*2.1. Molecular Dynamics*

Concerning the MD simulation method, we adopted the same method as our previous work [19], except that the specimens have a free surface. Therefore, the calculation method is only outlined as follows. The Ce and O atoms were laid out to form the fluorite structure unit cell. Then, the unit cell was cloned 6 times each in <010> and <100> directions and 4 times in the <001> direction. The dimensions of the CeO<sup>2</sup> crystal were 3.25 nm × 3.25 nm × 2.33 nm, whereas those of the calculation region were 3.25 nm × 3.25 nm × 3.25 nm. There was a free space on the upper and lower sides of the CeO<sup>2</sup> crystal along the <001> direction to mimic the free surface. Figure 1a shows the unit structure of the CeO<sup>2</sup> fluorite structure and Figure 1b shows the whole system for calculation.

**Figure 1.** (**a**) The unit structure of CeO2 fluorite. (**b**) The whole system for calculation. **Figure 1.** (**a**) The unit structure of CeO<sup>2</sup> fluorite. (**b**) The whole system for calculation.

In order to calculate the interaction between the atoms, we used the potential type proposed by Inaba et al. [28]. This is the Born–Mayer–Huggins potential type, which has a function form as In order to calculate the interaction between the atoms, we used the potential type proposed by Inaba et al. [28]. This is the Born–Mayer–Huggins potential type, which has a function form as

<001> direction to mimic the free surface. Figure 1a shows the unit structure of the CeO2

fluorite structure and Figure 1b shows the whole system for calculation.

$$\phi\_{i}(r\_{i\bar{j}}) = \frac{z\_{i}z\_{j}e^{2}}{r\_{i\bar{j}}} + f\_{0}(b\_{i} + b\_{\bar{j}})\exp\left(\frac{a\_{\bar{i}} + a\_{\bar{j}} - r\_{i\bar{j}}}{b\_{\bar{i}} + b\_{\bar{j}}}\right) - \frac{c\_{i}c\_{\bar{j}}}{r\_{i\bar{j}}^{6}}\tag{1}$$

where *rij* is the distance between ions *i* and *j*, *zi* is the effective valence of an ion *i*, *e* is the electron charge, *f*<sup>0</sup> is a constant to adjust the unit, *ci* and *cj* are the parameters of the molecular interaction term, and *ai* and *bi* are the parameters of the repulsion term. For the electrostatic interactions, the Ewald method was applied. The potential cut-off was 1.37 nm for short range interaction and for the real part of the Ewald summation. The potential parameters were determined to reproduce the lattice parameters at various temperatures and the bulk modulus of CeO2. The fitted parameters for CeO2 are presented in Table 1. where *rij* is the distance between ions *i* and *j*, *z<sup>i</sup>* is the effective valence of an ion *i*, *e* is the electron charge, *f* <sup>0</sup> is a constant to adjust the unit, *c<sup>i</sup>* and *c<sup>j</sup>* are the parameters of the molecular interaction term, and *a<sup>i</sup>* and *b<sup>i</sup>* are the parameters of the repulsion term. For the electrostatic interactions, the Ewald method was applied. The potential cut-off was 1.37 nm for short range interaction and for the real part of the Ewald summation. The potential parameters were determined to reproduce the lattice parameters at various temperatures and the bulk modulus of CeO2. The fitted parameters for CeO<sup>2</sup> are presented in Table 1.

**Table 1.** Potential parameters for CeO2 [28]. **Table 1.** Potential parameters for CeO<sup>2</sup> [28].


A cylindrical region with a diameter of 1.0 nm located at the center of the calculation region in the <001> direction was considered as the irradiation beam trajectory. Using a MD method, we initially relaxed the structure at the temperature of 298 K before a high thermal

energy was applied to the cylindrical region. This temperature was set as typical ambient temperature, or room temperature. Thermal energy is a part of the energy deposited by the heavy-ion irradiation. The present study names this thermal energy as an effective stopping power, and it is described as *gSe*. The parameter *g* in *gSe* signifies the ratio of thermal energy transferred from stopping power to the lattice as the vibration energy. In the MD simulation, the high thermal energy was applied to the cylindrical region by setting the velocity of the atoms in the region to values ranging from *gSe* = 0.0 to 1.6 keV/nm with 0.1 keV/nm energy bins according to the Maxwell distribution. According to the experiment by Ishikawa et al. [3], *Se* = 32.0 keV/nm for 200 MeV Au ion. The maximum value of *gSe* for our simulation is *gSe* = 1.6 keV/nm, thus the energy transfer ratio *g* = 0.05. According to TREKIS, a 167 MeV Xe ion gives *Se* = 21, 24.9 and 25 keV/nm for MgO, Al2O<sup>3</sup> and YAG, respectively, and it gives a radial distribution of the excess lattice energy density around the trajectory of the irradiated ion (see Figure 2 in reference [25]). A rough estimation of *gSe* can be made from the excess lattice energy of MgO and Al2O<sup>3</sup> as *gSe* = 2 keV/nm and the value of g is the same order as ours. The temperature of the region outside the cylindrical region with a diameter of 2.0 nm was kept at 298 K by the velocity scaling method [29]. In <010>, <001> and <100> directions, the periodic boundary condition was considered. The simulation duration was 10,000 molecular dynamic intervals, each taking 0.3 fs, summing up to 3 ps. thermal energy was applied to the cylindrical region. This temperature was set as typical ambient temperature, or room temperature. Thermal energy is a part of the energy deposited by the heavy-ion irradiation. The present study names this thermal energy as an effective stopping power, and it is described as *gSe*. The parameter *g* in *gSe* signifies the ratio of thermal energy transferred from stopping power to the lattice as the vibration energy. In the MD simulation, the high thermal energy was applied to the cylindrical region by setting the velocity of the atoms in the region to values ranging from *gSe* = 0.0 to 1.6 keV/nm with 0.1 keV/nm energy bins according to the Maxwell distribution. According to the experiment by Ishikawa et al. [3], *Se* = 32.0 keV/nm for 200 MeV Au ion. The maximum value of *gSe* for our simulation is *gSe* = 1.6 keV/nm, thus the energy transfer ratio *g* = 0.05. According to TREKIS, a 167 MeV Xe ion gives *Se* =21, 24.9 and 25 keV/nm for MgO, Al2O3 and YAG, respectively, and it gives a radial distribution of the excess lattice energy density around the trajectory of the irradiated ion (see Figure 2 in reference [25]). A rough estimation of *gSe* can be made from the excess lattice energy of MgO and Al2O3 as *gSe* = 2 keV/nm and the value of g is the same order as ours. The temperature of the region outside the cylindrical region with a diameter of 2.0 nm was kept at 298 K by the velocity scaling method [29]. In <010>, <001> and <100> directions, the periodic boundary condition was considered. The simulation duration was 10,000 molecular dynamic intervals, each taking 0.3 fs, summing up to 3 ps.

A cylindrical region with a diameter of 1.0 nm located at the center of the calculation region in the <001> direction was considered as the irradiation beam trajectory. Using a MD method, we initially relaxed the structure at the temperature of 298 K before a high

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#### *2.2. Structure Analysis 2.2. Structure Analysis*

A vacancy is defined as the vacant site in the CeO<sup>2</sup> fluorite structure, such that the distance between the vacant site and the nearest atom from the site is larger than the atomic radius of O or Ce originally set at the vacant site, 0.097 nm (for O) and 0.138 nm (for Ce). This is what we call the "Lindemann criterion", and it may overestimate the number of vacancies. An oxygen Frenkel pair is defined as the pair of a vacant site and the oxygen atom which escaped from the site. Oxygen Frenkel pairs can be classified according to the distance between the vacant site and the oxygen atom that was originally located at the site, such as 1NN (1st nearest neighbor) Frenkel pair, 2NN Frenkel pair, and so on [30]. Figure 2 shows an example of a 1NN Frenkel pair, and Table 2 lists the distance between the vacant site and the oxygen atom for *i*NN Frenkel pairs (*i* = 1, 2, . . . , 7). We evaluated the number of vacancies and Frenkel pairs in the irradiated specimen as a function of time and as a function of *gSe*. A vacancy is defined as the vacant site in the CeO2 fluorite structure, such that the distance between the vacant site and the nearest atom from the site is larger than the atomic radius of O or Ce originally set at the vacant site, 0.097 nm (for O) and 0.138 nm (for Ce). This is what we call the "Lindemann criterion", and it may overestimate the number of vacancies. An oxygen Frenkel pair is defined as the pair of a vacant site and the oxygen atom which escaped from the site. Oxygen Frenkel pairs can be classified according to the distance between the vacant site and the oxygen atom that was originally located at the site, such as 1NN (1st nearest neighbor) Frenkel pair, 2NN Frenkel pair, and so on [30]. Figure 2 shows an example of a 1NN Frenkel pair, and Table 2 lists the distance between the vacant site and the oxygen atom for *i*NN Frenkel pairs (*i* = 1, 2, …, 7). We evaluated the number of vacancies and Frenkel pairs in the irradiated specimen as a function of time and as a function of *gSe*.

**Figure 2.** An example 1NN Frenkel pair. **Figure 2.** An example 1NN Frenkel pair.


**Table 2.** Distance between the vacant site and oxygen atom for *i*NN Frenkel pairs (*i* = 1, 2, . . . , 7).

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#### **3. Results and Discussion 3. Results and Discussion**

Structural changes of the CeO<sup>2</sup> systems after the irradiation, as viewed from <001> and <100> directions, are shown in Figures 3 and 4 for the values of *gSe* = 0.0 and 0.8 keV/nm, respectively. The number of ejected atoms increased with increasing *gSe* value. Structural changes of the CeO2 systems after the irradiation, as viewed from <001> and <100> directions, are shown in Figures 3 and 4 for the values of *gSe* = 0.0 and 0.8 keV/nm, respectively. The number of ejected atoms increased with increasing *gSe* value.

**Figure 3.** Structural change of the CeO2 systems viewed from (**a**) <001> and (**b**) <100> directions after the irradiation and for *gSe* = 0.0 keV/nm. **Figure 3.** Structural change of the CeO<sup>2</sup> systems viewed from (**a**) <001> and (**b**) <100> directions after the irradiation and for *gSe* = 0.0 keV/nm.

diameter was smaller than that of *gSe* = 0.8 keV/nm.

Figure 3 indicates that the surface of the specimen showed disorder even when *gSe* = 0.0 keV/nm. This means that the disorder on the surface is not generated by irradiation. Therefore, we neglect the surface atoms for the structural analysis in order to clarify the formation process of the nanopore. In this case, no defects were found in the interior of the specimen; however, there is the possibility that defects are created at gSe = 0.0 keV/nm. The thermal energy provides the system necessary for the creation of defects. However, the ambient temperature is room temperature that is around 1/40 eV, which is low compared with the defect formation energy. Therefore, the defect creation event is rare, and no defects were found in the calculated system. We see the creation of defects among many specimens prepared, but this is out of our scope in the present study. As can be seen from Figure 4, a nanopore was formed in the specimen by ejection of the atoms located in the central region. In the case of *gSe* = 0.4 keV/nm, a nanopore was also produced but its

**Figure 4.** Structural change of the CeO2 systems viewed from (**a**) <001> and (**b**) <100> directions after the irradiation and for *gSe* = 0.8 keV/nm. **Figure 4.** Structural change of the CeO<sup>2</sup> systems viewed from (**a**) <001> and (**b**) <100> directions after the irradiation and for *gSe* = 0.8 keV/nm.

Figure 3 indicates that the surface of the specimen showed disorder even when *gSe* = 0.0 keV/nm. This means that the disorder on the surface is not generated by irradiation. Therefore, we neglect the surface atoms for the structural analysis in order to clarify the formation process of the nanopore. In this case, no defects were found in the interior of the specimen; however, there is the possibility that defects are created at *gSe* = 0.0 keV/nm. The thermal energy provides the system necessary for the creation of defects. However, the ambient temperature is room temperature that is around 1/40 eV, which is low compared with the defect formation energy. Therefore, the defect creation event is rare, and no defects were found in the calculated system. We see the creation of defects among many specimens prepared, but this is out of our scope in the present study. As can be seen from Figure 4, a nanopore was formed in the specimen by ejection of the atoms located in the central region. In the case of *gSe* = 0.4 keV/nm, a nanopore was also produced but its diameter was smaller than that of *gSe* = 0.8 keV/nm.

It is interesting to compare this figure with our previous result for computer simulation of irradiation of CeO<sup>2</sup> single crystals with no free surface. The Ce sublattice was stable up

to *gSe* = 1.5 keV/nm, and even at *gSe* = 2.0 keV/nm only several interstitial atoms were found as defects (see Figure 4 in reference [19]). It can be considered that the Ce sublattice damage is confined near the surface of the irradiated region compared with the O sublattice damage. To elaborate the nanopore formation process, the time change of the total number of vacancies for the various values of *gSe* was calculated. *Quantum Beam Sci.* **2021**, *5*, x FOR PEER REVIEW 7 of 14 It is interesting to compare this figure with our previous result for computer simula-

Figure 5a,b show the results for *gSe* = 0.1–0.8 and 0.8–1.6 keV/nm, respectively. (No defect was found in the interior of the specimen at *gSe* = 0.0 keV/nm because of the low temperature 298 K.) For all these *gSe* cases, the number of vacancies increased abruptly up to 0.2 ps after the irradiation. It increased further and reached the maximum, then gradually decreased until 1.5 ps and equilibrated to a constant value after that. The equilibrated number of vacancies increased as *gSe* increased for the low values of *gSe* less than 0.8 keV/nm and converged to a constant value around 200 when *gSe* was larger than 0.8 keV/nm. It is noteworthy that the number of atoms in the region where vacancies were counted is 1296. In the range of *gSe* = 0.1–0.8 keV/nm, the more thermal energy given, the more disordered the structure. However, in the range of 0.8–1.6 keV/nm, the additional thermal energy cannot change the structure drastically because the structure is already disordered. If the same amount of energy is given to an ordered and a disordered structure, the increment of entropy of the ordered structure is much larger than that of the disordered structure. tion of irradiation of CeO2 single crystals with no free surface. The Ce sublattice was stable up to *gSe*= 1.5 keV/nm, and even at *gSe*= 2.0 keV/nm only several interstitial atoms were found as defects (see Figure 4 in reference [19]). It can be considered that the Ce sublattice damage is confined near the surface of the irradiated region compared with the O sublattice damage. To elaborate the nanopore formation process, the time change of the total number of vacancies for the various values of *gSe* was calculated. Figure 5a,b show the results for *gSe* = 0.1–0.8 and 0.8–1.6 keV/nm, respectively. (No defect was found in the interior of the specimen at *gSe* = 0.0 keV/nm because of the low temperature 298 K.) For all these *gSe* cases, the number of vacancies increased abruptly up to 0.2 ps after the irradiation. It increased further and reached the maximum, then gradually decreased until 1.5 ps and equilibrated to a constant value after that. The equilibrated number of vacancies increased as *gSe* increased for the low values of *gSe* less than 0.8 keV/nm and converged to a constant value around 200 when *gSe* was larger than 0.8

The distributions of vacancies in the CeO<sup>2</sup> systems viewed from <001> and <100> directions are shown in Figures 6 and 7 for the values of *gSe* = 0.2 and 0.8 keV/nm, respectively. The nanopore was formed abruptly at around 0.3 ps after the irradiation and grew to its maximum size at 0.5 ps, then it shrank during the time to 1.0 ps and finally became equilibrated. The size of the nanopore increased as *gSe* increased, but it saturated when *gSe* was 0.8 keV/nm or more. This finding will provide useful information for precise control of the size of the nanopore. keV/nm. It is noteworthy that the number of atoms in the region where vacancies were counted is 1296. In the range of *gSe* = 0.1–0.8 keV/nm, the more thermal energy given, the more disordered the structure. However, in the range of 0.8–1.6 keV/nm, the additional thermal energy cannot change the structure drastically because the structure is already disordered. If the same amount of energy is given to an ordered and a disordered structure, the increment of entropy of the ordered structure is much larger than that of the disordered structure.

**Figure 5.** *Cont*.

**250**

**300**

keV/nm.

keV/nm.

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0.8 keV/nm 0.9 keV/nm 1.0 keV/nm 1.1 keV/nm 1.2 keV/nm 1.3 keV/nm 1.4 keV/nm 1.5 keV/nm 1.6 keV/nm

(**b**)

**Figure 5.** Time change of the total number of vacancies for the values of (**a**) *gSe* = 0.1–0.8 keV/nm and (**b**) *gSe* = 0.8–1.6 **Figure 5.** Time change of the total number of vacancies for the values of (**a**) *gSe* = 0.1–0.8 keV/nm and (**b**) *gSe* = 0.8–1.6 keV/nm. trol of the size of the nanopore.

*gSe* was 0.8 keV/nm or more. This finding will provide useful information for precise con-

(**a**)

**Figure 6.** *Cont*.

(**a**)

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**Figure 6.** Distributions of vacancies in the CeO2 systems viewed from (**a**) <001> and (**b**) <100> directions for the values of *gSe* = 0.2 keV/nm. **Figure 6.** Distributions of vacancies in the CeO<sup>2</sup> systems viewed from (**a**) <001> and (**b**) <100> directions for the values of *gSe* = 0.2 keV/nm. **Figure 6.** Distributions of vacancies in the CeO2 systems viewed from (**a**) <001> and (**b**) <100> directions for the values of *gSe* = 0.2 keV/nm.

**Figure 7.** Distributions of vacancies in the CeO2 systems viewed from (**a**) <001> and (**b**) <100> directions for the values of *gSe* = 0.8 keV/nm. **Figure 7.** Distributions of vacancies in the CeO<sup>2</sup> systems viewed from (**a**) <001> and (**b**) <100> directions for the values of *gSe* = 0.8 keV/nm.

(**b**)

**Figure 7.** Distributions of vacancies in the CeO2 systems viewed from (**a**) <001> and (**b**) <100> di-

rections for the values of *gSe* = 0.8 keV/nm.

Figure 8 shows the time change of oxygen Frenkel pairs for *gSe* = 0.4, 0.8, 1.2, and 1.6 keV/nm. The number of oxygen Frenkel pairs could be separated into two categories, higher *gSe* (0.8, 1.2 and1.6 keV/nm) and lower *gSe* (0.4 keV/nm). The number of oxygen Frenkel pairs in the higher *gSe* category was larger than that of the lower *gSe* category; however, there was no clear correlation between the Frenkel pairs and *gSe* in the higher category. This behavior was similar to that for the number of vacancies and the radius of the nanopore. It should be noted that the number of oxygen Frenkel pairs is one order of magnitude lower than the total number of vacancies shown in Figure 5. This difference comes from the difference of definition of vacancy and Frenkel pairs. As shown in Table 2, the distance between the vacant site and the escaped oxygen atom should be larger than 0.234 nm. In contrast, the corresponding distances defined for the vacancy are much smaller than that for oxygen Frenkel pairs, 0.097 nm (for O) and 0.138 nm (for Ce). The time averaged distribution of the *i*NN oxygen Frenkel pairs, where *i* = 1, 2, . . . , 7, is shown for the values of *gSe* = 0.4, 0.8, 1.2, and 1.6 keV/nm in Figure 9a–d, respectively. As seen from these figures, the short-distance Frenkel pairs, 1NN to 4NN, were in the majority for lower *gSe* (0.4 keV/nm), whereas the long-distance Frenkel pairs were found for the higher values of *gSe* (0.8, 1.2 and1.6 keV/nm). Figure 8 shows the time change of oxygen Frenkel pairs for *gSe* = 0.4, 0.8, 1.2, and 1.6 keV/nm. The number of oxygen Frenkel pairs could be separated into two categories, higher *gSe* (0.8, 1.2 and1.6 keV/nm) and lower *gSe* (0.4 keV/nm). The number of oxygen Frenkel pairs in the higher *gSe* category was larger than that of the lower *gSe* category; however, there was no clear correlation between the Frenkel pairs and *gSe* in the higher category. This behavior was similar to that for the number of vacancies and the radius of the nanopore. It should be noted that the number of oxygen Frenkel pairs is one order of magnitude lower than the total number of vacancies shown in Figure 5. This difference comes from the difference of definition of vacancy and Frenkel pairs. As shown in Table 2, the distance between the vacant site and the escaped oxygen atom should be larger than 0.234 nm. In contrast, the corresponding distances defined for the vacancy are much smaller than that for oxygen Frenkel pairs, 0.097 nm (for O) and 0.138 nm (for Ce). The time averaged distribution of the *i*NN oxygen Frenkel pairs, where *i* = 1, 2, …, 7, is shown for the values of *gSe* = 0.4, 0.8, 1.2, and 1.6 keV/nm in Figure 9a–d, respectively. As seen from these figures, the short-distance Frenkel pairs, 1NN to 4NN, were in the majority for lower *gSe* (0.4 keV/nm), whereas the long-distance Frenkel pairs were found for the higher values of *gSe* (0.8, 1.2 and1.6 keV/nm). Figure 8 shows the time change of oxygen Frenkel pairs for *gSe* = 0.4, 0.8, 1.2, and 1.6 keV/nm. The number of oxygen Frenkel pairs could be separated into two categories, higher *gSe* (0.8, 1.2 and1.6 keV/nm) and lower *gSe* (0.4 keV/nm). The number of oxygen Frenkel pairs in the higher *gSe* category was larger than that of the lower *gSe* category; however, there was no clear correlation between the Frenkel pairs and *gSe* in the higher category. This behavior was similar to that for the number of vacancies and the radius of the nanopore. It should be noted that the number of oxygen Frenkel pairs is one order of magnitude lower than the total number of vacancies shown in Figure 5. This difference comes from the difference of definition of vacancy and Frenkel pairs. As shown in Table 2, the distance between the vacant site and the escaped oxygen atom should be larger than 0.234 nm. In contrast, the corresponding distances defined for the vacancy are much smaller than that for oxygen Frenkel pairs, 0.097 nm (for O) and 0.138 nm (for Ce). The time averaged distribution of the *i*NN oxygen Frenkel pairs, where *i* = 1, 2, …, 7, is shown for the values of *gSe* = 0.4, 0.8, 1.2, and 1.6 keV/nm in Figure 9a–d, respectively. As seen from these figures, the short-distance Frenkel pairs, 1NN to 4NN, were in the majority for lower *gSe* (0.4 keV/nm), whereas the long-distance Frenkel pairs were found for the higher values of *gSe* (0.8, 1.2 and1.6 keV/nm).

**Figure 8.** Time change of oxygen Frenkel pairs for *gSe* = 0.4, 0.8, 1.2, and1.6 keV/nm. **Figure 8.** Time change of oxygen Frenkel pairs for *gSe* = 0.4, 0.8, 1.2, and1.6 keV/nm. **Figure 8.** Time change of oxygen Frenkel pairs for *gSe* = 0.4, 0.8, 1.2, and1.6 keV/nm.

**Figure 9.** *Cont*.

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**Figure 9.** The time averaged distribution of *i*NN oxygen Frenkel pair, where *i* = 1, 2, …, 7, is shown for the values of *gSe* = 0.4, 0.8, 1.2, and1.6 keV/nm in (**a**–**d**), respectively. **Figure 9.** The time averaged distribution of *i*NN oxygen Frenkel pair, where *i* = 1, 2, . . . , 7, is shown for the values of *gSe* = 0.4, 0.8, 1.2, and1.6 keV/nm in (**a**–**d**), respectively.

In the case of CeO2, the displacement threshold energy for O is 20–30 eV, much smaller than that for Ce 50–60 eV [30]. Therefore, oxygen-induced defects are a sensitive indicator for the ion irradiation process. Oxygen Frenkel pairs are important because the aggregation of multiple oxygen Frenkel pairs acts as a source of dislocation loops [31]. The In the case of CeO2, the displacement threshold energy for O is 20–30 eV, much smaller than that for Ce 50–60 eV [30]. Therefore, oxygen-induced defects are a sensitive indicator for the ion irradiation process. Oxygen Frenkel pairs are important because the aggregation of multiple oxygen Frenkel pairs acts as a source of dislocation loops [31]. The dislocation

dislocation loops create a high-strain energy region, which becomes an initial point of

loops create a high-strain energy region, which becomes an initial point of crack formation and propagation in the irradiated specimen.

As can be seen from Figure 4, no hemispherical protrusion was observed on the surfaces; only disordered atoms were seen. In our simulation, the outside of the CeO<sup>2</sup> is a free space, thus cooling by adiabatic extension could not occur. As Schattat et al. [1] pointed out, cooling by adiabatic extension is critical to obtain the hemispherical protrusion, especially for crystallizing as in the CeO<sup>2</sup> case. We considered that the protrusion was in a vapor-like state initially and then was cooled significantly by adiabatic expansion into a hemispherical-shaped crystal (see Figures 2 and 5 in reference [3]). This process is significant and resembles the cluster formation process in the cluster-beam deposition method; the cylindrical region (the nanopore) acts similar to a nozzle in the cluster-beam deposition apparatus [32,33].

The animation of the formation process of the nano-hillock of CaF<sup>2</sup> presented by Rymzhanov et al. [26] suggests that a strong Coulomb interaction plays an important role in forming the nano-hillock structure with single crystalline form. Karluši´c et al. [27] simulated the formation process of the nano-hillock of MgO and Al2O<sup>3</sup> with two different charge states, equilibrium charge state *Zeff* = +8.47 and fixed charge state *Zeff* = +6. They showed that the nano-hillock was formed for the equilibrium charge state, whereas a very small nano-hillock (Al2O3) or no nano-hillock (MgO) was formed for the fixed charge state. This finding also supports the importance of the strong Coulomb interaction. For the present case of CeO2, the equilibrium charge state of Ce is *Zeff* = +2.7 and the fixed charge state is *Zeff* = +4, indicating a weak Coulomb interaction. In addition, it should be noted that hillock formation experiment/simulation for the system with a free surface was done with 23 MeV I ion giving *Se*= 8.53 keV/nm (MgO) and 9.1 keV/nm (Al2O3), estimated by the SRIM code [27], whereas the irradiation experiment/simulation for the bulk system was done with 167 MeV Xe ion giving *Se*= 21 keV/nm (MgO), 24.9 keV/nm (Al2O3) [25]. This leads to another possible mechanism for nano-hillock formation: moderately strong beam irradiation to a strong Coulomb interaction system with a free surface.
