## 1 Introduction  
In 1923 the \(\sigma\)-phase was first observed by [PERSON] as a hard non-magnetic intermediate phase which was formed during a long-time high-temperature annealing in the Fe-Cr alloy with a nearly equiatomic composition. Being extremely brittle, the \(\sigma\)-phase substantially degrades the mechanical properties of different types of alloys, mainly stainless steels, Fe- and Ni-based superalloys. The precipitation of this phase decreases ductility and corrosion resistance, while it increases brittleness and promotes the formation of cracks leading to high temperature failure, metal dusting and loss of toughness during heat treatment or welding [1].  
The \(\sigma\)-phase has a topologically closed packed structure, which is defined by a tetragonal unit cell (space group \(P4_{2}/mann\)) containing 30 atoms. These atoms are located in five nonequivalent groups of sites or sublattices (A, B, C, D, E) [2,3]. One often uses the following formula \(A_{2}^{12}B_{4}^{15}C_{8}^{14}D_{8}^{12}E_{8}^{14}\) to describe the \(\sigma\)-phase structure, where bottom and top indices denote the site multiplicities and coordination numbers for each sublattice, respectively. The crystal structure of the \(\sigma\)-phase can be determined as a stacking of the kagome tiles with the A(0,0,0), B(\(x,x\),0), C(\(x,y\),0) and D(\(x,y\),0) atoms lying in the tile plane and the E(\(x,x,z\)) atoms placed between the pseudo-hexagonal rings of kagome tile [3] as shown in Figure 1.  
It is known that the nonequivalent \(\sigma\)-phase sites are partially occupied by all the alloy components, but with certain preferences. Already in 1955 using neutron and X-ray diffraction studies in Fe-V, Ni-V and Mn-Cr alloys [PERSON] and [PERSON] suggested a general scheme for the ordering in binary \(\sigma\)-phases of first row transition metal elements and those containing Mo [4]. They found that the highly coordinated B sites are predominately occupied by the elements to the left of Mn in the periodic table (V, Cr, Mo), whereas the A and D sublattice with an icosahedral environment are almost exclusively occupied by elements to the right of manganese (Fe, Co, Ni). In contrast, the C and E sites are partially occupied by both components and the proportions depend on the alloy composition and external conditions. Although some preferences in the site occupation could be estimated for some binary \(\sigma\)-phases, the knowledge of the atomic site distribution in a wide temperature and composition range is still missing. Such information is useful for studying the \(\sigma\)-phase stability and its precipitation. However, it is quite  
Figure 1: The tetragonal unit cell of the \(\sigma\)-phase structure with 5 crystallographically inequivalent sites (A, B, C, D, E).  
hard to obtain these data experimentally due to sluggish kinetics of the transformation, difficulties in getting high purity samples and satisfying some other technological requirements.  
In this context, an accurate theoretical study based on state-of-the-art first-principles methods combined with a proper statistical model can be used in order to determine the site preferences of the alloy components in the \(\sigma\)-phase. Recently the site occupation in the \(\sigma\)-phase has been widely studied by such methods [5; 6; 7; 8; 9; 10; 11; 12; 13]. However, most of these methods were based on a formalism, which is quite computationally demanding. Alternatively, we have recently proposed a simpler approach to calculate the site preference in the binary \(\sigma\)-phases [14; 15; 16]. It is based on a simple single-site mean-field model, which recently has demonstrated to be able to produce quite accurate results for the site occupancy of Fe and Cr in the Fe-Cr \(\sigma\)-phase. In this paper we briefly describe this method and apply it to study the distribution of alloy components in other binary \(\sigma\)-phase, namely, Co-Cr, Re-W and Fe-V, for which experimental and theoretical information is available.