## 3 Results and Discussion
### Mechanical Properties
If \(E\) is higher, it implies the material is stiffer. This can be seen from the definition of \(E\) as the ratio between stress and strain in the linear elastic region. The calculated [PERSON]'s modulus was larger than 170 GPa, except for Co\({}_{2}\)CrAl in the regular structure. The calculated [PERSON]'s modulus of Co\({}_{3}\)Al was 188 GPa and became 247 or 174 GPa when one Cr atom was added to the B or C site, respectively. Although the difference in \(E\) was not large, it also implies that \(E\) correlates with the atomic position. The regular Cr\({}_{2}\)CoAl had the lowest Young's modulus of 95 GPa. Similarly, the bulk modulus for Cr\({}_{3}\)Al was 133 GPa. If one Cr atom was replaced with a Co atom at the B or C site, the bulk modulus became 133 or 211 GPa, respectively. The \(G_{R}\) of regular Cr\({}_{2}\)CoAl was negative. It resulted in a significant decrease in shear modulus \(G\), and hence, the lowest [PERSON]'s modulus \(E\) occurred. [PERSON]'s ratio \(\
u\) is a quantity to understand the bonding characteristics of a solid. It describes the volume change in the direction perpendicular to the external force's direction. Typically, Poisson's ratio is numerically between \(-1\) and 0.5. The present Poisson's ratio ranged between 0.30 to 0.39, which suggests these materials are intermetallic compounds, except for Cr\({}_{3}\)Al, where the Poisson's ratio was 0.18. Young's modulus, the shear modulus and Poisson's ratio satisfy the relation \(G=0.5E/(1+\
u)\). The ratio of the bulk to the shear modulus, \(B/G\), known as Pugh's ratio, describes the brittle or ductile behavior of materials. A high \(B/G\) ratio, \(>\)1.75, is associated with ductility, whereas a low value ratio corresponds to brittleness. We concluded that Co\({}_{3-x}\)Cr\({}_{x}\)Al Heusler alloys could be classified as ductile materials since their \(B/G\) ratios were larger than 1.75, except for Cr\({}_{3}\)Al. The brittleness and ductility of a solid can also be verified by using Poisson's ratio, where \(\
u<1/3\) and \(\
u>1/3\), represent brittleness and ductility, respectively. To describe the thermodynamic properties of a material, the Debye temperature \(\Theta_{\rm D}\) is a fundamental and useful parameter, correlating the elastic and thermodynamic properties of a solid. In the Debye model, it assumes that all sound waves travel at the same velocity regardless of their wavelength. The Debye temperature can be estimated from the average elastic-wave velocity by the following equation:  
\[\Theta_{D}=\frac{h}{k_{B}}\sqrt[3]{\frac{3 nN_{A}\rho}{M}}v_{m}\]  
where \(h\) is [PERSON]'s constant, \(k_{B}\) is [PERSON]'s constant, \(N_{A}\) is the Avogadro constant, \(n\) is the number of atoms per molecule, M is the molecular weight and \(\rho\) is the density, respectively. To obtain the average sound velocity \(v_{m}\), we first calculated the longitudinal and transverse sound velocities using [PERSON]'s equation; see [30] for details. The calculated density, longitudinal, transverse and average sound velocity and Debye temperature are listed in Table 3. It is clear that the longitudinal sound velocity was almost double the transverse sound velocity. Cr\({}_{3}\)Al had the largest Debye temperature of 600 K.