## 3 Results and Discussion  
### Crystal Structure and Magnetic Properties  
Table 1 shows the calculated lattice constants and total and atomic spin magnetic moments. We can see that the calculated lattice constant increased as the Cr concentration increased for both regular and inverse Heusler structures. The lattice constant was in the range between 5.672 and 5.908 A. We can note that the lattice constants were 5.672, 5.701, 5.885 and 5.980 A for Co\({}_{3}\)Al, Co\({}_{2}\)CrAl, Cr\({}_{2}\)CoAl and Cr\({}_{3}\)Al, respectively, in the regular structure. Obviously, this increase in lattice constant can be related to the larger Wigner-Seitz of the Cr atom. Nonetheless, the lattice constants in the \(XA\) structure became 5.723 and 5.721 A for Co\({}_{2}\)CrAl and Cr\({}_{2}\)CoAl, respectively. In Co\({}_{2}\)CrAl, the total spin magnetic moments in the \(XA\) structure were smaller than in the regular structure. However, the atomic spin magnetic moment was larger in the \(XA\) structure, resulting in a small increase in lattice constants. In Cr\({}_{2}\)CoAl, the lattice constant in the \(XA\) structure was smaller because it is nonmagnetic, spin-unpolarized and in equilibrium. In all our considered Co\({}_{3-x}\)Cr\({}_{x}\)Al compounds, Co\({}_{2}\)CrAl was the only half-metal theoretically predicted by integer spin magnetization and the density of states at the Fermi level. The theoretical spin magnetic moment for Co\({}_{2}\)CrAl is 3.00 \(\upmu_{\text{B}}\); however, the experimental result was only half of this [23]. In the regular Heusler alloys, the atomic positions for the two X atoms are identical, and, thus, the spin magnetic moments for the two X atoms are identical. Previous theoretical studies on Co\({}_{3}\)Al and Co\({}_{2}\)CrAl [24; 25] reported that they are metals with spin magnetic moments of 3.90 and 3.00 \(\upmu_{\text{B}}\), respectively, which is in good agreement with our present study. We found that the total spin moments were decreased significantly in the \(XA\) structure. The total spin magnetic moments for Co\({}_{2}\)CrAl and Cr2 CoAl in the \(XA\) structure were 0.82 and 0.00 \(\upmu_{\text{B}}\), respectively. The reason for the decreases in the total spin moments for Co\({}_{2}\)CrAl, is that in the \(XA\) structure, the Cr atom is antiferromagnetically coupled with both Co atoms. The atomic spin magnetic moment for the Al atom is negative and one order of magnitude lower than the X and Y atoms. Our calculated lattice constants and magnetic moments for Cr\({}_{3}\)Al were in very good agreement with our present study. We found that the total spin moments were decreased significantly in the \(XA\) structure. The total spin magnetic moments for Co\({}_{2}\)CrAl and Cr2 CoAl in the \(XA\) structure were 0.82 and 0.00 \(\upmu_{\text{B}}\), respectively. The reason for the decreases in the total spin moments for Co\({}_{2}\)CrAl, is that in the \(XA\) structure, the Cr atom is antiferromagnetically coupled with both Co atoms. The atomic spin magnetic moment for the Al atom is negative and one order of magnitude lower than the X and Y atoms. Our calculated lattice constants and magnetic moments for Cr\({}_{3}\)Al were in very good agreement with our present study. We found that the total spin moments were decreased significantly in the \(XA\) structure. The total spin magnetic moments for Co\({}_{2}\)CrAl and Cr2 CoAl are in  
agreement with a previous theoretical study [26]. Nonetheless, Cr\({}_{2}\)CoAl was reported to be ferrimagnetic, with a very small spin magnetic moment of 0.009 \(\mu\)B [27]. To obtain a better understanding of the effect of Cr doping, as well as the atomic position of Cr doping on the spin magnetic moment, we can consider Co\({}_{2}\)CrAl as an example. In regular X\({}_{2}\)YZ Heusler alloys, the Y atom is surrounded by eight X atoms. That is to say, the Cr atom sits at the B site surrounded by eight Co atoms; see Figure 1a. In the \(a\), \(b\) and \(c\) directions, a Z atom sits between the Cr atoms.