## 3 Results and Discussion
### Mechanical Properties
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(\mathbf{X_{2}YZ}\) & **Structure** & \(C_{11}\) & \(C_{12}\) & \(C_{44}\) & \(B\) & \(G_{V}\) & \(G_{R}\) & \(G\) & \(E\) & \(A\) & \(\
u\) & \(B/G\) \\ \hline \multicolumn{11}{c}{**(GPa)**} & \multicolumn{1}{c}{**arb. Unit**} \\ \hline Co\({}_{3}\)Al & regular & 214 & 160 & 132 & 178 & 90 & 52 & 71 & 188 & 4.89 & 0.32 & 2.51 \\ Co\({}_{2}\)CrAl & regular & 268 & 166 & 145 & 200 & 107 & 83 & 95 & 247 & 2.86 & 0.30 & 2.11 \\ \multicolumn{11}{c}{_XA_} & \multicolumn{1}{c}{191} & 150 & 135 & 164 & 90 & 42 & 66 & 174 & 6.59 & 0.32 & 2.48 \\ \multicolumn{11}{c}{_Cr\({}_{2}\)CoAl_} & regular & 133 & 141 & 135 & 138 & 80 & \(-11\) & 34 & 95 & - & 0.39 & 4.06 \\ \multicolumn{11}{c}{_XA_} & \multicolumn{1}{c}{326} & 153 & 109 & 211 & 102 & 101 & 100 & 258 & 1.26 & 0.30 & 2.11 \\ Cr\({}_{3}\)Al & regular & 254 & 72.4 & 122 & 133 & 110 & 107 & 107 & 256 & 1.35 & 0.18 & 1.24 \\ \hline \end{tabular}
\end{table} 
Table 2: Our calculated elastic constants \(C_{11}\) (GPa), \(C_{12}\) (GPa) and C\({}_{44}\) (GPa), bulk modulus \(\beta\) (GPa), shear modulus \(G\) (GPa), Young’s modulus \(E\) (GPa), anisotropy factor A (arb. unit) and [PERSON]’s ratio \(\
u\) (arb. unit) of all considered Co\({}_{3-x}\)Cr\({}_{x}\)Al Heusler compounds. Cr\({}_{3}\)Al was 254 GPa and became 326 or 133 GPa depending on the crystal site of Co for Cr\({}_{2}\)CoAl. This trend is very interesting. Let us take bulk modulus as an example. The bulk modulus for Co\({}_{3}\)Al was 178 GPa. In the \(XA\) structure, when Cr was doped to the C site, the calculated bulk modulus was decreased. It was significantly increased if it became Cr\({}_{2}\)CoAl. This trend can be roughly explained by the anisotropy factor. If the anisotropy factor is smaller, meaning that the materials have more elastic isotropy, the bulk modulus becomes larger. Nonetheless, \(C_{12}\) was also positive, ranging between 72.4 and 166 GPa. Unlike \(C_{11}\), we found that \(C_{12}\) and \(C_{44}\) were not sensitive to Cr doping, except for C\({}_{12}\) in Cr\({}_{3}\)Al. In Co\({}_{3-x}\)Cr\({}_{x}\)Al Heusler compounds, the criterion \(C_{11}>C_{12}\) was always met, except for Cr\({}_{2}\)CoAl in the regular structure. This implies that regular Cr\({}_{2}\)CoAl is mechanically unstable. The Zener anisotropy factor \(A=2C_{44}/(C_{11}-C_{12})\) is a dimensionless number used in the cubic system to quantify whether the material is isotropic or anisotropic. If the Zener anisotropy is 1, the material has isotropy. Our calculated anisotropy was greater than 1, meaning that the Co\({}_{3-x}\)Cr\({}_{x}\)Al Heusler compounds were all elastically anisotropic, except for regular Cr\({}_{2}\)CoAl. Beyond this, [PERSON] et. al. [29] proposed a universal anisotropy index \(A^{U}\) instead of Zener anisotropy. The Zener anisotropy index is not a universal index, but is valid for the cubic phase. Since \(B_{V}=B_{R}\), we also calculated \(A^{U}\left(=5\frac{G_{V}}{G_{R}}-5\right)\); the results were 3.65, 1.45, 5.71, 0.05 and 0.14 for regular Co\({}_{3}\)Al, regular Co\({}_{2}\)CrAl, \(XA\) Cr\({}_{2}\)CoAl and \(XA\) Cr\({}_{3}\)Al, respectively. It is clear that the trend for both anisotropy factors \(A\) and \(A^{U}\) was the same. In the cubic structure, if the anisotropy factor is negative, which indicates either \(C_{11}<C_{12}\) or \(C_{44}<0\), the structure is unstable. The bulk modulus is a measurement of the volume change with respect to the external pressure. Our calculated bulk modulus was positive and ranged from 133 to 211 GPa. The ratio of the largest bulk modulus to the smallest one was 1:6, where the corresponding ratio for the shear modulus was 3:15. The bulk modulus was larger, which implies a strong resistance to volume change under pressure. Since all stability criteria were satisfied, except for regular Cr\({}_{2}\)CoAl, we can conclude that Co\({}_{3-x}\)Cr\({}_{x}\)Al Heusler compounds are mechanically stable. [PERSON]'s modulus \(E\) can also be used to determine the stiffness of a material.