## 3 Results and Discussions
In addition, the effects of the US on the \(E_{bg}\) and \(E_{HM}\) are also shown in Figures 8 and 9 for the ZrRhTiZ (Z = Al, Ga) compounds, respectively. The obtained results revealed both \(E_{HM}\) and \(E_{bg}\) first increase and then decrease with the increasing lattice constant. Figure 5: Schematic diagram of possible d-d hybridization between the transition-metal elements Zr-dd, Rh-dd, and Ti-3d in the ZrRhTiZ (Z = Al, Ga) compounds. The main group of elements Z (Z = Al, Ga) are not taken into account because the sp-bands are located at deep energy levels and barely contribute to the gap formation. Figure 6: \(Cont.\)  
Figure 6: Plots for total spin magnetic moment and atomic spin magnetic moment as a function of lattice constant for ZrRhTiAl (**a**) and ZrRhTiGa (**b**) compounds. Figure 1:0 Conduction band minimum (CBM) and valence band maximum (VBM) in the spin-down channel as a function of lattice constant (US) for ZrRhTiAl (**a**) and ZrRhTiGa (**b**). The half-metallicity of most Heusler compounds will be influenced and even be broken by the tetragonal distortion (TD) of the lattice. To analyze the effect of TD on half-metallicity and magnetic moments, the unit-cell volume was fixed at the equilibrium bulk volume and then the c/a ratio changed. In Figures 10 and 11, the TD effects on the half-metallicity properties and magnetic moments are described by the magnetic moment and the CBM and VBM dependence on c/a ratio. It is clear that the total and atomic magnetic moments of ZrRhTiZ (Z = Al, Ga) are nearly unchanged, and the half-metallicity can be kept in the c/a ratio range of 0.96-1.06 for ZrRhTiAl and 0.92-1.12 for ZrRhTiGa, respectively. We further plotted the curves of \(E_{bg}\) and \(E_{HM}\) as a function of c/a ratio, as shown in Figures 12 and 13, respectively. The \(E_{bg}\) and \(E_{HM}\) have a similar variation tendency and showed a rise-fall characteristic with increasing c/a ratio. The largest \(E_{bg}\) and \(E_{HM}\) occur in the cubic structure (c/a = 1) instead of tetragonal phase. Figure 1:1 Band gaps as a function of the lattice constant (US) for ZrRhTiAl and ZrRhTiGa. Figure 1:2 The half-metallic band-gap as a function of the lattice constant (US) for ZrRhTiAl and ZrRhTiGa. Figure 1:3 CBM and VBM in the minority channel as a function of c/a ratio (TD) for ZrRhTiAl (**a**) and ZrRhTiGa (**b**). Figure 1:4 Total spin magnetic moment and atomic spin magnetic moment as a function of c/a ratio (TD) for ZrRhTiAl (**a**) and ZrRhTiGa (**b**). Next, the elastic properties were investigated for ZrRhTiZ (Z = Al, Ga) compounds by calculating the single crystal elastic constants C\({}_{\rm ij}\). Heusler compounds belong to a cubic structure which needs only three independent elastic constants (C\({}_{11}\), C\({}_{12}\), and C\({}_{44}\)) for the complete description of their elastic properties. The following expressions are used to depict the elastic behavior of a material given in Reference [19]:  
Figure 1:5  
Figure 1:6  
Figure 1:7  
Figure 1:8  
Figure 1:9  
Figure 2:0 Band gaps as functions of the c/a ratio (TD) for ZrRhTiAl and ZrRhTiGa. Figure 2:1 The half-metallic band-gaps as functions of the c/a ratio (TD) for ZrRhTiAl and ZrRhTiGa. Figure 2:2  
where B, G, E, and A stands for bulk modulus, shear modulus, Young's modulus, and anisotropy factor, respectively. The estimated elastic constants and the mechanical properties for ZrRhTiAl and ZrRhTiGa are given in Figure 2:3  
For cubic structures, the following criteria for the mechanical stability must be satisfied [20]:  
Figure 2:4  
Figure 2:5  
Figure 2:6  
Figure 2:7  
Therefore, from Table 3, one can easily see that ZrRhTiZ (Z = Al, Ga) compounds are mechanically stable. The calculated results of [PERSON]'s modulus (E = 57.9 and 47.8 for ZrRhTiAl and ZrRhTiGa, respectively) indicate that ZrRhTiAl is stiffer than ZrRhTiGa. The Pugh's index (B/G) and Cauchy pressure (C\({}_{12}-C_{44}\)) are usually used to characterize the brittleness or ductility of a compound [21; 22; 23].