**1. Introduction**

With the increasing consumption of energy on Earth, the development of nuclear energy has attracted considerable research attention from all walks of life. Nuclear fission is a critical way to generate clean energy, and uranium dioxide (UO2) is the standard nuclear fuel used in pressurised water reactors [1]. Fission gases, such as Xe and Kr, are among the essential fission products in UO2 fuel, which can exacerbate the fuel swelling, thereby leading to the interaction between the fuel and the cladding [2–6]. As the fission products deposit energy in the surrounding material, point defects that control the microstructural evolution of the fuel occur. The point defects that survive the initial damage from irradiation in nuclear fuel form extended defects, such as vacancy clusters, dislocation loops, and voids [7]. Numerous experimental and modeling studies have been conducted to improve the understanding of the behaviour of Xe and Kr gases [7–14].

Among all volatile fission products, Xe and Kr have the highest concentration and are mainly studied herein. Previous literature mainly focused on stable configurations with a constant Xe-vacancy ratio; for example, Moore et al. [15] found that clusters of Xe atoms are formed by single Xe atoms occupying Schottky positions, which is caused by the supersaturation of Schottky vacancies in UO2. Due to the complexity of the behavioural characteristics of UO2 fuel materials and Xe bubbles, it is difficult to determine the behavioural mechanism of Xe gases. Consequently, several separation effect experiments have been proposed to simplify complex material systems by describing the physical processes of one or more fission gases to elucidate the underlying behavioural mechanisms. Thus, Zhang et al. [16] briefly explained the mechanism of UO2 by simulating molybdenum.

**Citation:** Wang, L.; Wang, Z.; Xia, Y.; Chen, Y.; Liu, Z.; Wang, Q.; Wu, L.; Hu, W.; Deng, H. Effects of Point Defects on the Stable Occupation, Diffusion and Nucleation of Xe and Kr in UO2. *Metals* **2022**, *12*, 789. https://doi.org/10.3390/ met12050789

Academic Editor: Alain Pasturel

Received: 16 March 2022 Accepted: 2 May 2022 Published: 4 May 2022

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They simulated the stable configuration of Mo by adding Xe atoms and found that Mo was most stable when the Xe-to-vacancy ratio was unity. This study compares the stable occupation of Xe/Kr clusters in perfect and varying defect-containing systems.

The diffusion of the Xe atoms in bulk UO2 or Xe-vacancy clusters formed by Xe atoms in Schottky vacancies has been studied in previous literature, and even the self-diffusion behaviour of U and O in UO2 has been studied [17–19]. Yun et al. [20] have investigated the vacancy-assisted diffusion of Xe in UO2, and calculated the incorporation, binding, and migration energies. They found that the tri-vacancy is a significant diffusion pathway of Xe in UO2. Lawrence [21] discussed the uncertainty in fission gas diffusion coefficients as a function of temperature. Higher activation energies in computing diffusion are usually compensated by higher pre-exponential factors [22]. The diffusion of cations in UO2 and other related compounds is very slow, at <10<sup>15</sup> or <1017 cm2/s [23,24], even at high temperatures from 1800 to 2000 K, which is one of the highest temperatures achieved in crystal correlation experiments. One of the most commonly used models in fuel performance codes was published by Massih and Forsberg [25–27]. Turnbull et al. [7] analysed this model and other models, and then computed the bulk fission gas diffusion rates, which capture both intrinsic and radiation-enhanced diffusion. This model divides the diffusivity into three regimes. Davies et al. [28] experimentally studied the diffusivity of UO2 at high temperatures (*D*1, T > 1650 K) and concluded that its activation energy (*Ea*) and pre-exponential factor (*D*0) were 3.04 eV and 7.6 × <sup>10</sup>−<sup>10</sup> <sup>m</sup>2/s, respectively. This study provides significant guidance for the subsequent diffusion studies of UO2 by many researchers [29–31]. The in-pile diffusion coefficient of UO2 is close to the intrinsic diffusion coefficient, so it is considered that the radiation-enhanced diffusion coefficient has high uncertainty [32]. However, due to the complex diffusion of Xe at interstitial sites, the diffusion of interstitial clusters has not presently been well described. Therefore, it is necessary to explore the diffusion behaviour of Xe/Kr clusters at octahedral interstitial sites in UO2 and to explain the relationship between the interstitial and vacancy diffusion mechanisms.

There are two main nucleation mechanisms of fission gases, such as homogeneous and heterogeneous nucleation [33]. Transmission electron microscopy (TEM) images of UO2–irradiated bubbles show that they are characterised by their high density and small, almost uniform bubble size. Nelson [34] predicted that the nucleation density of bubbles was almost independent of irradiation temperature and fission rate. Evans [35] observed bubbles in Kr- and Xe-irradiated UO2 using TEM and found that the threshold temperature for bubble nucleation was in the range of 350 ◦C–500 ◦C. Michel et al. [36] found subnanometer Xe bubbles in polycrystalline UO2 under low flux irradiation at 600 ◦C. Previous studies focused on the vital role of temperature and irradiation dose in the nucleation and growth of bubbles in UO2, but there are few studies on defect concentration. Hence, studying the formation of Xe/Kr clusters in systems with different defect concentrations is important for subsequent nucleation studies.

Thus, the influence of defects in materials on Xe/Kr gas clusters is worth further investigation. Therefore, it is crucial to investigate the effect of point defects on Xe/Kr clusters in UO2 to understand the evolution of fuels.
