## 3 Results and Discussion
### The Four Systems {Zr,Hf}-{Mo,W}-Re (\(\chi\) Phase)  
The four iso-electronic systems Zr-Mo-Re [16], Zr-W-Re [16], Hf-Mo-Re [17], and Hf-W-Re [18] share a common feature. They all present the Frank-Kasper \(\chi\) phase not only extending from one to the other binary systems, but also deep inside the ternary system along a completely different direction toward the third binary system. From the isothermal section of the Zr-W-Re system [16] presented in Figure 7, it is important to note both the homogeneity domain extending from Zr\(s\)Re\({}_{24}\) to the W\(s\)Re\({}_{24}\) phase, therefore approximately parallel to the Zr-W axis, and the homogeneity domain of the Laves phase (in C14 and C15 structures) extending parallel to the other W-Re axis. This dual character of substitutional behavior (Zr by W and W by Re) is further demonstrated by the odd shape of the homogeneity domain of the \(\chi\) phase that extends in both directions, _i.e._, has limiting compositions parallel to both the Zr-W and W-Re binary borders. Figure 6: Ground state at 0 K of the \(C14\) phase in the Zr–Nb–Fe system, obtained by DFT calculation (new results). Red lines represent an equilibrium tie-triangle, green lines are equilibrium tie-lines. The four iso-electronic systems listed above show very similar features concerning the shape of the \(\chi\) phase and only minor differences in their phase diagrams. Among these four systems, Zr-W-Re has been chosen for a deeper experimental study because it is more suited to the site occupancy determination by using combined XRD and ND (the contrast factors are more favorable). Our experimental and calculation results have been published in [3]. Their significance relative to our present study is developed below. The \(\chi\) phase has four inequivalent sites of multiplicity 2, 8, 24, and 24 with CN 16, 16, 13, and 12, respectively (for a detailed review on the crystal chemistry of this phase, see [19]). In binary systems, it tends to order at the composition \(A_{10}B_{48}\) and \(A_{34}B_{24}\). The homogeneity domain of this phase in this system has been explained by the stability of the ordered configurations ZrZr\({}_{2}\)Zr\({}_{8}\)Re\({}_{24}\), Zr\({}_{2}\)W\({}_{8}\)Re\({}_{24}\)Re\({}_{24}\), W\({}_{2}\)W\({}_{8}\)Re\({}_{24}\)Re\({}_{24}\), and Zr\({}_{2}\)Zr\({}_{8}\)W\({}_{24}\)Re\({}_{24}\), and to the fact that W\({}_{2}\)W\({}_{8}\)Re\({}_{24}\)Re\({}_{24}\) is close to stability. This indicates the ability of W to substitute in Zr\({}_{8}\)Re\({}_{24}\) on both the Zr and Re sites. This is opposite to the behavior of the Laves phases, for which W substitutes only for Re. There are two reasons for this: the weak stability of both the \(C14\) and \(C15\) phases in the W-Re system, and the absence of ternary configurations able to stabilize the phase in the other direction in the ternary composition range (the C15 structure, having only two sites, does not generate ternary configurations, and C14 does not have configurations on the line joining ZrRe\({}_{2}\) and WRe\({}_{2}\)). Experimental site occupancies by joint Rietveld analysis of XRD and ND data have been compared to those obtained from the DFT calculations and are shown in [3]. The good agreement has confirmed the reliability of the DFT calculations to obtain this quantity. The W site reversal can therefore be studied from DFT data because the experimental measurements were not chosen to specifically study the reversal and because measurements obviously cannot extend, contrary to DFT calculations, to the metastable range of the phase. Site occupancies are plotted in Figure 8 between the compositions Zr\({}_{5}\)W\({}_{24}\) and W\({}_{5}\)Re\({}_{24}\). The section clearly shows the reversal of W atoms switching from low CN to high CN sites. Figure 7: Isothermal section at 1350 \({}^{\circ}\)C of the Zr–W–Re system (redrawn from [16], Copyright 1970, Academy Science Ukraine).