*3.2. Investigation of Results Obtained from Ultrasonic Pulse-Echo Technique of CaTiO3*

Taking into account the Christoffel equation, the connection between ultrasonic phase velocity and the stiffness matrix is given as follows:

$$\left(\mathbf{C}\_{\mathrm{ijkl}}\mathbf{l}\_{\mathrm{j}}\mathbf{l}\_{\mathrm{l}} - \rho \mathbf{V}^{2} \delta\_{\mathrm{ik}}\right) \mathbf{a}\_{\mathrm{k}} = \mathbf{0}$$

where V is the ultrasonic phase velocity, Cijkl is the general stiffness matrix, ρ is the material density, l is the orientation of propagation, α<sup>k</sup> is the polarization direction and δik is the Kronecker delta (note that i, j, k, I = 1 to 3). For the extraction and calculation of elastic constants from ultrasonic measurements based on the Christoffel equation, with the propagation in X1, X2 and X3 directions, all of the diagonal elements of the stiffness matrix are obtained. For the determination of whole constants, we cut the specimen on the edges of the surfaces perpendicular to principal directions (bezel) and the velocity was measured from the propagation of ultrasound wave normal to these planes.

Based on Equations (1)–(5) [26,27] and the measured velocity according to the Table 1, stiffness constants values were obtained. C11 is in the agreement with longitudinal distortion and longitudinal compression/tension, so C11 can be described as the hardness. Moreover, the transverse distortion is connected to the C12, and C12 is obtained from the transverse expansion correlated to the Poisson's ratio. C44 is based on the shear modulus, as well as C44 is in the settlement with C11 and C12 [26].

$$\mathbf{C}\_{11} = \rho \mathbf{V}\_{\frac{1}{\tau}}^2 \tag{1}$$

$$\text{C}\_{22} = \text{\textquotedblleft}\_{\frac{2}{2}}^{2} \right. \tag{2}$$

$$\mathbf{C}\_{\text{66}} = \rho \mathbf{V}\_{\frac{1}{2}}^2 = \rho \mathbf{V}\_{\frac{2}{1}}^2 \tag{3}$$

$$\mathbf{C}\_{12} = \sqrt{(\mathbf{C}\_{11} + \mathbf{C}\_{66} - 2\rho \mathbf{V}\_{\frac{12}{12}}^2)(\mathbf{C}\_{22} + \mathbf{C}\_{66} - 2\rho \mathbf{V}\_{\frac{12}{12}}^2)} - \mathbf{C}\_{66} \tag{4}$$

$$\mathbf{C\_{44}} = \rho \mathbf{V\_{\frac{2}{3}}^2} = \rho \mathbf{V\_{\frac{3}{2}}^2} \tag{5}$$



After substitution and calculation, C11, C12 and C44 were registered at 330.89, 93.03 and 94.91 GPa respectively. These values of CaTiO3 were in good agreement with the values submitted in the [28–30]. Moreover, with the ultrasonic technique, longitudinal and transverse waves can be utilized for determining Young's modulus quantity [31,32]. The longitudinal and transverse waves of CaTiO3 sample are shown in Figure 7. In this method, by measuring the waves velocity and density of specimen, the determination of Young's modulus quantity was carried out (Equation (6)).

$$\mathcal{E} = \frac{4\rho \left(\frac{1}{t\_\*}\right)^2 \left(3t\_\*^2 - 4t\_1^2\right)}{t\_\*^2 - t\_1^2} \tag{6}$$

where, ts and tl are differences between two echo in longitudinal and transverse waves, respectively [33,34]. According to the results shown in Figure 7, ts and tl values are calculated as 5.75 and 3.01 μs, respectively. In addition, the density of the specimen is recorded as 3857.30 Kg <sup>m</sup><sup>3</sup> , and the length of the specimen after powder pressing reached 11.21 mm. After

calculation, Young's modulus value of CaTiO3 was 153.87 GPa. This value corresponds with the value reported by Ramajo et al. [35].

**Figure 7.** Recorded signals extracted via (**a**) longitudinal waves and (**b**) transverse waves of CaTiO3 specimen.

*3.3. Calculations: Relationship between Elastic Stiffness-Compliance Constants, Young's Modulus and Planar Density Extracted through the Unit Cell, Super Cells (2* × *2* × *2) and Symmetry Cells of CaTiO3 Lattice*

Three elastic constants of CaTiO3 were calculated via the ultrasonic technique. For the cubic CaTiO3 system, the relationship between stiffness (Cij) and compliance constant (Sij) are provided in Equations (7)–(9) [27,36]. The values resulted via Equations (7)–(9) are 0.0034, −0.0007 and 0.0105 GPa for S11, S12 and S44, respectively. Furthermore, Young's modulus of each diffracted plane of CaTiO3 can be written as Equation (10) [37].

$$\mathbf{S}\_{11} = \frac{\mathbf{C}\_{11} + \mathbf{C}\_{12}}{(\mathbf{C}\_{11} - \mathbf{C}\_{12})(\mathbf{C}\_{11} + 2\mathbf{C}\_{12})} \tag{7}$$

$$\mathbf{S}\_{12} = \frac{-\mathbf{C}\_{12}}{(\mathbf{C}\_{11} - \mathbf{C}\_{12})(\mathbf{C}\_{11} + 2\mathbf{C}\_{12})} \tag{8}$$

$$\mathbf{S}\_{44} = \frac{1}{\mathbf{C}\_{44}}\tag{9}$$

$$\frac{1}{\mathbf{E\_{hkl}}} = \mathbf{S\_{11}} - 2\left[ (\mathbf{S\_{11}} - \mathbf{S\_{12}}) - \frac{1}{2}\mathbf{S\_{44}} \right] \left[ \frac{\mathbf{h^2k^2} + \mathbf{k^2l^2} + \mathbf{l^2h^2}}{\left(\mathbf{h^2} + \mathbf{k^2} + \mathbf{l^2}\right)}\right] \tag{10}$$

The planar density and Young's modulus values related to the each diffracted plane of the unit, super (2 × 2 × 2), symmetry and super (8 × 8 × 8) cells of CaTiO3 lattice are tabulated in Table 2.

**Table 2.** Planar density and Young's modulus values of the unit cell, super cells (2 × 2 × 2) and symmetry cells of CaTiO3.


### **4. Discussion**

According to Table 2 and Figures 3–6, the expanded cells have an optimum matrix, and in this case, achieving the optimum matrix is introduced as the symmetry cells. An optimum matrix is the minimum extension for a specific plane of the unit cell to a super cell from which the density plane of that plane does not change. For example, symmetry cell (optimum matrix) of (311) plane is (3 × 3 × 3), which means that after extending to a greater matrix such as (4 × 4 × 4) or (8 × 8 × 8), planar density values will be similar (Figure 5a–c). Real planar density values of each plane are obtained from the symmetry cell, because once the symmetry of each plane is reached, with the extension of that plane to infinity (real plane), the planar density does not change. In addition, to recognize the symmetry cell, knowing some parameters such as crystal lattice, locations of atoms in the planes and index of planes is essential. Therefore, to determine Young's modulus values based on the planar density of CaTiO3, the symmetry cells should be found. It is very interesting that symmetrical or real values are related to the super cells of the (8 × 8 × 8) matrix, because in matrix (8 × 8 × 8), lattice correspondence in all directions is available; therefore, real planar density values should be calculated for the super cell of (8 × 8 × 8) matrix. To confirm this, calculations of real planar density and geometry of atoms and planes of (211), (221), (311) and (222) in super cells (8 × 8 × 8) are presented in Figures 3b, 4b, 5c and 6b, respectively. It is clear that finding the exact situation of planes and geometries is sophisticated, but with when they are known, the results obtained from Young's modulus values will have fewer errors. The basic supposition is that when the planar density is raised, the motion of atoms with the mechanism of dislocation movement needs high forces. Dislocations are regions in the lattice where an additional plane of atoms have been included abstracted from an ideal crystal (without imperfections). Dislocations are caused by losing acoustic energy, and this matter will affect the values of wavelength and time of ultrasonic waves [38].

The force (W), which is needed for the movement of atoms in each plane, is obtained from Equation (11) [39].

$$\mathbf{W} = \frac{\mathbf{E}}{2(1+\mathbf{v})} \mathbf{b}^2 \mathbf{l} \tag{11}$$

In Equation (11), E is Young's modulus, b is Burgers vector, l is dislocation length and ν is Poisson's ratio. The higher value of force is in accordance with the modulus of elasticity (Young's modulus), which would be higher.

To compare Young's modulus values of CaTiO3 in a unit cell, super cells (2 × 2 × 2) and symmetry cells, the fitting of Young's modulus values extracted by each diffracted plane versus planar density values is presented in Figure 8. According to the results (shown in the Figure 8) and the straight fitting line, Young's modulus values of unit cell, super cells (2 × 2 × 2) and symmetry cells were calculated as 162.62 ± 0.4 GPa, 151.71 ± 0.4 GPa and 152.21 ± 0.4 GPa, respectively. As expected, the Young's modulus value of symmetry cells of CaTiO3 (152.21 ± 0.4 GPa) is in good agreement with experimental Young's modulus value extracted via ultrasonic-echo technique (153.87 ± 0.2 GPa). Moreover, Young's modulus value of unit cell (162.62 ± 0.4 GPa) has a greater difference with experimental Young's modulus value, and as a result, the unit cell of CaTiO3 cannot be represented as whole cells. This is because in a unit cell of CaTiO3, crystalline defects are not considered and is especially controlling of deformation, and displacement of atoms in the planes is related to the dislocation networks [40]. Further, a unit cell of CaTiO3 is not involved in imperfections (such as dislocations, Frenkel defect and Schottky defect) with respect to the super cell [41]; therefore, the slope line value of the unit cell is reported (37.23) to be less than the slope line value of super cells (2 × 2 × 2) (63.67) and symmetry cells (62.41). Consequently, the effect of imperfections in expanded cells (super cells) is very impressive, so the unit cell of CaTiO3 is considered as the ideal lattice, while symmetry cells such as (8 × 8 × 8) of CaTiO3 are real lattices [42]; this is consistent with the experimental Young's modulus. It is clear that each imperfection will be caused by a decreasing Young's modulus [43], and in Figure 8, this matter is confirmed when the Young's modulus value (intercept) in the unit cell of CaTiO3 is higher than in super cells (2 × 2 × 2) and symmetry cells. Apparently, a

unit cell of CaTiO3 is represented by the volume of a real crystal, so the unit cell is useful to acquire theoretical density. Nevertheless, calculations of planar density based on the unit cell were obtained, but with errors.

**Figure 8.** Young's modulus versus planar density values of each diffracted plane related to the (**a**) symmetry cells, (**b**) super cells (2 × 2 × 2) and (**c**) unit cell of CaTiO3.

### **5. Conclusions**


**Supplementary Materials:** The following are available online at https://www.mdpi.com/1996-1 944/14/5/1258/s1, Figure S1: Synthesis route of CaTiO3, Table S1: Crystallographic parameters of each individual XRD pattern related to the CaTiO3, Figure S2: Linear plot of modified Scherrer equation related to the CaTiO3, Figure S3: TEM image of CaTiO3 powder, Figure S4: Geometry of planes and calculations of planar density of (a) (100), (b) (110), (c) (111), (d) (200), (e) (210), (f) (211), (g) (220), (h) (221), (i) (310), (j) (311) and (k) (222) related to the unit cell of CaTiO3, Figure S5: Geometry of planes and calculations of planar density of (a) (100), (b) (110), (c) (111), (d) (200), (e) (210), (f) (211), (g) (220), (h) (221), (i) (310), (j) (311) and (k) (222) related to the super cells (2 × 2 × 2) of CaTiO3, Figure S6: Geometry of planes and calculations of planar density of (a) (100), (b) (110), (c) (111), (d) (200), (e) (210), (f) (220), (g) (310) (4 × 4 × 4) and (h) (310) (8 × 8 × 8) related to the super cells (8 × 8 × 8) of CaTiO3, Figure S7: Geometry of planes and calculations of planar density of (a) (211) super cell (4 × 4 × 4) and (b) (211) super cell (8 × 8 × 8), Figure S8: Geometry of planes and calculations of planar density of (a) (221) super cell (4 × 4 × 4) and (b) (221) super cell (8 × 8 × 8), Figure S9: 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), Figure S10: Geometry of planes and calculations of planar density of (a) (222) super cell (3 × 3 × 3), (b) (222) super cell (8 × 8 × 8).

**Author Contributions:** Conceptualization, M.R. and A.P.; methodology, M.R. and S.N.; investigation, A.V. and A.D.; data curation, A.D.; writing—original draft, M.R.; writing—review and editing, S.N. and G.J.; resources, G.J.; supervision and validation A.P. and G.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by a grant No. S-MIP-19-43 from the Research Council of Lithuania.

**Data Availability Statement:** Data supporting the findings of this study are available from the corresponding author upon request.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


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