## 3 Results & Discussion  
### Structural and Thermal Properties  
Phase composition of Ag\({}_{1-x}\)Sb\({}_{1+x}\)Te\({}_{2+x}\) alloys was verified using PXRD experiments coupled with DTA thermal analysis. PXRD patterns of as-cast samples are presented in Figure 2. Main reflections were assigned to the \(\beta\) phase with NaCl-type crystal structure. The presence of small amounts of secondary phases is revealed. The slow cooling process does not allow obtaining single-phase materials in the whole analyzed compositional region, in agreement with the phase diagram [10] (Figure 1). The existence of Ag\({}_{2}\)Te and Sb\({}_{2}\)Te\({}_{3}\) precipitations after slow cooling, especially in samples from the compositional region of the \(\beta\) phase, is the result of the eutectoid decomposition of the latter in the reaction \(\beta\leftrightarrow\alpha+\varepsilon\) at 633 K [10]. However, the rate of this process is so small that even slow cooling can impede it. PXRD analysis of samples after quenching reveals that this procedure effectively minimizes the quantity of secondary phases and in the case of alloys with \(x=0.19\) and 0.17 (compositions from homogeneity region of the \(\beta\) phase) enables us to prepare single-phase materials (Figure 2). The strongest reflection from Ag\({}_{2}\)Te (\(2\theta=31.06^{\circ}\)) can be overlooked and alloys with nominal compositions of \(x=0.1\) and \(x=0\) can be considered as single-phase materials. Therefore, DTA analysis was performed to verify our PXRD data (Figure S1). Each expected thermal effect corresponding to the \(\beta\) phase was detected, together with a small endothermic peak at 418 K associated with the polymorphic phase transition of Ag\({}_{2}\)Te (Figure S1a, Table S1). A tiny endothermic effect was found at 693 K, which was ascribed to the incongruent melting of Ag\({}_{5}\)Te\({}_{3}\)[31]. However, no reflections of that phase are visible in PXRD pattern (Figure 2). The concentration of this phase seems to be below the detection limit of PXRD analysis. The results prove that rate of the decomposition of the \(\beta\) phase is low and impurities are present in the form of the mixture of Ag-Te phases, in agreement with our previous investigations [12]. Thermal analysis of the slowly cooled \(x=0.19\) sample shows only peaks associated with decomposition and incongruent melting of the \(\beta\) phase (Figure S1b). Lattice parameters of the \(\beta\) phase for Ag\({}_{1-x}\)Sb\({}_{1+x}\)Te\({}_{2+x}\) alloys are presented in Figure 3.  
Figure 2: Powder XRD patterns of slowly cooled Ag\({}_{1-x}\)Sb\({}_{1+x}\)Te\({}_{2+x}\) samples. For comparison, patterns of samples with \(x=0.17\) and 0.19 after quenching are shown in the bottom; LaB\({}_{6}\) was used as an internal standard. Assuming no interstitials and anti-site defects (Ag/Sb at 4\(a\) sites and Te at 4\(b\) sites only), increase of \(x\) (0 < \(x\) < 1) in the formula Ag\({}_{1-x}\)Sb\({}_{1+x}\)Te\({}_{2+x}\) enhances the concentration of vacancies at 4\(a\) site and reduces the lattice parameter of the \(\beta\) phase. For the non-annealed samples, the lattice parameter increases with Ag-content for the lower silver concentrations and remains almost constant for more than 45 mol.% of Ag\({}_{2}\)Te. After annealing, the lattice parameter practically does not change with concentration (Figure 3). This behavior suggests a wider homogeneity range of the \(\beta\) phase at elevated temperatures. Particular temperature treatment (cooling rate, annealing) of Ag-Sb-Te samples results in the formation and certain ordering of vacancies at 4\(a\) crystallographic sites. Kinetic hindering of the formation and decomposition reaction of the \(\beta\) phase at 633 K inhibits the complete equilibration of the samples, even after long term annealing. Small discrepancies in values, between respective samples after annealing (from different series), are visible and show the importance of the thermal history of Ag-Sb-Te alloys.