## 3 Results and Discussion  
### Physical Properties  
Density measurements (\(\rho\)) were experimentally conducted based on Archimedes' principle using distilled water as the immersion liquid. The reported results represent the average value of five distinct measurements. The experimental data obtained for the density of the studied samples were used to calculate the molar volume (\(V_{m}\)), the packing density of oxygen atoms (\(OPD\)), and the molar volume of oxygen (\(V_{a}\)). Evaluating these physical parameters is extremely important, providing the opportunity to study the compositional effect on the vitreous network structure. The \(\rho\) and \(\mathrm{V_{m}}\) values obtained for the studied systems are graphically represented as a function of the compositional content in Figure 1.  
It is observed that for the PS system, \(\rho\) and \(\mathrm{V_{m}}\) decrease drastically with the increase in SiO\({}_{2}\) content, in agreement with the data presented in reference [24]. The \(\rho\) values obtained for the PS system range between 6.75 and 5.29 g/cm\({}^{3}\), while the \(\mathrm{V_{m}}\) values range between 35.43 and 25.57 cm\({}^{3}\)/mol. For the PSC, PSE, and PSY systems, \(\rho\) follows an opposite trend compared to \(\mathrm{V_{m}}\), with the density slightly increasing and \(\mathrm{V_{m}}\) decreasing as the rare earth (RE) content is introduced and increased. As \(\mathrm{V_{m}}\) decreases, the glass structure becomes more compact due to the reduction of interatomic spacing and/or the bond length between atoms. These results align with those reported for similar ternary systems doped with RE ions [1; 25]. The \(\rho\) values obtained range between 6.71 and 6.96 g/cm\({}^{3}\), 6.71 and 6.85 g/cm\({}^{3}\), and 6.71 and 7.13 g/cm\({}^{3}\) for the PSC, PSE, and PSY systems, respectively, while the \(\mathrm{V_{m}}\) corresponding to the PSC, PSE, and PSY systems ranges from 30.31 to 29.13 cm\({}^{3}\)/mol, 30.31 to 29.90 cm\({}^{3}\)/mol, and 30.31 to 28.79 cm\({}^{3}\)/mol, respectively. The evolution of the density and \(\mathrm{V_{m}}\) depends on the composition of the systems, with their values influenced by different molecular masses, the type and length of the formed bonds, and the ionic radii of the components, reflecting the structural transformations in the amorphous network as the composition changes [26]. SiO\({}_{2}\) is a classic network former, forming BO (bridging oxygen) bonds, specifically Si-O-Si, and thus strengthens the network. PbO\({}_{2}\) has the ability to behave both as a network former and a network modifier depending on its coordination number and content. At high concentrations, PbO\({}_{2}\) forms a network through Pb-O-Pb bonds, and as its concentration decreases, it acts as a network modifier, with the fraction of BO decreasing in favor of NBO (non-bridging oxygen) atoms, along with the gradual depolymerization of the vitreous network. Network formation for the binary PbO\({}_{2}\)-SiO\({}_{2}\) system is governed by the interaction between PbO\({}_{n}\) polyhedra, where n = 3, 4, and SiO\({}_{4}\) tetrahedra as network formers [4] so that the lead-dioxide-based vitreous network is gradually replaced, as the SiO\({}_{2}\) content increases, by the silicate network, accompanied by an increase in BO fractions. The formation of BO leads to a decrease in V\({}_{m}\) values. In Figure S1, the variations of OPD and V\({}_{0}\) with the composition of the samples from the PS, PSC, PSE, and PSY systems are graphically represented. It can be observed that for all systems, the values obtained for V\({}_{0}\) exhibit an opposite behavior compared to the OPD values. The increase in OPD is associated with the rise in the number of oxygen atoms in the unit chemical composition, which leads to changes in the number of bonds in the network and the average stretching force constant.