## 3 Results and Discussion
[PERSON] [17] calculated the elastic constants of \(\rm C_{11},C_{12},\) and \(\rm C_{44}\), as 286.71,  
Figure 2: Total energy of LaFeO\({}_{3}\), EuFeO\({}_{3}\), and GdFeO\({}_{3}\) structures as a function of strain under the influence of three types of deformation tensor (E\({}_{0}\) is total energy of equilibrium structure). 116.46, and 125.61 for BaVO\({}_{3}\), respectively; all of which are at the same order of the elastic constants of RFeO\({}_{3}\) (R = La, Eu, Gd). In addition, the mechanical properties of SrTiO\({}_{3}\) and SrZrO\({}_{3}\) have been investigated by [PERSON] et al. [18]. They reported the elastic constants C\({}_{11}=318.35\), C\({}_{12}=100.04\), and C\({}_{44}=110.20\) and C\({}_{11}=309.71\), C\({}_{12}=71.96\), and C\({}_{44}=73.78\) for SrTiO\({}_{3}\) and SrZrO\({}_{3}\), respectively [18]. The difference in the results (the values of the elastic constants) can be caused due to the influence of some structural factors in these materials such as structural and relative density, metal-oxygen bond strength and interactions between central cation with the scaffold. Born stability criteria [19] is a valid criterion to investigate mechanical stability of elastic stiffness constants. For cubic structures these criteria are written as:  
\[\text{C}_{11}>0,\text{ }\text{C}_{11}-\text{C}_{12}>0,\text{ }\text{C}_{11}+2\text{C}_{12}>0,\text{ }\text{C}_{44}>0 \tag{11}\]  
According to the Equation (11), all our obtained elastic constants of RFeO\({}_{3}\), as reported in Table 2, are satisfied by Born stability criterion. Furthermore, the calculated bulk, Young's, and shear modulus and Poisson's ratio for all structures using Equations (3)-(10) are listed in Table 3. As can be seen in Table 3, it is clear that with changing the central cation from the left to the right in the periodic table, the bulk modulus decreases. In each period of the periodic table from left to right, the effective nuclear charge on the outer electrons, and accordingly, the nuclear attraction on the valence band electrons increase. Reducing the atomic radius of the central cation increases the compressibility, which has an inverse relation with the bulk modulus. In Table 4, shear velocity (V\({}_{\text{t}}\)), compressional velocity (V\({}_{\text{l}}\)), sound velocity (V\({}_{\text{m}}\)), and the Debye Temperature (\(\theta_{\text{D}}\)). The average sound velocity (\(\
u_{\text{m}}\)) is calculated using Equation (12):  
\[\
u_{\text{m}}=\left[\frac{1}{3}\left(\frac{2}{\
u_{1}^{3}}+\frac{1}{\
u_{ \text{t}}^{3}}\right)\right]^{-\frac{1}{3}} \tag{12}\]  
where \(\
u_{\text{t}}\) and \(\
u_{\text{l}}\) are the shear velocity, and compressional velocity, respectively, and are dependent on bulk (B) and shear (G) modulus as follows:  
\[\rm B_{H}=\frac{\rm B_{V}+B_{R}}{2} \tag{7}\]0  
\[\rm B_{H}=\frac{\rm B_{V}+B_{R}}{2} \tag{7}\]1  
The Debye temperature is another physical quantity, which is related to elastic constants, specific heat, and melting point. The relation between Debye temperature and the average sound velocity (\(\
u_{\text{m}}\)) is described in Equation (15):  
\[\rm B_{H}=\frac{\rm B_{V}+B_{R}}{2} \tag{7}\]2  
where h, k, NA, n, M, and \(\rho\) represent the [PERSON]'s constant, [PERSON]'s constant, [PERSON]'s number, atoms in the cation, molecular mass, and density, respectively. The Debye temperature of RFeO\({}_{3}\) (R = La, Eu, Gd) ranged from 503.1 K for GdFeO\({}_{3}\) to 792 K for LaFeO\({}_{3}\), as can be seen in \[\rm B_{H}=\frac{\rm B_{V}+B_{R}}{2} \tag{7}\]3 The anisotropy of the Young's and shear modulus of LaFeO\({}_{3}\), EuFeO\({}_{3}\), and GdFeO\({}_{3}\) is illustrated in the form of 3D surface contours in Figure 3 using ELATE code [20]. For an isotropic crystal, one sees a \begin{table}
\begin{tabular}{c c c c} \hline  & **C\({}_{\text{11}}\) (GP)** & **C\({}_{\text{12}}\) (GP)** & **C\({}_{\text{44}}\)(GP)** \\ \hline LaFeO\({}_{3}\) & 278.06 & 154.63 & 92.92 \\ EuFeO\({}_{3}\) & 242.64 & 148.81 & 45.34 \\ GdFeO\({}_{3}\) & 215.56 & 156.80 & 34.77 \\ \hline \end{tabular}
\end{table}