## 3 Results and Discussion
Red, gray, and dark green spheres represent Cu, Ti, and Sn atoms, respectively. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Space Group**} & \multicolumn{3}{c}{**Lattice Constants (Å)**} & \multicolumn{3}{c}{**Angle (\({}^{\circ}\))**} & \multicolumn{3}{c}{**Total Energy**} \\ \cline{3-10}  & & \(a\) & \(b\) & \(c\) & \(\alpha\) & \(\beta\) & \(\gamma\) & **(eV)** \\ \hline Model 1 & _P6/mmm_ & 5.110 & 5.110 & 4.523 & 90.00 & 90.00 & 120.05 & \(-\)33.094 \\ Model 2 & _Cmmm_ & 5.134 & 4.980 & 4.650 & 90.00 & 90.00 & 119.01 & \(-\)33.926 \\ Model 3 & _Cmmm_ & 5.113 & 5.113 & 4.728 & 90.00 & 90.00 & 123.22 & \(-\)33.099 \\ Model 4 & _P-3m1_ & 4.419 & 4.419 & 5.636 & 90.00 & 90.00 & 120.00 & \(-\)35.232 \\ Model 5 & _P3m1_ & 4.520 & 4.520 & 5.445 & 90.00 & 90.00 & 120.00 & \(-\)34.730 \\ Model 6 & _P-3m1_ & 4.425 & 4.425 & 5.665 & 90.00 & 90.00 & 120.00 & \(-\)35.239 \\ \hline \end{tabular}
\end{table} 
Table 3: Space groups, lattice constants, and total energies of Models 1–6, showing Model 6 is preferred. Cu\({}_{2}\)SnTi\({}_{3}\) (space group: 65 \(Cmmm\)) with an energy gain of 0.832 eV from a metastable state. The transition phase of the orthorhombic Cu\({}_{2}\)SnTi\({}_{3}\) finally becomes the most stable phase of the trigonal structure Cu\({}_{2}\)SnTi\({}_{3}\) (space group: 164 \(P\)-3\(m\)1), and the energy gain obtained from the transition state is 1.313 eV. Note that the energy barrier of the Cu\({}_{2}\)SnTi\({}_{3}\) phase transition is ignored and only a schematic diagram is drawn. A simulated SAD of Cu\({}_{2}\)SnTi\({}_{3}\) along the [001], [\(-\)101], and [221] in Figure 5 using the optimized parameters and atomic structures of Models 1, 2, and 6 (see Figure 3) shows that the simulated SAD pattern along the [221] zone axes is very similar to the experimental SAD pattern (see Figure 2b), which represents orthorhombic Cu\({}_{2}\)SnTi\({}_{3}\). This finding is remarkably similar to that predicted by our detailed simulations as shown in the energy diagram in Figure 4.  
In order to understand the XRD analysis of the CaCu\({}_{5}\)-like (InNi\({}_{2}\)-like) Cu\({}_{2}\)SnTi\({}_{3}\) models, we simulate XRD curves of Models 1-6 to compare the experimental one as shown in Figure 6. The main diffraction peaks at 40.10 (40.70) degrees 20 of Model 2 (Model 6) and 40.52 degrees 20 of experimental XRD have only a slight error of 1.03% (0.44%). To compare the experimental XRD depicted in Figure 6, the diffraction peaks at 34.90 (58.10) degrees 20 of Model 2 (Model 6) has a maximum error of 3.0% (2.0%), and the other characteristic peak errors are less than 1.5%. In view of the fact that hexagonal, orthorhombic, and trigonal structure Cu\({}_{2}\)SnTi\({}_{3}\) all have a relatively small difference in XRD, it is perhaps not surprising that all three phases exist. Figure 7 shows the ternary Cu-Sn-Ti phase diagram, including Phases A, B, C, D, and M. Three phases of Cu-23 Ti-17 Sn, Phases A, B, and M, exist in equilibrium with each other at the temperature of 900 \({}^{\circ}\)C. Phase M decomposes into Phases C and D under slow cooling conditions. The composition triangle becomes a quadrangle under the furnace cooling conditions, i.e., Phases A, B, C, and D.  
Figure 5: Density functional theory (DFT) simulated SAD of Models 1, 2, and 6 along the [001], [\(-\)101], and [221].