## 3 Results and Discussion
This indicates that the tetragonal phase becomes a more stable phase with the expanding volume, and it becomes easier to transform from the cubic phase to the tetragonal phase. So, from the \(\Delta\)E\({}_{\text{total}}\)-c/a curves with different volume of Mn\({}_{3}\)Ga, it is clear that a very small change of the volume can lead to a great change of the shape of the \(\Delta\)E\({}_{\text{total}}\)-c/a curves. In other word, the phase stability of Mn\({}_{3}\)Ga Heusler alloysis very sensitive to the change of the volume. For the Mn\({}_{3}\)Ga systems, we can adjust the volume to achieve the alloys with different structures as well as possible martensitic transformations. A good way to adjust the volume is to dope similar elements into the matrix. Therefore, next, we extend the research scope to all the other Mn\({}_{3}\)Z (Z = Al, In, Tl, Si, Ge, Sn, Pb) alloys. We perform systematical investigations on \(\Delta\)E\({}_{\text{total}}\)-c/a curves for these alloys under their respective equilibrium cell volumes, which are achieved by their equilibrium lattice constant in the cubic structure. The equilibrium lattice constants in the cubic structure are 5.6 A, 5.95 A, 6.01 A, 5.53 A, 5.61 A, 5.87 A, and 6.01 A for Mn\({}_{3}\)Z (Z = Al, In, Tl, Si, Ge, Sn, Pb) alloys respectively. Their \(\Delta\)E\({}_{\text{total}}\)-c/a curves are shown in Figure 2b. For a clear analogy, the \(\Delta\)E\({}_{\text{total}}\)-c/a curve of Mn\({}_{3}\)Ga is also replotted in Figure 2b, in which one can see that, similar to Mn\({}_{3}\)Ga alloy, there are two local energy minima in the \(\Delta\)E\({}_{\text{total}}\)-c/a curves for all the other Mn\({}_{3}\)Z alloys. One energy minimum is at c/a = 1 (cubic phase), and the other is at c/a = 1.35 (tetragonal phase), except for Mn\({}_{3}\)Ge, where it is at c/a = 1.4. From Figure 2b, we can observe that when Z is cognate element, the \(\Delta\)E\({}_{\text{total}}\) of the tetragonal phase at c/a = 1.35 (for Mn\({}_{3}\)Ge at c/a = 1.4) decreases gradually with the increase of atomic number. It is also clear that the smaller the atomic number, the smaller the volume of compound. The cubic phase is more stable than the tetragonal phase for the compounds with a small volume, such as Mn\({}_{3}\)Al, Mn\({}_{3}\)Ga, Mn\({}_{3}\)Si, and Mn\({}_{3}\)Ge. And the tetragonal phase is more stable in energy for the compounds with bigger atomic number, such as Mn\({}_{3}\)In, Mn\({}_{3}\)Tl, Mn\({}_{3}\)Sn and Mn\({}_{3}\)Pb. The energy barrier between the two local energy minima is crucial to the occurrence of martensitic transformation (or reverse transformation) from the cubic (tetragonal) to tetragonal (cubic) phase. When the energy barrier is higher than the driving forces of phase transformation, the compound is stable in one of two local energy minima, and the martensitic transformation can not occur in the compound. Conversely, when the energy barrier is lower than the driving force of phase transformation,  
Figure 1: Total energy difference (per formula unit) relative to the cubic phase as a function of c/a for Mn\({}_{3}\)Ga alloy (\(\Delta\)E\({}_{\text{total}}\) = E\({}_{\text{total}}\)(c/a) \(-\) E\({}_{\text{total}}\)(c/a = 1). The lower insets show the corresponding crystal structures and the upper insets show the band structures and densities of states for the cubic (c/a = 1) and distorted (c/a = 1.35) phases. martensitic transformation may occur in the compound. In addition, from Figure 2b, it can be found that when the Z atom varies from Al to Tl (Si to Pb), the energy barrier exhibits a maximum at Mn\({}_{3}\)Ga (Mn\({}_{3}\)Sn) for \(\Delta\)E\({}_{\text{total}}\), and the local energy minimum of tetragonal phase changes from positive to negative value with the increasing atomic number. All the above results imply that the atomic radius of the main group element Z has a great influence on the volume. We can mix different Z elements to obtain Mn\({}_{3}\)Z alloys with the different volumes and stable phases.