## 3 Results and Discussion
It should be noted that a thermoelastic martensitic transformation from cubic to tetragonal phase may also occur in the Mn\({}_{3}\)Z alloys, since the energy barrier can be flexibly regulated by the mixture of different Z elements. So, the Mn\({}_{3}\)Z alloys have the potential to be developed into a series of magnetic shape memory alloys originating from thermoelastic martensitic transformation. It is well known that with the increase of the distance between the main group Z atom and the nearest neighbor Mn(A) and Mn(C), the hybridization strength of the p-d orbitals between Z and Mn(A/C) atoms is weakened [34]. Before we start to analyze the magnetic properties, we perform an investigation on the change of the relative position of the atoms in Mn\({}_{3}\)Z alloys during the tetragonal distortion. As shown in Figure 3a, the distance between Z and Mn(B) and the distance between Mn(A) and Mn(C) along the c axis increase linearly with the increase of the c/a ratio in the process of  
Figure 2: (a) Total energy as functions of c/a ratio for Mn\({}_{3}\)Ga alloy with different volume (V = 17.0 nm\({}^{3}\), 18.0 nm\({}^{3}\), 19.0 nm\({}^{3}\), 20.0 nm\({}^{3}\), 21.0 nm\({}^{3}\), 22.0 nm\({}^{3}\)). (b) Total energy difference (per formula unit) relative to the cubic phase as a function of c/a for Mn\({}_{3}\)Z (Z = Al, Ga, In, Tl) and (c) for Mn\({}_{3}\)Z (Z = Ge, Sn, Pb) alloys. tetragonal distortion, while the distance between \(Z\) and Mn(B) and the distance between Mn(A) and Mn(C) along the a or b axis decreases linearly with the increase of the c/a ratio. Figure 3b show the changing curve of the distance between two nearest-neighbor atoms with the change of c/a ratio. It is clear that the distance of between two nearest neighbor atoms decreases first, and then increases with the increase of the c/a ratio and get a minimum at c/a = 1. Next, we will compare and analyze the atomic magnetic moments for the Mn\({}_{3}\)Ga alloy with different volumes, and all the other Mn\({}_{3}\)Z alloys with the equilibrium volume, which are shown in Figures 4 and 5. As we know, with the increase of the lattice constant, the distance between atoms increases, which leads to the weakening of covalent hybridization between atoms. The weakened covalent hybridization will result in an increase of atomic magnetic moments in the Heusler alloys [35; 36]. When we compress the lattice along the c-axis (c/a < 1), we see that the dependence of the magnetic moment of Mn(A/C) on c/a ratio shows three different tendency ranges with the change of the volume for Mn\({}_{3}\)Ga alloy, as shown in Figure 4a. (1) When the volume is small (V = 17 nm\({}^{3}\)), the magnetic moment of Mn(A/C) atom shows a sharp decrease with the increase of c/a. (2) When the volume is in the range of 18 nm\({}^{3}\)-20 nm\({}^{3}\), the magnetic moment of Mn(A/C) almost remains constant with the increase of c/a, which indicates that the moment is very stable against the compressive strain along the c-axis. (3) When the volume is higher than 21 nm\({}^{3}\), the magnetic moment of Mn(A/C) increases slowly with the increase of c/a. Figure 4: The atomic magnetic moments of Mn(A/C) (**a**) and Mn(B) (**b**) as functions of c/a ratio for Mn\({}_{3}\)Ga alloy with different lattice constants. Figure 3: The change of interatomic distance in the process of tetragonal distortion for Mn\({}_{3}\)Ga alloy (the lattice constant of cubic structure is 5.65 Å): (**a**) The red circle represents the distance between Ga and Mn(B) and the distance between Mn(A) and Mn(C) along c axis. The blue triangle represents the distance between Ga and Mn(B) and the distance between Mn(A) and Mn(C) along a or b axis. (**b**) The changing curve of the distance between two nearest neighbor atoms with the change of c/a. The thick lines between the atoms in the inset indicate the corresponding interatomic distance). The three different changing trends of the Mn(A/C) moment are due to the change of covalent hybridization between these atoms with the changing tetragonal distortion ratio [37].