## 4 Results and Discussion
### The Re-W and Co-Cr \(\sigma\)-Phase  
In order to test the applicability of our model to other binary \(\sigma\)-phases we first chose the Re-W and Co-Cr systems, which have a small mismatch between the atomic radii similar to the Fe-Cr system. The calculated effective on-site interactions and magnetic moments of these two systems are given in Table 2. According to [29] the Co-Cr system is a possible third candidate for a magnetic \(\sigma\)-phase in addition to the Fe-Cr and Fe-V \(\sigma\)-phases. The magnetism of the Co-Cr \(\sigma\)-phase at 0 K was theoretically studied in [10]. In present work it was found that both the Co and Cr atoms in the Co-Cr \(\sigma\)-phase have finite magnetic moments, even at the relevant high temperatures. In contrary the Re-W system is non-magnetic. Using these parameters the distributions of the alloy components in the Re\({}_{0.5}\)W\({}_{0.5}\) and Co\({}_{0.5}\)Cr\({}_{0.5}\) alloys were calculated and the corresponding results are shown in Figure 3 in comparison with the Fe\({}_{0.5}\)Cr\({}_{0.5}\)\(\sigma\)-phase. Taking into account the small mismatch between the atomic radii of the alloy components in the Re-W and Co-Cr systems and the absence of corresponding experimental data, in the present work we neglect structural changes with respect to the alloy composition and temperature within the range of the \(\sigma\)-phase formation. Nevertheless these effects can be approximately estimated based on the corresponding study of the Fe-Cr \(\sigma\)-phase. As can be seen from Figure 3 the site occupancies in the Re\({}_{0.5}\)W\({}_{0.5}\) system behave similarly to the Fe\({}_{0.5}\)Cr\({}_{0.5}\) alloy. However, one can notice that at low temperatures the C and E sites in the Re\({}_{0.5}\)W\({}_{0.5}\)\(\sigma\)-phase are occupied by the W atoms in reverse order in comparison to the Cr atoms in the Fe\({}_{0.5}\)Cr\({}_{0.5}\)\(\sigma\)-phase. This is in agreement with earlier published results [30; 31]. Both theoretical results agree well with the experimental data from [30]. The reversed occupation of the C and E sites seems to be caused by the big difference between the Cr and W atomic radii. Thus the occupation of these sites (C and E) can change with respect to each other depending on the alloy component. For the Co\({}_{0.5}\)Cr\({}_{0.5}\)\(\sigma\)-phase the values of the Cr site occupancies are almost the same as in the Fe\({}_{0.5}\)Cr\({}_{0.5}\) system, but the slightly pronounced dominance of a particular alloy component in the occupation of a particular site occurs and can be explained by the slightly larger size mismatch between the alloy components in the Co-Cr than in Fe-Cr system. The Co-Cr \(\sigma\)-phase forms more on the right hand side of the phase diagram (larger Cr content) and is usually observed for a concentration range 54%-67% of Cr in a temperature interval 600-1,280 \({}^{\circ}\mathrm{C}\)[29]. As in the Fe-Cr system (see [14; 16]), one can use the parameters obtained for the Co\({}_{0.5}\)Cr\({}_{0.5}\) in order to predict the Cr site occupancies in the \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & & **A** & **B** & **C** & **D** & **E** \\ \hline \(J_{\alpha}^{(1)}\) (mRy/cell) & ReW & 15.188 & \(-\)8.156 & \(-\)1.594 & 12.656 & 0.0 \\  & CoCr & 15.656 & \(-\)1.500 & 0.563 & 12.656 & 0.0 \\ \(\mu_{\alpha}^{Co}(\mu_{B})\) & Co & 0.802 & 1.167 & 1.064 & 0.854 & 0.875 \\ \(\mu_{\alpha}^{Cr}(\mu_{B})\) & Cr & 0.491 & 0.903 & 0.673 & 0.570 & 0.564 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: On-site parameters \(J_{\alpha}^{(1)}\) (mRy/cell) calculated for the Re\({}_{0.5}\)W\({}_{0.5}\) and Co\({}_{0.5}\)Cr\({}_{0.5}\) systems within the DLM-LSF model at \(T\) = 1,400 K and \(T\) = 1,500 K respectively. \(\mu_{\alpha}^{Co}\) and \(\mu_{\alpha}^{Cr}\) are the local magnetic moments of the Co and Cr atoms in the Co\({}_{0.5}\)Cr\({}_{0.5}\)\(\sigma\)-phase, while the Re-W \(\sigma\)-phase is nonmagnetic. full composition range of the \(\sigma\)-phase formation.