## 3 Results and Discussion
Figure 2: The total energies of GaFe(CN)\({}_{6}\) in ferromagnetic (FM), non-magnetic (NM), and antiferromagnetic (AFM) states. Ga-N, and Fe-C in the compounds decrease with the increase of pressure, which is mainly due to the compression of the volume of the compounds under pressure and the reduction of the spacing between atoms. The pressure from 0 to 40 GPa does not cause structural transition of GaFe(CN)\({}_{6}\), because GaFe(CN)\({}_{6}\) still presents a cubic phase structure. Except for the x-direction coordinates of C and N atoms, the positions or coordinates of other atoms in compounds have not changed. The elastic constants are important parameters reflecting the mechanical stability of the compounds [44; 45]. At 0 GPa, the elastic constants C\({}_{11}\), C\({}_{12}\), and C\({}_{44}\) of GaFe(CN)\({}_{6}\) are 206.7, 53.2, and 54.6 GPa, respectively. The mechanical stability Born-Huang criteria of cubic crystal are expressed as [46; 47]:  
\[\mathrm{C}_{11}-\mathrm{C}_{12}>0,\mathrm{C}_{11}+2\mathrm{C}_{12}>0,\mathrm{ C}_{44}>0. \tag{3}\]  
The elastic constants of GaFe(CN)\({}_{6}\) at 0 GPa satisfy the above conditions, which means that GaFe(CN)\({}_{6}\) has stable mechanical properties in an equilibrium state. At the same time, it was noted that the unidirectional elastic constant C\({}_{11}\) is higher than C\({}_{44}\), which indicates that GaFe(CN)\({}_{6}\) has weaker resistance to the pure shear deformation compared to the resistance of the unidirectional compression. Some mechanical parameters can be calculated by elastic constants according to some formulas, which can be obtained in our previous studies [48]. The elastic anisotropy factor A is calculated by the following formula:  
Figure 1:0  
The elastic anisotropy factor A of GaFe(CN)\({}_{6}\) is 0.71; it is usually used to quantify the elastic anisotropy and the degree of elastic anisotropy of the compound. In general, the elastic anisotropic factor for isotropic crystals is A = 1, while for anisotropic crystals A \(\
eq\) 1. According to this criterion, GaFe(CN)\({}_{6}\) is an anisotropic compound. The Poisson's ratio, which reflects the binding force characteristics, is often between 0.25 and 0.50. The Poisson's ratio of GaFe(CN)\({}_{6}\) is 0.25, which is just in the range of values, meaning that the inter-atomic forces are central for the compounds. The Debye temperature of the GaFe(CN)\({}_{6}\) is 738.4 K, which is calculated from a formula in [47; 49]. Under the isotropic pressure, the elastic constants are transformed into the corresponding stress-strain coefficients by the following expressions:  
Figure 1:1  
The mechanical stability of GaFe(CN)\({}_{6}\) under isotropic pressure is determined by the following formula [48; 50]:  
Figure 1:2  
The P in the formula above refers to the external pressure. The curves of \(\mathrm{B}_{11}-\mathrm{B}_{12}\), \(\mathrm{B}_{11}+2\mathrm{B}_{12}\), and \(\mathrm{B}_{44}\) with pressure are plotted in Figure 1:3 \(\mathrm{B}_{11}-\mathrm{B}_{12}\) and \(\mathrm{B}_{11}+2\mathrm{B}_{12}\) increase with the increase of pressure, and also meet the mechanical stability criterion under pressure. When the pressure is greater than 35 GPa, the value of \(\mathrm{B}_{44}\) is negative, and the stability condition of \(\mathrm{B}_{44}\) is not satisfied. Generally  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Pressure** & **a (Å)** & **C-N(Å)** & **Ga-N(Å)** & **Fe-C(Å)** & **C(x,0,0)** & **N(x,0,0)** \\ \hline
0 & 10.1883 & 1.160 & 2.155 & 1.780 & 0.32533 & 0.21148 \\
5 & 10.0706 & 1.156 & 2.118 & 1.762 & 0.32508 & 0.21029 \\
10 & 9.9649 & 1.152 & 2.085 & 1.745 & 0.32492 & 0.20928 \\
15 & 9.8695 & 1.149 & 2.057 & 1.729 & 0.32481 & 0.20843 \\
20 & 9.7830 & 1.145 & 2.028 & 1.  
speaking, when the external pressure of GaFe(CN)\({}_{6}\) is less than 35 GPa, its mechanical performance is stable. Once the external pressure exceeds 35 GPa, the mechanical performance of GaFe(CN)\({}_{6}\) is unstable.