*4.1. Amorphization of TiO<sup>2</sup> by High Energy Heavy Ion Irradiation*

Here, we refer the result of anatase TiO<sup>2</sup> by Ishikawa et al. as an example of the ion irradiation induced amorphization [21]. They irradiated TiO<sup>2</sup> films with 230 MeV Xe+15 ions and measured the x ray diffraction (XRD) spectra. They have shown that the XRD peak intensity decreases in an exponential manner as a function of ion fluence. This experimental result can be explained as a result of the overlapping of amorphous ion tracks. Along the ion beam path in TiO2, the crystal structure becomes amorphous, and the amorphous areas still remains amorphous irrespective of the number of impacts by ion-tracks. As only the non-amorphized area contributes to the x ray diffraction, the intensity of the XRD peak is proportional to the fraction of zero impact (*r* = 0) area, *A*(Φ, 0), which is not amorphized by the irradiation. By using Equation (4), the relative XRD peak intensity, *I*(Φ)/*I*(0), is related to the ion-fluence, Φ, by the following function,

$$I(\Phi)/I(0) = \, ^\prime A(\Phi, 0) \, = \, \exp(-\mathcal{S}\Phi) \tag{5}$$

or

$$\ln(I(\Phi)/I(0)) = \ln A(\Phi, 0) = -\mathbb{S}\Phi \tag{6}$$

where *I*(0) is the XRD peak intensity for the unirradiated target and *I*(Φ), the XRD peak intensity after the irradiation with the fluence of Φ. From the slope of ln(*I*(Φ)/*I*(0))−Φ plot, the cross section of the ion track, *S*, has been determined as 72.3 nm<sup>2</sup> , which corresponds to the track diameter of 9.6 nm. Because the total fraction of amorphized and non-amorphized areas is unity, the fraction of the amorphized area is expressed as,

$$1 - A(\Phi, 0) = 1 - \exp(-S\Phi),\tag{7}$$

which is well known as "the Poisson law".

The above model that TiO<sup>2</sup> sample is amorphized only inside the ion -track, and the amorphized region never contributes to the XRD diffraction seems to be too simple. The previous TEM (transmission electron microscope) observations, however, clearly show that just inside the ion-track in TiO2, the structure becomes amorphous, and outside the iontrack, the crystal structure is maintained [26,27]. The XRD peaks for TiO<sup>2</sup> samples which are almost completely amorphized by high energy heavy ion irradiation are much smaller than for the unirradiated crystalline TiO<sup>2</sup> and are scarcely observed [22]. These experimental results, therefore, justify the above model for the analysis of ion track overlapping in TiO2, and the fraction of amorphized area can be estimated by the decrease in XRD peak intensity.

The result of the Monte Carlo simulation for the Xe ion induced amorphization of TiO<sup>2</sup> is shown in Figure 1. The figure represents the two-dimensional images of amorphized (yellow) and non-amorphized (blue) areas for the ion-fluences of 5 <sup>×</sup> <sup>10</sup>11, 1 <sup>×</sup> <sup>10</sup><sup>12</sup> , <sup>2</sup> <sup>×</sup> <sup>10</sup><sup>12</sup> and 2.5 <sup>×</sup> <sup>10</sup><sup>12</sup> cm−<sup>2</sup> . The track diameter is 9.6 nm which has been determined by the experiment.

mined by the experiment.

1012, 2 × 1012 and 2.5 × 1012 cm−2. The track diameter is 9.6 nm which has been deter-

**Figure 1.** Two-dimensional images of amorphized (yellow) and non-amorphized (blue) areas for various ion fluences; (**a**) 0 cm2 (unirrradiated), (**b**) 5 × 1011 cm2, (**c**) 1 × 1012 cm2, (**d**) 2 × 1012 cm2 and (**e**) 2.5 × 1012 cm2. The diameter of ion track is assumed to be 9.6 nm. **Figure 1.** Two-dimensional images of amorphized (yellow) and non-amorphized (blue) areas for various ion fluences; (**a**) 0 cm<sup>2</sup> (unirrradiated), (**b**) 5 <sup>×</sup> <sup>10</sup><sup>11</sup> cm<sup>2</sup> , (**c**) 1 <sup>×</sup> <sup>10</sup><sup>12</sup> cm<sup>2</sup> , (**d**) 2 <sup>×</sup> <sup>10</sup><sup>12</sup> cm<sup>2</sup> and (**e**) 2.5 <sup>×</sup> <sup>10</sup><sup>12</sup> cm<sup>2</sup> . The diameter of ion track is assumed to be 9.6 nm.

Figure 2 shows the fraction of the non-amorphized area as a function of ion-fluence, which has been calculated from the two-dimensional images. The figure also shows the experimental result [21] and the result calculated using Equation (6). The result of the Monte Carlo simulation well agrees with the experimental result and that estimated by the Poisson distribution function. Figure 3 confirms that the ion fluence dependence of the fraction of the amorphous area is expressed by the Poisson law (Equation (7)). Figure 2 shows the fraction of the non-amorphized area as a function of ion-fluence, which has been calculated from the two-dimensional images. The figure also shows the experimental result [21] and the result calculated using Equation (6). The result of the Monte Carlo simulation well agrees with the experimental result and that estimated by the Poisson distribution function. Figure 3 confirms that the ion fluence dependence of the fraction of the amorphous area is expressed by the Poisson law (Equation (7)). *Quantum Beam Sci.* **2021**, *5*, x FOR PEER REVIEW 5 of 13

**Figure 2.** Fraction of non-amorphized area as a function of Xe ion fluence. Green circles, blue line, and red line represent the experimental result [21], result calculated by Equation (6) and the result from two-dimensional images, respectively. The logarithmic scale is used on the vertical axis. **Figure 2.** Fraction of non-amorphized area as a function of Xe ion fluence. Green circles, blue line, and red line represent the experimental result [21], result calculated by Equation (6) and the result from two-dimensional images, respectively. The logarithmic scale is used on the vertical axis.

**Figure 3.** Fraction of amorphized area as a function of Xe ion fluence. Green circles, blue line, and red line represent the experimental result [21], result calculated by Equation (7) and the result

Next, one of the experimental results of the lattice structure transformation by high energy ion irradiation is referred. Benyagoub et al. irradiated monoclinic zirconia with 135 MeV Ni ions, and the lattice structures of the irradiated samples were characterized by XRD [16]. They have found that the lattice structure of monoclinic ZrO2 gradually changes to the tetragonal structure by the ion irradiation. They have also shown that the evolution of the fraction of the tetragonal phase with the ion fluence shows a sigmoidal shape. This experimental result suggests that only one impact of the ion track does not cause the lattice structure transformation, but two or more impacts of the ion tracks are needed for the lattice structure transformation. This phenomenon can be explained by

*4.2. Lattice Structure Change of ZrO2 by High Energy Heavy Ion Irradiation* 

from two-dimensional images, respectively.

**Figure 3.** Fraction of amorphized area as a function of Xe ion fluence. Green circles, blue line, and red line represent the experimental result [21], result calculated by Equation (7) and the result from two-dimensional images, respectively. **Figure 3.** Fraction of amorphized area as a function of Xe ion fluence. Green circles, blue line, and red line represent the experimental result [21], result calculated by Equation (7) and the result from two-dimensional images, respectively.

#### *4.2. Lattice Structure Change of ZrO2 by High Energy Heavy Ion Irradiation*  Next, one of the experimental results of the lattice structure transformation by high *4.2. Lattice Structure Change of ZrO<sup>2</sup> by High Energy Heavy Ion Irradiation*

**Figure 2.** Fraction of non-amorphized area as a function of Xe ion fluence. Green circles, blue line, and red line represent the experimental result [21], result calculated by Equation (6) and the result from two-dimensional images, respectively. The logarithmic scale is used on the vertical axis.

energy ion irradiation is referred. Benyagoub et al. irradiated monoclinic zirconia with 135 MeV Ni ions, and the lattice structures of the irradiated samples were characterized by XRD [16]. They have found that the lattice structure of monoclinic ZrO2 gradually changes to the tetragonal structure by the ion irradiation. They have also shown that the evolution of the fraction of the tetragonal phase with the ion fluence shows a sigmoidal shape. This experimental result suggests that only one impact of the ion track does not cause the lattice structure transformation, but two or more impacts of the ion tracks are needed for the lattice structure transformation. This phenomenon can be explained by Next, one of the experimental results of the lattice structure transformation by high energy ion irradiation is referred. Benyagoub et al. irradiated monoclinic zirconia with 135 MeV Ni ions, and the lattice structures of the irradiated samples were characterized by XRD [16]. They have found that the lattice structure of monoclinic ZrO<sup>2</sup> gradually changes to the tetragonal structure by the ion irradiation. They have also shown that the evolution of the fraction of the tetragonal phase with the ion fluence shows a sigmoidal shape. This experimental result suggests that only one impact of the ion track does not cause the lattice structure transformation, but two or more impacts of the ion tracks are needed for the lattice structure transformation. This phenomenon can be explained by using the Poisson distribution function (Equation (4)) as follows. Since the total fraction of areas which are affected and not affected by ion tracks is unity:

$$\sum\_{r=0}^{\Phi} A(\Phi, r) = A(\Phi, 0) + A(\Phi, 1) + \sum\_{r=2}^{\Phi} A(\Phi, r) = 1. \tag{8}$$

According to Equation (4),

*A*(Φ, 0) = exp(−*S*Φ) (9)

and

$$A(\Phi, 1) \,=\, \mathcal{S}\Phi \exp(-\mathcal{S}\Phi) \tag{10}$$

Therefore, the fraction of two or more impacted areas, or the area of the tetragonal structure is,

$$\sum\_{r=2}^{\Phi} A(\Phi, r) = 1 - \exp(-S\Phi) - S\Phi \exp(-S\Phi). \tag{11}$$

The result of the Monte Carlo simulation is shown in Figure 4 for the lattice structure transformation of ZrO2. The figure represents the two-dimensional images of monoclinic (blue) and tetragonal (yellow) areas for the ion-fluences of 2.5 <sup>×</sup> <sup>10</sup>12, 5 <sup>×</sup> <sup>10</sup>12, 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 assumed to be 4.4 nm. This value was determined by the comparison of the experimental data and the result calculated by Equation (11).

using the Poisson distribution function (Equation (4)) as follows. Since the total fraction

0 2 <sup>Φ</sup> <sup>=</sup> <sup>Φ</sup> <sup>+</sup> <sup>Φ</sup> <sup>+</sup> <sup>Φ</sup> <sup>=</sup>

*r*= *r*

( , ) ( ,0) ( ,1) ( , ) 1

Therefore, the fraction of two or more impacted areas, or the area of the tetragonal

( , ) 1 exp( ) exp( )

The result of the Monte Carlo simulation is shown in Figure 4 for the lattice structure transformation of ZrO2. The figure represents the two-dimensional images of monoclinic (blue) and tetragonal (yellow) areas for the ion-fluences of 2.5 × 1012, 5 × 1012, 1.5 × 1013 and 3 × 1013 cm−2. The track diameter is assumed to be 4.4 nm. This value was determined by the comparison of the experimental data and the result calculated by Equation (11).

<sup>Φ</sup> <sup>=</sup> <sup>−</sup> <sup>−</sup> <sup>Φ</sup> <sup>−</sup> <sup>Φ</sup> <sup>−</sup> <sup>Φ</sup>

*A r S S S*

Φ =

*A r A A A r* . (8)

*A*(Φ,0) = exp(−*S*Φ) (9)

*A*(Φ,1) = *S*Φexp(−*S*Φ) (10)

. (11)

of areas which are affected and not affected by ion tracks is unity:

Φ

2

*r*

Φ =

According to Equation (4),

and

structure is,

**Figure 4.** Two-dimensional images of tetragonal (yellow) and monoclinic (blue) areas for various ion fluences; (**a**) 0 cm2 (unirrradiated), (**b**) 2.5 × 1012 cm2, (**c**) 5 × 1012 cm2, (**d**) 1.5 × 1013 cm2 and (**e**)3 × 1013 cm2. The diameter of ion track is assumed to be 4.4 nm. **Figure 4.** Two-dimensional images of tetragonal (yellow) and monoclinic (blue) areas for various ion fluences; (**a**) 0 cm<sup>2</sup> (unirrradiated), (**b**) 2.5 <sup>×</sup> <sup>10</sup><sup>12</sup> cm<sup>2</sup> , (**c**) 5 <sup>×</sup> <sup>10</sup><sup>12</sup> cm<sup>2</sup> , (**d**) 1.5 <sup>×</sup> <sup>10</sup><sup>13</sup> cm<sup>2</sup> and (**e**) 3 <sup>×</sup> <sup>10</sup><sup>13</sup> cm<sup>2</sup> . The diameter of ion track is assumed to be 4.4 nm.

*Quantum Beam Sci.* **2021**, *5*, x FOR PEER REVIEW 7 of 13

Figure 5 shows the fraction of the tetragonal structure area as a function of ion-fluence, which has been calculated from the two-dimensional images. The figure also shows the experimental result and the result estimated using Equation (11). The result of the Monte Carlo simulation well agrees with the experimental result and that calculated by using the Poisson distribution function with the track cross section, *<sup>S</sup>*, of 1.5 <sup>×</sup> <sup>10</sup>−<sup>13</sup> cm<sup>2</sup> , or the track diameter of 4.4 nm. Benyagoub et al. have shown that the similar equation to Equation (11) can reproduce their experimental result for ZrO<sup>2</sup> [16]. The equation used in ref. [16] was deduced through a quite complicated manner [28]. Meanwhile, in our case, Equation (11) can be simply given as an approximated formula of the binomial distribution function. Figure 5 shows the fraction of the tetragonal structure area as a function of ion-fluence, which has been calculated from the two-dimensional images. The figure also shows the experimental result and the result estimated using Equation (11). The result of the Monte Carlo simulation well agrees with the experimental result and that calculated by using the Poisson distribution function with the track cross section, *S*, of 1.5 × 10−13 cm2, or the track diameter of 4.4 nm. Benyagoub et al. have shown that the similar equation to Equation (11) can reproduce their experimental result for ZrO2 [16]. The equation used in ref. [16] was deduced through a quite complicated manner [28]. Meanwhile, in our case, Equation (11) can be simply given as an approximated formula of the binomial distribution function.

**Figure 5.** Fraction of tetragonal structure area of ZrO2 as a function of ion fluence. Green circles, blue line, and red line represent the experimental result [16], result calculated by Equation (11) and the result from two-dimensional images, respectively. The value of S used in Equation (11) is 1.5 × 10<sup>−</sup>13 cm2 which corresponds to the track diameter of 4.4 nm. *4.3. Change in Magnetic States of CeO2 by High Energy Heavy Ion Irradiation*  **Figure 5.** Fraction of tetragonal structure area of ZrO<sup>2</sup> as a function of ion fluence. Green circles, blue line, and red line represent the experimental result [16], result calculated by Equation (11) and the result from two-dimensional images, respectively. The value of S used in Equation (11) is 1.5 <sup>×</sup> <sup>10</sup>−<sup>13</sup> cm<sup>2</sup> which corresponds to the track diameter of 4.4 nm.

Concerning the appearance of magnetism in CeO2 at room temperature, a lot of ex-

suggested that defects of O anions and Ce3+ state of cations somehow contribute to the magnetism of CeO2 [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 CeO2 [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 CeO2 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
