**1. Introduction**

It is well known that, in a lot of polymers, the energetic ion irradiation and the subsequent chemical etching produce one-dimensional holes with small diameters [1]. Resultant perforated membranes have been used as filters for small particles [1,2]. In some ceramics irradiated with swift heavy ions, one-dimensional areas, in which the lattice structures and the physical properties are strongly modified, are produced along the ion beam path. Such one-dimensional structures are called "ion-tracks" [3]. The iontracks originate from the energetic ion induced high-density electronic excitation, and their production mechanisms have been explained by using the thermal spike models [4–6] and by the Coulomb explosion model [7–9]. A lot of studies have been performed in order to investigate the individual ion-track structures and their dependence on the electronic stopping power, Se, of the irradiating ions [10–15]. The effects of the ion-track overlapping on lattice structures of target materials, which appear for the high fluence irradiation, have also been investigated so far [10,16–22].

Ishikawa et al. have explained the ion track overlapping effect on the TiO<sup>2</sup> amorphization by using the binomial distribution function [23]. In our recent study, we analyzed the ion-track overlapping effect on the magnetization of CeO<sup>2</sup> by using the Poisson distribution function, and succeeded in the reproduction of the experimentally observed ion-fluence dependence of the magnetization [24]. As shown later in this report, the Poisson distribution function, which is the approximated formula of the binomial distribution function under some extreme condition, can also describe the ion-track overlapping effects on the evolution of the amorphization in TiO<sup>2</sup> and the lattice structure change in ZrO2. Such analytical methods, however, only reproduce the total fraction of the areas modified by

**Citation:** Iwase, A.; Nishio, S. Simulation of Two-Dimensional Images for Ion-Irradiation Induced Change in Lattice Structures and Magnetic States in Oxides by Using Monte Carlo Method. *Quantum Beam Sci.* **2021**, *5*, 13. https://doi.org/ 10.3390/qubs5020013

Academic Editor: Lorenzo Giuffrida

Received: 9 March 2021 Accepted: 10 May 2021 Published: 13 May 2021

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high energy ions as a function of ion fluence. In the present study, we used the Monte Carlo method in order to simulate the two-dimensional images of the lattice structures and the magnetic states as a result of the ion -track overlapping for TiO2, ZrO<sup>2</sup> and CeO2. The total fractions of the modified area were calculated from the two-dimensional images, and were compared with the experimental results and those estimated by using the Poisson distribution function.
