## 3 Results and Discussion
### Structural Stability
Overall, all parameters indicate that the Co-Cr-Fe-Ni HEAs studied in this work may be a single FCC solid-solution phase, and the structural stability decreases with increasing Cr and Ni contents and increases with increasing Co and Fe contents. It is worth noting that the range of VEC, \(\Delta H_{mix}\), \(\Omega\), and \(c^{\prime}\) of the Cr element is the largest, which is consistent with previous work [38], indicating that Cr has a significant impact on structural stability. In addition, the Fe element has a significant effect on \(\delta\), and the influence on \(c^{\prime}\) is almost the same as that of Cr. The equilibrium lattice constants and mass density of the Co-Cr-Fe-Ni HEAs based on the FCC supercell structure are presented in Figure 3. The arithmetic mean cell length of the supercell \(a_{avg}\) is used as the lattice constants of the alloy for comparison with experimental data. \(a_{avg}\) is calculated as follows [57]:  
\[a_{avg}=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{3}(a_{i}+b_{i}+c_{i}) \tag{7}\]  
where \(n\) represents the number of unit cells in the supercell, and \(a_{i}\), \(b_{i}\), and \(c_{i}\) are lattice constants. The lattice constant for the equimolar ratio is 0.35 nm, consistent with previous works, which found it as 0.3498-0.3587 nm [27; 28; 29; 32; 33; 36; 58]. In Figure 3a, the lattice constant decreases with increasing Co and Fe content and increases with increasing Cr and Ni content. The lattice constant of the Fe element has the largest variation range, which  
Figure 2: Structural stability parameters of Co-Cr-Fe-Ni HEAs with the \(x\) value of elements. (**a**) Valence electron concentration (VEC), (**b**) atomic size difference \(\delta\), (**c**) enthalpy of mixing \(\Delta H_{mix}\), (**d**) entropy of mixing \(\Delta S_{mix}\), (**e**) parameter \(\Omega\), and (**f**) tetragonal shear modulus \(c^{\prime}\). is consistent with that of \(\delta\), indicating that Fe has the most significant influence on the equilibrium lattice constant. The lattice constants and atomic size difference of Co-Cr-Fe-Ni HEAs vary with the content of each element in the opposite direction. Fe has the smallest atomic radius, which can explain the change in lattice constants and atomic size difference with Fe content. However, the lattice constants decrease with Co contents even though Co has the largest atomic radius. This shows that the atomic radius does not determine the lattice constant of the alloy. This result is consistent with previous work [54], which demonstrated that lattice distortion cannot be accurately described by empirical parameter \(\delta\) based on the atomic radius, especially for 3d FCC alloys. The variation in VEC can cause charge transfer resulting in distortions even for atoms of FCC Co-Ce-Fe-Ni HEAs with similar atomic sizes [59; 60]. Chemical interactions between the elements, electronic spin-flip, and magnetovolume effects in HEAs can maximize the effect of lattice distortion caused by Cr [33; 34]. For Co and Fe atoms, the (\(Fe\leftrightarrow Cr\)) and (\(Co\leftrightarrow Cr\)) bonds are stronger than (\(Fe\leftrightarrow Ni\)) and (\(Co\leftrightarrow Ni\)) bonds (\(A\leftrightarrow B\) means the atomic bond of A and B elements), hence (\(Fe\leftrightarrow Cr\)) and (\(Co\leftrightarrow Cr\)) bonds will be preferred over (\(Fe\leftrightarrow Ni\)) and (\(Co\leftrightarrow Ni\)) bonds [33]. For Co-Cr-Fe-Ni HEAs, the increase in Cr-related bonds will enhance the lattice distortion effect caused by Cr. This may be the reason why the influence of the Cr element on the equilibrium lattice constant is almost the same as that of Ni. Therefore, the lattice distortion caused by the increase in Cr and Ni content or decrease in Co and Fe content may be the cause of the increase in the lattice constant. As shown in Figure 3b, the mass density of Co-Cr-Fe-Ni HEAs increases with Co, Fe, and Ni contents and decreases with Cr contents. The Cr element has the most significant influence on the density.