In our previous calculations for the FeCr σ-phase, we have neglected the effect of local atomic relaxations. As has been shown in [14], the contribution from local lattice relaxations to the formation enthalpy is about 2.5 mRy / 30 at. cell in the FeCr case, which was expected due to small size mismatch of Fe and Cr. In case of the CoCr, ReW and FeV σ-phases we show in Table 1 the results for the energy of local lattice relaxations E_relax, which have been obtained for the room temperature values of an unit cell volume V and a ratio of the lattice cell constants c/a by the full-potential linearized augmented plane wave method [23], implemented in the WIEN2k code [24]. Obviously, E_relax is small in the case of the CoCr and ReW σ-phases, but not negligible in the case of the FeV σ-phase.

In order to account for the effects of the lattice relaxations in the Fe-V system, we introduce in our model an additional term, called J_α^(relax) , such that the total energy has the following form:

(6)

These J_α^(relax) parameters can be interpreted as the energy gain due to the atomic relaxations with respect to substitution of one Fe atom by one V atom in each sublattice α. To calculate these energies we use the WIEN2k code and simulate a cell of 30 atoms, in which we exchange one Fe atom from each of the A, B, C and D sublattices with one V atom from the E sublattice. Thus the relaxation parameter is defined as J_α^(relax) = E_α^(relax) – E_E^(relax), where E_α^(relax) is the energy of relaxation of α = A, B, C and D site with respect to substitution of one Fe atom and E_E^(relax)is the energy of relaxation of the E site with respect to substitution of one V atom.

In [14,16] we analyzed the possible effects, which may play a role in the behaviour of the atomic site distribution in the σ-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.

Taking these effects into account, we have calculated the Cr site occupancies in the Fe_0.5Cr_0.5σ-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 σ-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 σ-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 σ-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 σ-phase in any system is found in literature.

In order to test the applicability of our model to other binary σ-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 σ-phase in addition to the Fe-Cr and Fe-V σ-phases. The magnetism of the Co-Cr σ-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 σ-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_0.5W_0.5 and Co_0.5Cr_0.5 alloys were calculated and the corresponding results are shown in Figure 3 in comparison with the Fe_0.5Cr_0.5σ-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 σ-phase formation. Nevertheless these effects can be approximately estimated based on the corresponding study of the Fe-Cr σ-phase.

As can be seen from Figure 3 the site occupancies in the Re_0.5W_0.5 system behave similarly to the Fe_0.5Cr_0.5 alloy. However, one can notice that at low temperatures the C and E sites in the Re_0.5W_0.5σ-phase are occupied by the W atoms in reverse order in comparison to the Cr atoms in the Fe_0.5Cr_0.5σ-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_0.5Cr_0.5σ-phase the values of the Cr site occupancies are almost the same as in the Fe_0.5Cr_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 σ-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 °C [29]. As in the Fe-Cr system (see [14,16]), one can use the parameters obtained for the Co_0.5Cr_0.5 in order to predict the Cr site occupancies in the full composition range of the σ-phase formation. In Figure 4 the Cr site distribution calculated versus the alloy composition is displayed together with the experimental data from [29] measured at various temperatures. Taking into account the scattering of the experimental data, good agreement is found between the present calculated results and the experimental measurements (probabilities), justifying our theoretical model also for this case.

The formation probability of the σ-phase in the Fe-V alloy is many times larger than in the Fe-Cr system due to a much wider temperature and compositional ranges, where this phase is stable [32]. Despite the fact that the σ-phase in the Fe-V system was found many years ago, the concentration and temperature ranges of its existence are still not precisely known. In general it is difficult to accurately determine the phase boundaries due to several factors such as: the slow kinetics of the transformation, the dependence on the purity of the alloys, the mechanical state (strained or strain-free state) and the annealing temperature. According to the phase diagrams analyzed in [32] the upper temperature limit for the σ-phase stability equals to 1, 321 °C for an alloy composition x_V = 46.5%, and it depends rather strongly on composition: ∼ 1, 000 °C for x_V = 34% and ∼ 1, 050 °C for x_V = 63%. On the other hand, the lower temperature limit of formation does not depend too much on the concentration and is around 647–650 °C.

As mentioned above, the quite large size mismatch of the alloy components in the Fe-V system causes that the effect of local atomic relaxations might not be negligible in contrast to the other studied systems (Fe-Cr, Re-W and Co-Cr) (see Table 1). In order to account for these effects we introduced a new parameter in our model, namely J_α^(relax) , which we calculated for each sublattice in the Fe_0.5V_0.5 σ-phase (assuming the experimental lattice parameters V = 0.37074 nm³ and c/a = 0.517) and list them in Table 3. As can be seen, the local lattice relaxations change the effective on-site interactions only moderately and consequently the corresponding site occupancies will be hardly affected. Therefore one might expect that the effect of local atomic relaxations in binary σ-phases near the equiatomic composition is negligible for all binary systems, for which the component size mismatch is less than that in the Fe-V system.

Let us start from the equiatomic composition x_V = 0.50 and consider a possible increase in volume from V_exp = 0.37074 nm³ to the high temperature value V = 0.38319 nm³ due to thermal expansion. The calculated effective on-site interactions and corresponding local magnetic moments of Fe and V atoms are given in Table 4 at three different temperatures.

Using the J_α⁽¹⁾ parameters and the corresponding Fe and V magnetic moments (obtained at T = 1,000 K), we calculated the distribution of the V atoms among the five σ-phase sublattices in the Fe_0.5V_0.5 alloy with respect to temperature and alloy composition (see Figure 5). The results obtained at high temperatures agree reasonably well with the experimental values measured for an alloy composition x_V = 0.48. At the low temperature we find that for example the C sublattice is equally occupied by both alloy components. Using the WIEN2k code we also calculated the equilibrium atomic distribution at T = 0 K. In contrast to the Fe_0.5Cr_0.5 alloy (see Figure 3), the E sublattice of the Fe_0.5V_0.5σ-phase has a partial site occupancy by V atoms. There are 7 V + 1 Fe atoms in the E sublattice, while the C sublattice has an equal occupancy by both alloy components, i.e., 4 V + 4 Fe atoms. Thus the atomic site occupancies estimated within our model at low temperatures are found in good agreement with the ground state equilibrium distribution obtained at T = 0 K by the WIEN2k code.

Let us now consider the atomic site distribution in the Fe-V σ-phase with respect to the alloy composition, x_V (see Figure 5). The predicted site occupancies as function of the alloy composition reproduce well the experimental values for the C and E sublattices, while they differ in slope for the B and (A, D) sublattices. Due to the large size mismatch between the Fe and V atoms, the lattice parameters of the Fe-V σ-phase should depend significantly on the alloy composition. According to experimental measurements [27] an increase of the vanadium content by 10% from the equiatomic composition, Fe_0.5V_0.5, linearly increases the volume of the unit cell from 0.37074 nm³ (for x_V = 0.5) to 0.37634 nm³ (for x_V = 0.6), while the c/a values decrease from 0.517 to 0.516, respectively. On the other hand, with a decrease of the vanadium content by 10% to x_V = 0.4, the volume and c/a ratio change to 0.36514 nm³ and 0.518, correspondingly. Assuming a linear dependence, one can extrapolate the expanded volumes as V = 0.37747 nm³ for x_V = 0.4 and V = 0.38892 nm³ for x_V = 0.6. The optimization of the c/a ratio in the Fe-V σ-phase within the EMTO-CPA method is not really reliable due to the large size mismatch of the Fe and V atoms. Therefore, we stay at a constant c/a ratio, but keep in mind that the site occupancies might vary due to a change of c/a, as has been discussed for the Fe-Cr σ-phase.

The corresponding effective interactions J_α⁽¹⁾and magnetic moments µ_α^(Fe,V) obtained for x_V = 0.40 and x_V = 0.60 are given in Table 5. One can see that for x_V = 0.40 all four effective interactions are different (smaller in absolute values) from those obtained for x_V = 0.50, while for x_V = 0.60 only the changes for the B and C sublattices are remarkable. Nevertheless, when we compare the site occupancies for the parameters obtained for x_V = 0.40 and x_V = 0.60 with those predicted from the parameters obtained for x_V = 0.50, only small differences can be observed. The corresponding values of the site occupancies calculated at T = 1,000 K are given in Table 6. As can be seen only the B, C and E site occupancies change noticeably (those, which have predominant occupancy by V atoms) for x_V = 0.40, while for x_V = 0.60 all values are nearly the same. For x_V = 0.40 the C and E sublattices now have almost an equal occupation in accord with the experimental data (see Figure 5). The occupancy of the B site is a bit smaller, but still higher than the corresponding experimental value. This discrepancy could be due to several reasons: additional structural variations with respect to the alloy composition, temperature expansion, or local atomic relaxations. Particularly the latter might not be negligible for alloy compositions different from the equiatomic composition, especially for those sublattices with small deviations from integer occupation, e.g., the A and D sublattices for x_V = 0.60 or the B sublattice for x_V = 0.40. These effects could be taken into account similiar as for the equiatomic composition (see Equation (6)), but such calculations are not presented in this work.