## 3 Results and Discussion
### The Augmented SROC Model, ASROC
Since we inserted an additional node at x = 0.3, the new set of equations for the molar fraction of each particular borate entity \(B_{j}\), \(\widetilde{B}_{j}\) takes the following form:  
\[X_{\underline{M}}+X_{0.3}=1\]1 \[X_{\underline{M}}+X_{0.3}=1\]2 (17b) \[X_{\underline{M}}+X_{0.3}=1\]3  
The component values \(f_{\underline{q},j}\) for the SROCs N, M, 0.3, P, and Q are presented in \[X_{\underline{M}}+X_{0.3}=1\]4 The results for the molar fractions of the borate units, derived from Equation (17a-c) are depicted in \[X_{\underline{M}}+X_{0.3}=1\]5  
The molar fractions of Figure 5 coincide with the experimental results for the SRO structure at x = 0.2, x = 0.3, and x = 0.4 of our previous study [43] since we used this information as an input in the ASROC model. Nonetheless, it is still possible to compare  
\[X_{\underline{M}}+X_{0.3}=1\]6 Molar fractions of the borate entities in xBi\({}_{2}\)O\({}_{3}\)-(1-x)B\({}_{2}\)O\({}_{3}\) glasses for \(0\leq x\leq 0.40\). Straight lines represent the fractions calculated by the augmented SROC model (ASROC) with the three fundamental nodes \(\underline{N}\), (x = 0); \(\underline{M}\) (x = 0.25); and P (x = 0.4) and the new node at x = 0.3. \(X_{4}\), \(X_{3}\), \(X_{2}\), \(X_{1}\), and \(X_{0}\) denote the molar fractions of B\(\mathcal{O}_{4}\) –, B\(\mathcal{O}_{3}\), B\(\mathcal{O}_{2}\)O\({}^{-}\), B\(\mathcal{O}_{2}\)O\({}_{2}\)\({}^{-}\), and BO\({}_{3}\)\({}^{3-}\) units, respectively. Square and open triangles symbols correspond to the molar fraction of B\(\mathcal{O}_{4}\) – units from the NMR study of ref. [45] and from the neutron scattering study of ref. [28]. \begin{table}
\begin{tabular}{c c c c c c} \hline
**SROC** & \(f_{\underline{q},\underline{0}}\) & \(f_{\underline{q},\underline{1}}\) & \(f_{\underline{q},\underline{2}}\) & \(f_{\underline{q},\underline{3}}\) & \(f_{\underline{q},\underline{4}}\) \\ \hline N (x = 0) & 0 & 0 & 0 & 1 & 0 \\ M (x = 0.25) & 0 & 0 & 0.63 & 0 & 0.37 \\
0.3 (x = 0.3) & 0 & 0.29 & 0.30 & 0 & 0.41 \\ P (x = 0.4) & 0.43 & 0.14 & 0 & 0 & 0.43 \\ \hline \end{tabular}
\end{table} 
Table 3: Components of the SROCs for nodes N, M, 0.3, and P according to Equation (17). For details, see text. The subscript q represents an SROC node and q \(\in\) (\(\underline{N}\), M, 0.3, P). our results with the available experimental data for the molar fraction of B\(\mathcal{O}_{4}\)\({}^{-}\) tetrahedra from the NMR study of ref. [45], included in Figure 5 as square symbols, and from the neutron scattering study of ref. [28], inserted in Figure 5 as open triangles. As evidenced, there is an overall close agreement between the predictions of the ASROC model and the experimental NMR and neutron diffraction data. The ASROC model provides the molar fractions in a continuous way in the 0 \(\leq\) x \(\leq\) 0.40 interval and, more interestingly, by considering only the SRO structure of three distinct glass compositions, i.e., x = 0.25, x = 0.30, and x = 0.40. Another interesting trend from Figure 5 is the behavior of the molar fraction \(X_{1}\) of pyroborate triangles. As shown, \(X_{1}\) progressively increases after the metaborate composition at x = 0.25 up to x = 0.30. Then, \(X_{1}\) decreases up to the pyroborate composition at x = 0.40. This decrease is accompanied by a substantial increase in the population of orthoborate triangles and a slower increase in borate tetrahedra. These structural rearrangements can be fully understood by invoking the disproportionation reaction (14). Thus, the ASROC model, with the addition of only one node, incorporates the mechanism of these rearrangements. As a final remark, it would be tempting to try to extend the model above x = 0.40. In order to accomplish this task, first of all, the nominal orthoborate node at x = 0.50 has to be translated to higher composition values. Based on our previous structural findings in bismuth borate glasses [43], the glass with x = 0.80 is characterized by an SRO structure of orthoborate units only.