## 3 Results and Discussion  
Crystal structure characterization based on high resolution synchrotron radiation X-ray diffraction shows that the Prussian blue analogue of GaFe(CN)\({}_{6}\) is a cubic crystal with space group \(Fm\overline{3}m\), as shown in Figure 1. The structure of GaFe(CN)\({}_{6}\) is formed with FeC\({}_{6}\) and GaN\({}_{6}\) octahedrons, which are equivalent to ABX\({}_{3}\) type perovskite with vacancy in A site. In the structure of GaFe(CN)\({}_{6}\), the -Ga-N\(\equiv\)C-Fe- chain is formed between gallium, carbon, nitrogen, and iron atoms. Experimentally, the lattice constant of GaFe(CN)\({}_{6}\) was measured as 10.0641 A at 273 K [36], and the occupied positions of each atom in the structure are shown in Table 1.  
In order to obtain the theoretical equilibrium lattice constant and the ground state properties of GaFe(CN)\({}_{6}\), we constructed supercells based on experimental structural parameters and calculated the total energy of ferromagnetic (FM), non-magnetic (NM), and antiferromagnetic (AFM) states of GaFe(CN)\({}_{6}\) under different lattice constants. The ground state is determined based on the principle \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Atom**} & \multicolumn{3}{c}{**Exp.**} & \multicolumn{3}{c}{**Present**} \\ \cline{2-7}  & **x** & **y** & **z** & **x** & **y** & **z** \\ \hline Ga & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ Fe & 0.5 & 0.0 & 0.0 & 0.5 & 0.0 & 0.0 \\ C & 0.3043 & 0.0 & 0.0 & 0.3253 & 0.0 & 0.0 \\ N & 0.1883 & 0.0 & 0.0 & 0.2114 & 0.0 & 0.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Atomic occupied positions in GaFe(CN)\({}_{6}\). Figure 1: Crystal structure of GaFe(CN)\({}_{6}\). (**a**) Side view; (**b**) top view. that the lower the energy is, the more stable the structure is. The total energies of GaFe(CN)\({}_{6}\) in FM, NM, and AFM states are drawn in Figure 2. Obviously, FM states have lower total energy than NM and AFM states, which means the ferromagnetic state is the most stable for GaFe(CN)\({}_{6}\). The equilibrium lattice constant obtained at the same time was 10.1883 A. This result is slightly larger than the experimental result, and the deviation is 1.23% compared with the experimental result, which is within a reasonable range. The coordinates of the positions of the atoms in the equilibrium state of GaFe(CN)\({}_{6}\) are also listed in Table 1. Excepting that the x coordinates of C and N atoms deviate from the experimental data, the other results are consistent with the experimental values. In order to study the effect of pressure on the crystal structure of GaFe(CN)\({}_{6}\), the pressure measurement of GaFe(CN)\({}_{6}\) was carried out at intervals of 5.0 GPa under pressure of 0-40 GPa. The variation of relative lattice constant a/a\({}_{0}\) and relative volume V/V\({}_{0}\) with pressure was obtained, as shown in Figure 3. Among them, a\({}_{0}\) is the equilibrium lattice constant at 0 GPa and V\({}_{0}\) is the cell volume at 0 GPa. As can be seen from Figure 3, the lattice constant decreases gradually with the increase of external pressure, resulting in the corresponding decrease of volume V and relative volume V/V\({}_{0}\). In order to further understand the variation of structural parameters with pressure, the curve of Figure 3 is fitted and calculated, and the binary quadratic state equations of a/a\({}_{0}\) and V/V\({}_{0}\) of GaFe(CN)\({}_{6}\) and pressure are obtained, as shown below. \[\text{a/a}_{0}=0.99645-0.0017\text{P}+5.71387\times 10^{-5}\text{P}^{2} \tag{1}\]  
\[\text{V/V}_{0}=0.98777-0.00475\text{P}+4.05769\times 10^{-4}\text{P}^{2} \tag{2}\]  
Table 2 gives the structural parameters of GaFe(CN)\({}_{6}\) under pressure. The lattice constant at 40 GPa is 9.4828 A, which is only 93.1% of the lattice constant at 0 GPa. The bond lengths of C-N,  
Figure 3: The variation of relative lattice constant a/a\({}_{0}\) and relative volume V/V\({}_{0}\) with pressure.