*4.3. Change in Magnetic States of CeO<sup>2</sup> by High Energy Heavy Ion Irradiation*

Concerning the appearance of magnetism in CeO<sup>2</sup> at room temperature, a lot of experimental and theoretical studies have been performed [29]. Although the mechanism of the appearance of the magnetism has never been fully clarified, previous studies have suggested that defects of O anions and Ce3+ state of cations somehow contribute to the magnetism of CeO<sup>2</sup> [29]. The measurements of EXAFS (extended x-ray absorption fine structure) and XPS (X-ray photoelectron spectroscopy) using synchrotron radiation facilities have revealed that 200 MeV Xe ion irradiation induces the oxygen deficiency around Ce cations in CeO2, and the resultant change in valence state of cations from Ce4+ to Ce3+ [30,31]. The change in Ce valence state due to oxygen disorders has also been confirmed by the first principles calculation [32]. The SQUID (super quantum interference device) measurement shows that the irradiation with 200 MeV Xe ions induces the ferromagnetic state in CeO<sup>2</sup> [24,33]. These experimental and theoretical results imply that the appearance of the magnetism is attributed to the magnetic moment of localized 4f electrons on Ce3+ cations. Takaki and Yasuda have shown by the TEM observations that 200 MeV Xe ion irradiation produces one-dimensional defective regions (ion-tracks) in CeO<sup>2</sup> samples, and that only inside the ion-tracks, the arrangement of oxygen atoms is preferentially disordered [11]. Based on such previous results, we use the following model for the analysis of the ion-track overlapping effect on the magnetic state of Xe+14 ion irradiated CeO2. Only inside the ion track, disorders of oxygen atoms and the resultant Ce3+ valence state are produced, and the ferro magnetic state appears. Outside the ion track, the sample is still nonmagnetic. With increasing Xe ion fluence, arrangements of not only oxygen atoms, but also cerium atoms become disordered by the ion track overlapping, resulting in the decreases in magnetization [24,33]. The effect of the ion track overlapping on the magnetic states for CeO<sup>2</sup> irradiated with 200 MeV Xe+14 ions is, therefore, more complicated than the cases of TiO<sup>2</sup> or ZrO2. Our previous paper has shown that if the following effect of track overlapping on the saturation magnetization is assumed, the ion fluence dependence of the saturation magnetization calculated by using the Poisson distribution function well reproduces the experimental result [24]. The saturation magnetization is M<sup>0</sup> = 0 emu/g, M<sup>1</sup> = 0.1 emu/g, M<sup>2</sup> = 0.05 emu/g, M<sup>3</sup> = 0.025 emu/g and M<sup>4</sup> = 0.005 emu/g for the nonimpacted area (r = 0), one-impacted area (r = 1), two-impacted area (r = 2), three-impacted area(r = 3), and the area for four or more impacts (r > =4), respectively. From Equation (4), *A*(Φ,*r*) for r = 0, 1, 2, 3 and for r of 4 or more impacts, are given by,

$$\begin{aligned} A(\Phi, 0) &= \exp(-S\Phi) \\ A(\Phi, 1) &= \left( S\Phi \right) \exp(-S\Phi) \\ A(\Phi, 2) &= \frac{\left( S\Phi \right)^2}{2} \exp(-S\Phi) \\ A(\Phi, 3) &= \frac{\left( S\Phi \right)^3}{6} \exp(-S\Phi) \\ A(\Phi, r > 4) &= 1 - \sum\_{r=0}^{3} A(\Phi, r) \end{aligned} \tag{12}$$

and the irradiation induced saturation magnetization as a function of ion fluence is given as,

$$M(\Phi) = M\_1 \cdot A(\Phi, 1) + M\_2 \cdot A(\Phi, 2) + M\_3 \cdot A(\Phi, 3) + M\_4 \cdot A(\Phi, r > -4) \tag{13}$$

Equation (13) will be used later for the comparison with the result of the Monte Carlo simulation.

The result of the Monte Carlo simulation is shown in Figures 6 and 7 for the transition of the magnetic states of CeO2. The figure represents the two-dimensional images of areas having different saturation magnetization for the ion-fluence of 2.5 <sup>×</sup> <sup>10</sup>12, 5 <sup>×</sup> <sup>10</sup><sup>12</sup> , 1.5 <sup>×</sup> <sup>10</sup><sup>13</sup> and 3 <sup>×</sup> <sup>10</sup><sup>13</sup> cm−<sup>2</sup> . The track diameter is 4.7 nm which has been determined by the experiment [24].

*Quantum Beam Sci.* **2021**, *5*, x FOR PEER REVIEW 9 of 13

**Figure 6.** Two-dimensional images of irradiation-induced saturation magnetization of CeO2 for ion fluences of (**a**) 0 cm2 (unirradiated), and (**b**) 2.5 × 1012 cm2. The diameter of ion track is assumed to be 4.7 nm. The correspondence relationship for colors and the values of magnetization is shown in Table 1. **Table 1.** Correspondence relationship for colors in Figures 6 and 7, the number of ion track im-**Figure 6.** Two-dimensional images of irradiation-induced saturation magnetization of CeO<sup>2</sup> for ion fluences of (**a**) 0 cm<sup>2</sup> (unirradiated), and (**b**) 2.5 <sup>×</sup> <sup>10</sup><sup>12</sup> cm<sup>2</sup> . The diameter of ion track is assumed to be 4.7 nm. The correspondence relationship for colors and the values of magnetization is shown in Table 1.

pacts and saturation magnetization. **Color Number of Ion Track Impacts, r Saturation Magnetization (emu/g) Mi Table 1.** Correspondence relationship for colors in Figures 6 and 7, the number of ion track impacts and saturation magnetization.


**Figure 7.** Two-dimensional images of irradiation-induced magnetization for ion fluences of (**a**) 5 × 1012 cm2, (**b**) 1.5 × 1013 cm2 and (**c**) 3 × 1013 cm2. The diameter of ion track is assumed to be 4.7 nm. The correspondence relationship for colors and the values of magnetization is shown in Table 1. The correspondence relationship for colors in Figures 6 and 7, the number of ion track **Figure 7.** Two-dimensional images of irradiation-induced magnetization for ion fluences of (**a**) 5 <sup>×</sup> <sup>10</sup><sup>12</sup> cm<sup>2</sup> , (**b**) 1.5 <sup>×</sup> <sup>10</sup><sup>13</sup> cm<sup>2</sup> and (**c**) 3 <sup>×</sup> <sup>10</sup><sup>13</sup> cm<sup>2</sup> . The diameter of ion track is assumed to be 4.7 nm. The correspondence relationship for colors and the values of magnetization is shown in Table 1.

Monte Carlo simulation well agrees with the experimental result and that calculated by

Figure 8 shows the saturation magnetization of CeO2 as a function of ion-fluence,

impacts, and the saturation magnetization is shown in Table 1.

using the Poisson distribution function.

The correspondence relationship for colors in Figures 6 and 7, the number of ion track impacts, and the saturation magnetization is shown in Table 1.

Figure 8 shows the saturation magnetization of CeO<sup>2</sup> as a function of ion-fluence, which has been calculated from the two-dimensional images. The figure also shows the experimental result [24] and the result calculated using Equation (13). The result of the Monte Carlo simulation well agrees with the experimental result and that calculated by using the Poisson distribution function. *Quantum Beam Sci.* **2021**, *5*, x FOR PEER REVIEW 11 of 13

**Figure 8.** Saturation magnetization of CeO2 as a function of ion fluence. Green circles, blue line, and red line represent the experimental result [24], result calculated by Equation (13) and the result from two-dimensional images, respectively. The value of S used in Equation (13) is 1.7 × 10<sup>−</sup><sup>13</sup> cm2 corresponding to the track diameter of 4.7 nm. **Figure 8.** Saturation magnetization of CeO<sup>2</sup> as a function of ion fluence. Green circles, blue line, and red line represent the experimental result [24], result calculated by Equation (13) and the result from two-dimensional images, respectively. The value of S used in Equation (13) is 1.7 <sup>×</sup> <sup>10</sup>−<sup>13</sup> cm<sup>2</sup> corresponding to the track diameter of 4.7 nm.

#### **5. Discussion**  In the previous section, by using the Monte Carlo method, we have simulated the **5. Discussion**

two-dimensional images for the three kinds of the ion track overlapping effects on the oxides irradiated with high energy heavy ions. In the case of the amorphization of TiO2, as can be seen in Figure 1, the target is dotted with disk-shaped amorphized areas with the same diameter for small ion fluence, because only one impact of the ion-track can make a target amorphous. With increasing the ion fluence, the overlapping of amorphous tracks occurs more frequently, and for higher ion fluences, most part of the target becomes amorphous. For the crystal phase transformation of ZrO2, Figure 4 shows that the areas of the tetragonal phase, which are surrounded by the monoclinic area, appear for the small ion fluence. The tetragonal phase areas have various shapes, which are far from diskshape. This is due to the fact that only one ion-track impact does not cause the crystal phase transformation, but two or more ion-track impacts are needed for the crystal phase transformation. As can be seen in Figures 1, 4, 6, and 7, the overlapping of ion -tracks leads to the modulated lattice and magnetic structures with a nanometer scale. Therefore, the Monte Carlo simulation provides a good first approach for understanding the nanometersized two-dimensional structures of oxides, which are produced by the overlapping of the ion-tracks. Moreover, the complementary usage of the Monte Carlo simulation with some experimental techniques of precise imaging, such as TEM, AFM (atomic force microscope), MFM (magnetic force microscope), and PEEM (photo-emission electron microscope), will also be useful in order to promote the study of the effect of high energy ion irradiation in materials. In the present report, we have only mentioned the lattice structure and magnetic property changes by the ion-track overlapping. If the electrical conductivity appears inside the ion–tracks in insulators, and the overlapping of the ion–tracks affects their electrical conductivity, the formation of continuous electron paths or the conducting network In the previous section, by using the Monte Carlo method, we have simulated the two-dimensional images for the three kinds of the ion track overlapping effects on the oxides irradiated with high energy heavy ions. In the case of the amorphization of TiO2, as can be seen in Figure 1, the target is dotted with disk-shaped amorphized areas with the same diameter for small ion fluence, because only one impact of the ion-track can make a target amorphous. With increasing the ion fluence, the overlapping of amorphous tracks occurs more frequently, and for higher ion fluences, most part of the target becomes amorphous. For the crystal phase transformation of ZrO2, Figure 4 shows that the areas of the tetragonal phase, which are surrounded by the monoclinic area, appear for the small ion fluence. The tetragonal phase areas have various shapes, which are far from disk-shape. This is due to the fact that only one ion-track impact does not cause the crystal phase transformation, but two or more ion-track impacts are needed for the crystal phase transformation. As can be seen in Figure 1, Figure 4, Figure 6, and Figure 7, the overlapping of ion -tracks leads to the modulated lattice and magnetic structures with a nanometer scale. Therefore, the Monte Carlo simulation provides a good first approach for understanding the nanometer-sized two-dimensional structures of oxides, which are produced by the overlapping of the ion-tracks. Moreover, the complementary usage of the Monte Carlo simulation with some experimental techniques of precise imaging, such as TEM, AFM (atomic force microscope), MFM (magnetic force microscope), and PEEM (photo-emission electron microscope), will also be useful in order to promote the study of the effect of high energy ion irradiation in materials.

will drastically change the macroscopic electrical property of the insulators. The two-dimensional images of ion track overlapping, which are simulated by the Monte Carlo In the present report, we have only mentioned the lattice structure and magnetic property changes by the ion-track overlapping. If the electrical conductivity appears inside the

ion–tracks in insulators, and the overlapping of the ion–tracks affects their electrical conductivity, the formation of continuous electron paths or the conducting network will drastically change the macroscopic electrical property of the insulators. The two-dimensional images of ion track overlapping, which are simulated by the Monte Carlo method, may also be helpful for the understanding of such the percolation behavior in ion-irradiated insulators.

The ion-track overlapping effects have been analyzed so far by using complicated manners [20,28,34]. In the present study, the Monte Carlo method has confirmed that the effects of the ion-track overlapping can surely be described by a much simpler formula, the Poisson distribution function, which is the approximated formula of the binomial distribution function.
