## 3 Results and Discussion
### Mechanical Properties of MgBi\({}_{2}\)O\({}_{6}\)  
To investigate the mechanical properties of tetragonal MgBi\({}_{2}\)O\({}_{6}\), the six independent elastic constants \(C_{ij}\) are calculated by applying finite distortions of the lattice. The results are listed in Table 3. For the tetragonal system, the mechanical stability criterion [32] is given by the following relationships:  
\[(C_{11}-C_{12})>0,(C_{11}+C_{33}-2C_{13})>0,\]  
\[C_{11}>0,C_{33}>0,C_{44}>0,C_{66}>0,\]  
\[(2C_{11}+C_{33}+2C_{12}+4C_{13})>0, \tag{2}\]  
The calculated values of \(C_{ij}\) in Table 3 satisfy the stability criterion mentioned above, indicating that the tetragonal MgBi\({}_{2}\)O\({}_{6}\) has mechanical stability. As listed in Table 3, \(C_{11}\) is significantly smaller  
Figure 6: Calculated optical properties of MgBi\({}_{2}\)O\({}_{6}\): (**a**) complex refractive index \(n\), (**b**) extinction coefficient \(k\), (**c**) absorption coefficient \(\alpha\), (**d**) reflectivity \(R\), and (**e**) electron energy loss function \(L\). than \(C_{33}\), indicating that the chemical bonding strength in the (100) and (010) directions is significantly weaker than the bonding strength in the (001) direction. In addition, \(C_{44}\) is obviously smaller than \(C_{66}\), which demonstrates that it is easier for shear deformation to occur along the (001) direction in comparison with the (010) direction. The shear elastic anisotropy of the material can be estimated by the relation \(A=2C_{66}/(C_{11}-C_{12})\). Typically, if \(A\) has a value of 1, meaning that the material is isotropic. The more the value of \(A\) deviates from 1, the elastic anisotropy would be more prominent. For MgBi\({}_{2}\)O\({}_{6}\), the calculated \(A\) value is 4.6, indicating that MgBi\({}_{2}\)O\({}_{6}\) is highly anisotropic. For the tetragonal system, the bulk modulus \(B\) and the shear modulus \(G\) are calculated as follows:  
\[B=(2C_{11}+C_{33}+2C_{12}+4C_{13})/9, \tag{3}\]  
\[G=(2C_{11}+C_{33}-C_{12}-2C_{13}+6C_{44}+3C_{66})/15, \tag{4}\]  
[PERSON]'s modulus \(E\) and [PERSON]'s ratio \(v\) can be estimated from the bulk and shear moduli  
\[E=9 BG/(G+3B), \tag{5}\]  
\[v=(3B-2G)/[2(3B+G)], \tag{6}\]  
The calculated \(B\), \(G\), \(E\), and \(v\) are listed in Table 3. The \(B/G\) value is about 1.81. According to [PERSON]'s criteria of brittleness and ductility [33], MgBi\({}_{2}\)O\({}_{6}\) exhibits some toughness, but it is not obvious, which is consistent with the [PERSON]'s ratio \(v=0.27\). In addition, the elastic anisotropy of MgBi\({}_{2}\)O\({}_{6}\) can be directly determined by the direction-dependent [PERSON]'s modulus. The [PERSON]'s modulus in a specific direction can be expressed by the elastic compliances (\(s^{\prime}\)):  
\[(C_{11}-C_{12})>0,(C_{11}+C_{33}-2C_{13})>0,\]0  
with  
\[(C_{11}-C_{12})>0,(C_{11}+C_{33}-2C_{13})>0,\]1  
where \(A\) is the matrix associated with the change of axes:  
\[(C_{11}-C_{12})>0,(C_{11}+C_{33}-2C_{13})>0,\]2  
Using the reduction of \(s_{ijkl}\) for the tetragonal crystal class [34], the reduced Young's modulus with orientation can be expressed as follows:  
\[(C_{11}-C_{12})>0,(C_{11}+C_{33}-2C_{13})>0,\]3  
The calculated directional dependence of [PERSON]'s modulus depicted in Figure 7 is a significantly distorted spherical shape. The calculated values along the [001], [110], and [-110] directions are obviously larger than those along the [100] and [010] directions. Tetragonal MgBi\({}_{2}\)O\({}_{6}\) exhibits the highly elastic anisotropy. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Elastic Constants (GPa)**} & \multicolumn{4}{c}{**Mechanical Moduli (GPa)**} & \(\boldsymbol{v}\) \\ \hline \(\boldsymbol{C_{11}}\) & \(\boldsymbol{C_{12}}\) & \(\boldsymbol{C_{13}}\) & \(\boldsymbol{C_{33}}\) & \(\boldsymbol{C_{44}}\) & \(\boldsymbol{C_{66}}\) & \(B\) & \(\boldsymbol{G}\) & \(\boldsymbol{E}\) & \\ \hline
171.8 & 110.4 & 98.1 & 280.3 & 66.2 & 141.9 & 137.7 & 76.2 & 193.0 & 0.27 \\ \hline \hline \end{tabular}
\end{table}