## 3 Results And Discussion
### Mechanical Properties
In other words, typical isotropic materials have a Poisson's ratio of \(0.25<\
u<0.5\). The Poisson's ratio is 0.33 for copper and 0.5 for rubber. The present Poisson's ratio ranged between 0.28 and 0.34, which suggests these materials are intermetallic compounds. Another physical quantity, B/G, describes the brittle or ductile behavior of materials as suggested by Pugh and known as [PERSON]'s ratio. The critical value for the B/G ratio is 1.75. A high B/G ratio is associated with ductility, whereas a low value corresponds to brittleness. At zero pressure, our calculated results for all the APoO\({}_{3}\) perovskites can be classified as ductile materials since all B/G ratios were larger than 1.75. The Poisson's \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline  & \(C_{11}\) & \(C_{12}\) & \(C_{44}\) & B & G & E & \(A_{f}\) & \(\
u\) & B/G \\ \hline BePoO\({}_{3}\) & 237.0 & 46.8 & 18.8 & 110.2 & 38.50 & 103.5 & 0.198 & 0.344 & 2.86 \\ MgO\({}_{3}\) & 247.3 & 52.4 & 22.8 & 117.4 & 42.77 & 114.4 & 0.234 & 0.337 & 2.74 \\ CaPoO\({}_{3}\) & 261.1 & 56.3 & 26.8 & 124.6 & 47.54 & 126.5 & 0.262 & 0.330 & 2.62 \\ SrPoO\({}_{3}\) & 248.2 & 55.6 & 35.0 & 119.8 & 53.24 & 139.1 & 0.363 & 0.306 & 2.25 \\ BaPoO\({}_{3}\) & 232.9 & 57.9 & 44.8 & 116.2 & 58.78 & 150.9 & 0.512 & 0.284 & 1.98 \\ RaPoO\({}_{3}\) & 225.9 & 60.6 & 49.6 & 115.7 & 60.93 & 155.5 & 0.600 & 0.276 & 1.90 \\ \hline \end{tabular}
\end{table} 
Table 2: Calculated elastic constants \(C_{11}\) (GPa), \(C_{12}\) (GPa), \(C_{44}\) (GPa), bulk modulus B (GPa), shear modulus G (GPa), [PERSON]’s modulus E (GPa), anisotropy factor \(A_{f}\)(arb. unit), [PERSON]’s ratio \(\
u\) (arb. unit), and [PERSON]’s ratio B/G (arb. unit) of the APoO\({}_{3}\) perovskite. ratio can also be used to classify the brittleness and ductility as \(v<1/3\) and \(v>1/3\), respectively. Moreover, the so-called Cauchy parameter \(C_{12}-C_{44}\) also determines the brittleness and ductility of a material. The Cauchy relation's judgement is simple; the material is ductile if the Cauchy parameter is positive; otherwise, the material is brittle. Our calculated Cauchy parameters were all positive, which implies the ductile behavior of APoO\({}_{3}\) perovskites. The Zener anisotropy factor \(A_{f}\) can be used to predict whether the material is isotropic or anisotropic. It takes the value of 1 for isotropy material. As shown in Table 2, the calculated anisotropy was larger than one, indicating that APoO\({}_{3}\) perovskites are elastically anisotropic. Corresponding to the Debye theory, the Debye temperature \(\Theta_{D}\) of a material correlates to the elastic properties and the thermodynamic properties. The Debye model assumes that the solid is elastic continuum in which all sound waves travel at the same velocity independent of their wavelength. Indeed, \(\Theta_{D}\) is the temperature of a crystal's highest normal mode of vibration. The lattice vibrations also determine the thermodynamic properties such as thermal expansion and specific heat of a solid. Besides, it also corresponds to the structural stability and the strength of bonds. The Debye temperature can be estimated from the average elastic wave velocity:  
\[\Theta_{D}=\frac{h}{k_{B}}\sqrt[3]{\frac{3n}{4\pi V}}v_{m}, \tag{8}\]  
where \(h\) is the Planck constant, \(k_{B}\) is the Boltzmann constant, \(n\) is the number of atoms per formula, \(V\) is the atomic volume, and \(v_{m}\) is the average sound velocity. To obtain average sound velocity \(v_{m}\), we first calculate the longitudinal, \(v_{l}\), and transverse, \(v_{t}\), sound velocity from [PERSON]'s equation:  
\[v_{l}=\sqrt{\frac{3B+4G}{3\rho}},v_{t}=\sqrt{\frac{G}{\rho}} \tag{9}\]  
and  
\[v_{m}=\frac{1}{\sqrt[3]{\frac{1}{3}(\frac{1}{v_{l}^{2}}+\frac{2}{v_{l}^{2}})}}, \tag{10}\]  
Calculated \(v_{l}\), \(v_{t}\), \(v_{m}\), and \(\Theta_{D}\) values are listed in Table 3; indeed, there are no experimental and theoretical results in APoO\({}_{3}\) perovskite. The Debye temperatures for most crystals were around 200 to 400 Kelvin.