## 3 Results and Discussion
### Mechanical Properties  
Recall that the ideal X\({}_{2}\)YZ full-Heusler alloy is a cubic \(L2_{1}\) structure with point group 225. Our calculated mechanical properties, together with the related physical properties of the Co\({}_{3-x}\)Cr\({}_{x}\)Al Heusler compounds in both regular and inverse structures, are listed in Table 2. Due to the symmetric nature of the crystal structure, only three independent elastic constants--\(C_{11}\), \(C_{12}\) and \(C_{44}\)--must be taken into account. For the \(L2_{1}\) crystal structure, the elastic constants matrix, \(C_{ij}\), can be represented as:  
Figure 2: Left panel (**a,c**), scalar relativistic band structures; right panel(**b,d**), atom and spin decomposed density of states for Co\({}_{2}\)CrAl in the regular and \(XA\) structures. The solid blue lines in (**a,c**) are for the spin-up bands, and the red-dashed lines are for the spin-down bands. The solid blue, red, green, and orange lines in (**b,d**) are for the two Co, Cr, and Al atoms, respectively. \[C_{ij}= \begin{pmatrix}C_{11}&C_{12}&C_{12}&0&0&0\\ C_{12}&C_{11}&C_{12}&0&0&0\\ C_{12}&C_{12}&C_{11}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{44}&0\\ 0&0&0&0&0&C_{44}\end{pmatrix} \]  
We used the so-called stress-strain relations method to calculate these elastic constants in our density functional theoretical studies. To achieve this, depending on the symmetry of the \(L2_{1}\) crystal structure, the total energies for some finite lattice distortions were first calculated. The stress and strain corresponding to these lattice distortions were then obtained, as well as the elastic constants. Unlike the total energy self-consistent calculations, we used a denser k-mesh of 14 \(\times\) 14 \(\times\) 14, and the self-consistent total energy criterion was set to be as high as \(1.0\times 10^{-8}\) eV, to obtain more accurate elastic constants. In the \(L2_{1}\) crystal structure, the bulk modulus \(B\) in both Voigt (\(B_{V}\)) and Reuss (\(B_{R}\)) formalism, in terms of elastic constants, can be expressed as:  
\[B=B_{V}=B_{R}=\frac{C_{11}+2C_{12}}{3}\]  
In general, Voigt is the upper bond and Reuss is the lower bond of the bulk modulus. The bulk modulus is greater than zero; therefore, \(C_{11}+2C_{12}>0\) is one of the mechanical stability criteria in the \(L2_{1}\) structure. The shear moduli \(G_{V}\) and \(G_{R}\), in terms of the elastic constants in the \(L2_{1}\) structure, are:  
\[G_{V}=\frac{C_{11}-C_{12}+3C_{44}}{5}G_{R}=\frac{5(C_{11}-C_{12})C_{44}}{4C_{4 4}+3(C_{11}-C_{12})}\]  
[PERSON]'s modulus, \(E\), and [PERSON]'s ratio, \(\
u\), were calculated by the following equations in terms of the bulk and shear moduli:  
\[E=\frac{9 BG}{3B+G}\
u=\frac{1}{2}-\frac{E}{6B}=\frac{3B-2G}{6B+2G}\]  
[PERSON]'s ratio is a measurement of the materials that expand perpendicularly in the direction of the external compression. Our calculated nonzero elastic constants, bulk and shear moduli, [PERSON]'s modulus and [PERSON]'s ratio are all listed in Table 2.  
Since our calculated elastic constants \(C_{ij}\) were all positive, the mechanical stability criteria for the cubic structure could be reduced to \(C_{11}>B>C_{12}\). The calculated \(C_{11}\), \(C_{12}\) and \(C_{44}\) for the Co\({}_{3-x}\)Cr\({}_{x}\)Al Heusler compounds were positive, and ranged from 72.4 to 326 GPa. We found that C\({}_{11}\) was 214 GPa for Co\({}_{3}\)Al. If one Cr atom was doped to Co\({}_{3}\)Al, the resulting \(C_{11}\) either increased or decreased, depending strongly on the crystal site of the Cr of Co\({}_{2}\)CrAl. This crystal site dependency also occurred for Cr\({}_{3}\)Al. The C\({}_{11}\) for