## 2 Results and Discussion
As each of these peaks must be assigned to a different nanostructured phase, we can argue that phase A is very rich in gold, phase C is very rich in palladium, and phase B is a bimetallic alloy with some intermediate composition. For each phase, the position of the (111) diffraction line provides information on the lattice parameter of the involved crystallographic phase. [PERSON]'s law establishes that the lattice parameter, a\({}_{x}\), of a binary metallic alloy, A\({}_{x}\)B\({}_{1-x}\), varies linearly with the concentration, x, of the constituting elements of the alloy (a\({}_{x}\) = a\({}_{A}\) + x (a\({}_{B}\) - a\({}_{A}\)), with a\({}_{A}\) and a\({}_{B}\) being the lattice parameters of the pure compounds. The Scherrer law relates the full width at half maximum of a diffraction peak to the  
Figure 3: Analysis of A-XRD difference patterns for P50 (upper panels) and P75 (lower panels) samples at Au (left) and Pd (right) edges. The experimental points (red dots) and the best fit line (black line) are shown together, with peaks corresponding to the bimetallic phases A (orange shade), B (green shade), and C (blue shade). The residuals (experimental-fit) are shown at the bottom (blue dots), shifted for clarity. volume-weighted average particle diameter, D (A). The relative peak area is proportional to the phase fraction. The quantitative analysis of the \(\Delta\)I difference data yields two sets of results (one obtained from the data at the gold edge and the other from the data at the palladium edge). As illustrated in Table 1, the results from the two independent sets match significantly well to one another. Not only is the trend in the change of the composition from phase A to phases B and C the same, but also the respective Au and Pd atomic percentages are very close to each other. These results show that the quantitative analysis of the data is self-consistent and that the discrepancies between the two sets of data can be assumed as an estimate of the \"experimental uncertainty\" of the method. Looking at the results obtained at the Au edge (but similar considerations also apply for the Pd edge), as far as the chemical composition is concerned, phase A turns out to have atomic Au percentages of \(98.0\pm 1.4\%\) and \(94.4\pm 1.3\%\) in P75 and P50 respectively, so it can be considered to be formed by almost pure gold. As pure gold is not active in the DS reaction, this phase can be considered of little relevance in the catalytic behavior of P50 and P75. Moreover, as indicated by the TEM characterization, the amount of the largest MNPs is very low in both specimens; according to Scherrer analysis of the A-XRD data (see below), phase A is formed by the largest MNPs both in P50 and P75. This supports the hypothesis that phase A does not contribute to the overall activity of these catalysts in the DS reaction. Considering phase B, data analysis shows that this phase has an intermediate gold content, which is higher in P50 (\(83.6\pm 2.4\%\)) and lower in P75 (\(\sim\)65.9\(\%\)). This result is easily explained considering that in P50, the overall Au/Pd ratio is 1/1.85 while in P75 it is 1/5.57. It follows that P75 will be necessarily richer in Pd and lower in Au for phases B and C. Accordingly, the atomic percentage of Au present in phase C is \(\sim\)0.1\(\%\) for P75 and \(8.6\pm 0.1\%\) in P50. In summary, phase C is very rich in palladium (\(>\)90\(\%\) Pd) in both specimens, and it is almost pure palladium in P75. Within each phase, the size of the MNPs, as assessed from the Scherrer analysis of the respective (111) peaks, was strictly comparable in both P50 and in P75 (Table 2). Phase A always shows the largest MNPs, with diameters of about 11 nm for P50 and about 16 nm for P75. As mentioned above, TEM characterization suggests that the number of MNPs of phase A is very small. This is in agreement with the fact that these MNPs being so large and so rich in gold must be only a few, as the least abundant metal in the specimens (gold) is concentrated in this phase.