## 4 Results and Discussion  
### The Fe-Cr \(\sigma\)-Phase  
In [14,16] we analyzed the possible effects, which may play a role in the behaviour of the atomic site distribution in the \(\sigma\)-phase. These are the magnetic state, structural variations (volume and \(c/a\)), which might be present in the system due to effects of the neutron irradiation or thermal expansion, and temperature effects. It was found that the magnetic entropy contributes mainly to the icosahedral coordinated (A, D) and highly coordinated B sites. Usually these sites are predominantly occupied by only one sort of the alloy components. At the same time the structural variations lead to an additional atomic redistribution. In particular, a volume expansion promotes an accumulation of more Cr in the A and D sublattices, while the \(c/a\) ratio affects the occupation of the C sites.  
\begin{table}
\begin{tabular}{l c c c c} \hline
**System** & **Volume** & \(\boldsymbol{c/a}\) & \(\boldsymbol{\Delta_{AR}}\) & \(\boldsymbol{E_{relax}}\) \\ \hline Re\({}_{0.5}\)W\({}_{0.5}\) & 0.46482 nm\({}^{3}\)\({}^{a}\) & 0.521 \({}^{a}\) & 0.002 nm & 5 mRy/30 at. cell \\ Co\({}_{0.5}\)Cr\({}_{0.5}\) & 0.35235 nm\({}^{3}\)\({}^{b}\) & 0.517 \({}^{b}\) & 0.003 nm & 6 mRy/30 at. cell \\ Fe\({}_{0.5}\)V\({}_{0.5}\) & 0.37074 nm\({}^{3}\)\({}^{c}\) & 0.517 \({}^{c}\) & 0.008 nm & 30 mRy/30 at. cell \\ \hline \end{tabular} \({}^{a}\) Ref. [25]; \({}^{b}\) Ref. [26]; \({}^{c}\) Ref. [27].  
\end{table}
Table 1: The room temperature lattice parameters (\(V,c/a\)), the corresponding size (atomic radii) mismatch of the alloy components \(\Delta_{AR}\) and the energy of the local atomic relaxations \(E_{relax}\) in the Re\({}_{0.5}\)W\({}_{0.5}\), Co\({}_{0.5}\)Cr\({}_{0.5}\) and Fe\({}_{0.5}\)V\({}_{0.5}\)\(\sigma\)-phases.  
Taking these effects into account, we have calculated the Cr site occupancies in the Fe\({}_{0.5}\)Cr\({}_{0.5}\)\(\sigma\)-phase for the low and high temperature values that were obtained for the experimental room temperature lattice parameters and temperature expanded ones. The results are shown in Figure 2 and agree quite well with the available experimental data [2,28]. At very high temperatures the configurational entropy drives the system to a highly disordered distribution of the alloy components on the five \(\sigma\)-phase sublattices. By decreasing the temperature, Fe and Cr atoms exhibit a clear preference for different inequivalent sites. At temperatures of about 700 K, which is the lower border of the \(\sigma\)-phase stability range, the (A, D) sites become occupied only by Fe atoms (_i.e._, nearly no Cr atoms), the (B, E) sites mostly by Cr atoms, while the occupation of the C site remains almost unchanged and mixed. It is interesting to note that the preference of the Cr atoms between the B and E sites crosses around 650-700 K, at a temperature where the \(\sigma\)-phase formation starts. The occupation behavior for other sites also changes at this temperature as discussed above. This effect might be due to structural variations with temperature in our model, whereas no experimental data on thermal expansion of the \(\sigma\)-phase in any system is found in literature.  
Figure 2: The Cr site occupancies of the (A, B, C, D, E) sublattices _versus_ temperature including the magnetic and temperature effects in the Fe\({}_{0.5}\)Cr\({}_{0.5}\)\(\sigma\)-phase. The empty symbols denote the available experimental data (error bars within the symbol sizes) [2,28]. The vertical lines indicate the temperature range of the \(\sigma\)-phase formation in the Fe-Cr alloy. (Reprinted figure with permission from [16] as follows: [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] _Phys. Rev. B_**2011**, _84_, 184206. Copyright (2011) by the American Physical Society. [[http://link.aps.org/abstract/PRB/v84/p184206](http://link.aps.org/abstract/PRB/v84/p184206)]([http://link.aps.org/abstract/PRB/v84/p184206](http://link.aps.org/abstract/PRB/v84/p184206))).