2.3.3. Ultrasonic Pulse-Echo Technique of CaTiO3

An ultrasonic wave is a type of elastic wave spread in the medium with high frequency to obtain the Young's modulus value of samples. Mastering the ultrasonic parameters can be used to acquire more accurate values of mechanical properties [20]. Recently, different studies on mechanical properties have been done by ultrasonic techniques. Basically, the crossing of longitudinal and transverse waves in nano- or microstructures is performed at different velocities. Each returned velocity is considered as the represented properties.

For ultrasonic measurements based on the Christoffel procedure, the first cubic specimen of CaTiO3 was prepared by cold isostatic press. The schematics of ultrasonic measurement are depicted in Figure 1a. The main part of the ultrasonic system is the pulser-receiver, which creates an electric pulse and stimulates the probe. Furthermore, the produced pulses enter the specimen, and after a sweep, they can be received via a probe. In this measurement, some drops of water were utilized to prevent the depreciation of waves in the air, and the effect of hand pressure on the probe was decreased [21].

**Figure 1.** Schematic of (**a**) ultrasonic pulse instrument and (**b**) a sketch of prepared CaTiO3 sample.

At any position in the sample, a local coordinate is adjusted, such as X1, the radial coordinate; X2, the circumferential coordinate; and X3, the axial coordinate. Vi/j denotes the velocity of an ultrasound wave propagating in the Xi direction with particle displacements in the Xj direction. Vi/j with the same i and j is longitudinal, and with i = j is related to the transverse waves. For the measurement of quasi-longitudinal or quasi-transverse velocity (Vij/ij), the specimen should be cut (bezel) on the edges of the surfaces perpendicular to the X directions. A sketch of the sample is represented in Figure 1b.

### **3. Results**

### *3.1. X-ray Diffraction of CaTiO3 and Planar Density Calculations*

The XRD pattern of CaTiO3 is presented in Figure 2. The characteristic peaks of CaTiO3 correspond to the report in Ref [22]. The crystal structure of CaTiO3 is cubic, the atomic positions of Ti are at (000), Ca at ( <sup>1</sup> 2 , 1 2 , 1 <sup>2</sup> ) and O at ( <sup>1</sup> <sup>2</sup> , 0, 0), (0, <sup>1</sup> <sup>2</sup> , 0), (0, 0, <sup>1</sup> <sup>2</sup> ). According to X-ray powder diffraction results, the lattice parameter is 3.79 ± 0.02 Å, which is in good corresponds with the amount recorded in the Ref [23]. In addition, crystallographic parameters (Table S1) of CaTiO3 and analyzed data by X'Pert [24] nasiri are recorded as the cell volume = 54.44 ± 0.01 Å3 and crystal density = 4.14 ± 0.01 g/cm3, and the space group is Pm-3m. In addition, the crystal size of CaTiO3 was calculated by the Monshi–Scherrer equation (Figure S2) [25] and BET analysis. The crystal size values were registered at ~59.10 and 63.02 nm, respectively. The Monshi–Scherrer method is described in Section 2 of the supporting information. Furthermore, a TEM image of CaTiO3 is shown in Figure S3. According to the images shown in Figure S3, the size of CaTiO3 particles basically corresponds to the crystallite size, and it is clear that particles of powder have nanoscale and size can be reported almost ±50 nm.

**Figure 2.** X-ray diffraction of CaTiO3 (powder sample).

For the evaluation of cells as the results, the comprehensive calculations of the planar density of diffracted planes in the unit cell, super cells (2 × 2 × 2) and super cells (8 × 8 × 8) of CaTiO3 lattice are presented in Figures S4–S6 respectively. In addition, the locations of atoms, geometry of planes and calculations of planar density of (211) super cell (4 × 4 × 4), (211) super cell (8 × 8 × 8), (221) super cell (4 × 4 × 4), (221) super cell (8 × 8 × 8), (311) super cell (3 × 3 × 3), (311) super cell (4 × 4 × 4), (311) super cell (8 × 8 × 8), (222) super cell (3 × 3 × 3) and (222) super cell (8 × 8 × 8) are depicted briefly in Figures 3–6 respectively. Furthermore, the completed calculations with their figures are shown in Figures S7–S10.

$$\left[ \left( 2 \times \frac{78.52}{360} + 2 \times \frac{101.48}{360} + 4 \times \frac{1}{2} + 5 \right) \times \pi \left( \operatorname{r}\_{\operatorname{Ti}^{++}} \right)^2 \right] + \left( \left( \mathbb{B} \times \frac{1}{2} + 4 \right) \times \pi \left( \operatorname{r}\_{\operatorname{O}^{2-}} \right)^2 \right) \Big| + \left[ \left( \mathbb{B} \times \pi \left( \operatorname{r}\_{\operatorname{Ca+}} \right)^2 \right) \right] = 0$$
 
$$\left[ \left( \mathbb{B} \times \pi \left( 0.60 \right)^2 \right) + \left( \mathbb{B} \times \pi \left( 1.40 \right)^2 \right) + \left( \mathbb{B} \times \pi \left( 1 \right)^2 \right) \right] = 83.44$$

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**Figure 3.** *Cont.*

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**Figure 3.** Geometry of planes and calculations of planar density of (**a**) (211) super cell (4 × 4 × 4) and (**b**) (211) super cell (8 × 8 × 8) (which shows and emphasizes the symmetry of (8 × 8 × 8) super cells).

$$\mathbb{E}\left[\left(\left(2\times\frac{141.1}{640} + 4\times\frac{169.44}{360} + 4\times\frac{1}{2} + 3\right) \times \pi\left(\operatorname{r}\_{\widetilde{\Pi}}\right)^{2}\right) + \left(\left(12 + 4\times\frac{1}{2}\right) \times \pi\left(\operatorname{r}\_{\widetilde{\Pi}}\right)^{2}\right)\right] - \left[\left(7 \times \pi\left(0.60\right)^{2}\right) + \left(14 \times \pi\left(1.40\right)^{2}\right)\right] = 94.12\text{ W}$$

$$\text{Planar density} = \frac{\text{number of atoms in the plane (221)} \times \text{area of each atom in the plane (221)}}{\text{area of the plane (221)}} = \frac{94.12}{301.71} = 0.315$$

**Figure 4.** *Cont.*

$$\left[ \left( \left( 2 \times \frac{141.13}{\text{mol}} + 4 \times \frac{100.64}{\text{mol}} + 14 \times \frac{1}{2} + 19 \right) \times \pi \left( r\_{\text{Ti}^{1+\ast}} \right)^2 \right) + \left( \left( 52 + 8 \times \frac{1}{2} \right) \times \pi \left( r\_{\text{O}^{1-\ast}} \right)^2 \right) \right] \div \left[ \left( 2\text{B} \times \pi \left( 0.60 \right)^2 \right) + \left( 56 \times \pi \left( 1.40 \right)^2 \right) \right] = 376.49 \text{ J}$$

$$\text{Planar density} = \frac{\text{number of atoms in the plane (221)} \times \text{area of each atom in the plane (221)}}{\text{area of the planes (221)}} = \frac{376.49}{1206.65} = 0.315$$

**Figure 4.** Geometry of planes and calculations of planar density of (**a**) (221) super cell (4 × 4 × 4) and (**b**) (221) super cell (8 × 8 × 8) (which shows and emphasizes the symmetry of (8 × 8 × 8) super cells).

$$\left| \left( (2 \times \frac{95.74}{360} + 2 \times \frac{84.26}{360} + 2) \times \pi \left( \text{r}\_{\text{Tl}^{4+}} \right)^2 \right) \right| = (3 \times \pi \times (0.6)^2) = 3.39$$

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**Figure 5.** *Cont.*

$$\left[ \left( (2 \times \frac{95.74}{360} + 4 \times \frac{1}{2} + 3) \times \pi \left( \operatorname{r}\_{\text{Ti}^{4+}} \right)^2 \right) \right] = (5.53 \times \pi \times (0.6)^2) = 6.25$$

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$$\left[ \left( \left( 2 \times \frac{95.74}{360} + 8 \times \frac{1}{2} + 17 \right) \times \pi \left( \operatorname{tr}\_{\text{Tl}^{\dagger \text{t}}} \right)^{2} \right) \right] \\ = \{21.53 \times \pi \times \{0.6\}^{2} \} = 24.35$$
 
$$\text{Planar density} \\ = \frac{\text{number of atoms in the plane (311)} \times \text{area of each atom in the plane (311)}}{\text{min} \times \text{min}} = \frac{24.35}{\text{min}} = \frac{24.35}{\text{min}} = 1.15$$

**Figure 5.** The concept of a symmetry cell; geometry of planes and calculations of planar density of (**a**) (311) super cell (3 × 3 × 3), (**b**) (311) super cell (4 × 4 × 4) and (**c**) (311) super cell (8 × 8 × 8).

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**Figure 6.** Geometry of planes and calculations of planar density of (**a**) (222) super cell (3 × 3 × 3) and (**b**) (222) super cell (8 × 8 × 8).
