**1. Introduction**

When oxide metals are irradiated by high-energy heavy ions, nano-sized protrusions and pores are produced on the surface and inside the specimen, respectively. For example, in an irradiated NiO specimen, nano-sized protrusions and cylindrically shaped nanopores were generated simultaneously [1]. Using transmission electron microscopy (TEM), Jensen et al. [2] observed hillocks at both ends of tracks in Yttrium Iron Garnet (YIG) induced by high-energy -C<sup>60</sup> ions. They believed that the matter was emitted from the ion track, leaving the spherical hillocks on the sample surface at the entrance and exit. Recently, Ishikawa et al. [3] found that hemispherical protrusions and nanopores were also generated in CeO<sup>2</sup> by high-energy ion beam irradiation. Subsequently Ishikawa et al. [4] observed ion tracks and hillocks produced by swift heavy ions of different velocities in Y3Fe5O<sup>12</sup> by TEM. They found the dimensions of the hillocks increase as a function of stopping power, *Se*. The data can be interpreted by the lifetime of the melt region produced by irradiation. Ion-irradiated CeO<sup>2</sup> was studied extensively regarding its optical reflectivity [5] and spectroscopic characteristics [6], defects [7–10], grain size effects [11], X-ray Photoelectron Spectroscopy (XPS) [12], and effects of irradiation temperature on final structure [13].

Nanopore formation is an interesting phenomenon from the viewpoint of nano-order fabrication of materials and can lead to the realization of highly functional materials such as catalysts [14]. Therefore, it is crucial to clarify the formation mechanism of nano-sized protrusions and cylindrically shaped nanopores. Since high-energy beam irradiation is a non-equilibrium process and its relaxation time is very short, it is difficult to clarify such a mechanism only by an actual experiment. Molecular dynamics (MD) simulation provides

**Citation:** Sasajima, Y.; Kaminaga, R.; Ishikawa, N.; Iwase, A. Nanopore Formation in CeO<sup>2</sup> Single Crystal by Ion Irradiation: A Molecular Dynamics Study. *Quantum Beam Sci.* **2021**, *5*, 32. https://doi.org/10.3390/ qubs5040032

Academic Editor: Lorenzo Giuffrida

Received: 15 October 2021 Accepted: 9 November 2021 Published: 18 November 2021

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a useful tool for the analysis of such a process because it can calculate the trajectories of individual atoms during a short time period.

In our previous papers, using computer-aided simulations, we studied the effects of high-energy, heavy-ion irradiation [15–19] as a theme in the development of a new generation of nuclear fuels with a high burn-up ratio. Employing MD simulations, we evaluated the disorder of a single crystal structure of specimens irradiated by fast particles. Our MD simulations elaborated the structural change followed by the high-energy dissipation of a nano-scale region in a single crystal of CeO<sup>2</sup> after irradiation [18,19]. Yablinsky et al. [20] analyzed the structure of ion tracks and investigated thermal spikes in CeO<sup>2</sup> with energy depositions using MD simulation. Medvedev et al. [21] developed the Monte-Carlo code TREKIS (Time-Resolved Electron Kinetics in swift heavy-ion Irradiated Solid) which models how a penetrating swift heavy ion (SHI) excites the electron subsystems of various solids, and the generated fast electrons spread spatially. Subsequently, the same group proposed a hybrid approach that consisted of the Monte-Carlo code TREKIS and the classical molecular dynamics code LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) for lattice atoms to simulate the formation process of a cylindrical track of about 2 nm diameter in Al2O<sup>3</sup> irradiated by Xe 167 MeV ions [22]. Then, Rymzhanov et al. [23] simulated a structure in overlapping swift heavy-ion track regions in Al2O<sup>3</sup> and found good coincidence with observation results of irradiated Al2O<sup>3</sup> by high resolution TEM. Another paper by Rymzhanov et al. [24] examined swift heavy-ion irradiation of forsterite (Mg2SiO4) and the effects of the ion energy and its energy losses on the track radius were explained, and the track formation thresholds was determined [24]. Rymzhanov et al. [25] also clarified that different ion tracks were produced in MgO, Al2O3, and Y3Al5O<sup>12</sup> (YAG) by irradiation with Xe (176 MeV) ions, whereas no ion tracks in MgO, discontinuous distorted crystalline tracks in Al2O<sup>3</sup> and continuous amorphous tracks in YAG were detected. Recently, Rymzhanov et al. [26] studied the irradiation process of MgO, CaF2, and Y3Al5O<sup>12</sup> (YAG) with fast ions. They found MgO and CaF<sup>2</sup> showed recovery of transient damage in the surface region, forming a spherically shaped nano-hillock, whereas YAG showed almost no recovery of the transient disorder, forming an amorphous hillock. The movie they attached as supplemental material demonstrates nano-hillock formation of CaF<sup>2</sup> irradiated by 200 MeV Au. Some of the above same researchers teamed with Karluši´c et al. [27] to research Al2O<sup>3</sup> and MgO irradiated under grazing incidence with an I beam of 23 MeV. In this study, they found grooves surrounded with nano-hillocks on MgO surfaces and smoother, roll-like discontinuous structures on the surfaces of Al2O3.

In the present study, we used an MD method to simulate the nanopore structure formation process in a single crystal CeO<sup>2</sup> with two free surfaces by supplying a thermal spike. Structural analysis was done for the obtained specimens. We evaluated the number of Frenkel pairs as a function of the thermal energy deposited per unit length in the specimen, *gSe,* which represents the beam strength. We classified various types of oxygen Frenkel pairs by the distance between the vacancy and the corresponding oxygen atom.
