup>0</sup> is the lattice constant of the UO2 fluorite structure at different temperatures from 1800 to 2300 K) containing 12,000 atoms was used. The total simulation time is up to 5 ns, with a timestep of 1 fs.

The Arrhenius's equation can express the temperature dependence of the diffusivity,

$$D = D\_0 \exp\left(-\frac{E\_d}{k\_B T}\right) \tag{4}$$

where *D* is the diffusion coefficient, *Ea* is the diffusion barrier, *T* is the temperature, *D*<sup>0</sup> is a pre-diffusion factor, and *kB* is the Boltzmann constant. Taking the logarithms of both sides of the above equation give

$$
\ln D = \ln D\_0 - \frac{E\_a}{k\_B T} \tag{5}
$$

Therefore, if the ln *D* at different simulated temperatures is obtained, *Ea* can be obtained by linear fitting, whereas the diffusion coefficient *D* at different temperatures can be obtained by the MSD method:

$$D\_T = \frac{(MSD)\_T}{2dt} = \frac{<\Delta r(t)\_T^2>}{2dt} \tag{6}$$

In the simulation process, a long simulation time and short coordinate position output intervals are used to obtain the atomic coordinate information. The results are obtained by averaging the MSD trajectory segmentation severally.
