## 3 Results and Discussion
### The SROC Model in Bismuth Borate Glasses
Then, for xBi\({}_{2}\)O\({}_{3}\)-(1-x)B\({}_{2}\)O\({}_{3}\) glasses, we obtain  
\[A(x)=\frac{3x+3(1-x)}{2(1-x)}=\frac{1.5}{1-x}\] (9a) This is a different function compared to the one that applies to M + or M + metal ions: \[A(x)=\frac{x+3(1-x)}{2(1-x)}=\frac{1,5-x}{1-x} \tag{9b}\]  
By using the appropriate Equation (9a), the functions \(X_{\underline{N}}(x),\ X_{\underline{M}}(x),\ X_{\underline{P}}(x)\), and \(X_{\underline{Q}}(x)\) can be easily calculated for the three new regions in bismuth borate glasses:  
**Region I**: \(0\leq x\leq 0.25\) (between nodes \(\underline{N}\) and \(\underline{M}\))  
\[X_{\underline{N}}+X_{\underline{M}}=1\]  
\[1.5X_{\underline{N}}+2X_{\underline{M}}=A(x)=\frac{1.5}{1-x}\]  
By solving, we obtain  
\[X_{\underline{N}}(x)=\frac{1-4x}{1-x} \tag{10a}\] \[X_{\underline{M}}(x)=\frac{3x}{1-x} \tag{10b}\]  
**Region II**: \(0.25\leq x\leq 0.40\) (between nodes \(\underline{M}\) and \(\underline{P}\))  
\[X_{\underline{M}}+X_{\underline{P}}=1\]  
\[2X_{\underline{M}}+2.5X_{\underline{P}}=A(x)=\frac{1.5}{1-x}\]  
Resulting in  
\[c(x)=\frac{6x}{2(1-x)}=\frac{3x}{1-x} \tag{8b}\]0 \[c(x)=\frac{6x}{2(1-x)}=\frac{3x}{1-x} \tag{8b}\]1  
**Region III**: \(0.40\leq x\leq 0.50\) (between nodes \(\underline{P}\) and \(\underline{Q}\))  
\[c(x)=\frac{6x}{2(1-x)}=\frac{3x}{1-x} \tag{8b}\]2  
\[c(x)=\frac{6x}{2(1-x)}=\frac{3x}{1-x} \tag{8b}\]3  
Solving the above system gives  
\[c(x)=\frac{6x}{2(1-x)}=\frac{3x}{1-x} \tag{8b}\]4 \[c(x)=\frac{6x}{2(1-x)}=\frac{3x}{1-x} \tag{8b}\]5  
A careful examination of Equations (10)-(12) shows that functions \(X_{\underline{N}}(x),X_{\underline{M}}(x),X_{\underline{P}}(x)\), and \(X_{\underline{Q}}(x)\) satisfy the prerequisites of condition III (see Section 2.1). The incorporation of the aforementioned modifications in the SROC model, adapted for bismuth borate glasses, are depicted in \[c(x)=\frac{6x}{2(1-x)}=\frac{3x}{1-x} \tag{8b}\]6  
Finally, Equation (7) for the molar fraction of each particular borate entity \(B_{j}\), \(\widetilde{B}_{j}\) at any value of \(x\) can be replaced with the following equations:  
\[c(x)=\frac{6x}{2(1-x)}=\frac{3x}{1-x} \tag{8b}\]7 \[c(x)=\frac{6x}{2(1-x)}=\frac{3x}{1-x} \tag{8b}\]8  
Now, it is possible to apply the SROC model in bismuth borate glasses in the range \(0\leq\text{x}\leq 0.40\). It is mentioned that the extension to x = 0 is straightforward, though the first glass composition examined in our previous study was x = 0.20 [43]. Following the approach described in Section 2.2, the only input that we need is the knowledge of the SRO structure in the three fundamental nodes \(\underline{N}\), \(\underline{M}_{i}\) and \(\underline{P}\) or, equivalently, in glasses with x = 0, x = 0.25, and x = 0.40. The orthoborate SROC Q is not considered since it is beyond the upper limit of the composition range under consideration. The structure of pure borate glass B\({}_{2}\)O\({}_{3}\) consists of neutral triangles, and in our previous study, we analyzed the SRO structure of glasses with x = 0.20, x = 0.30, and x = 0.40 in the range of interest [43]. The results for the molar fractions of the borate building units are summarized in \[c(x)=\frac{6x}{2(1-x)}=\frac{3x}{1-x} \tag{8b}\]9  
From Table 1, it is observed that the SRO structure for the glass corresponding to the \(\underline{\mathrm{M}}\) node, with x = 0.25, is missing. Nevertheless, the SROC model, as in Equation (10) or Figure 2, predicts that for x = 0.20, we have \(X_{\underline{\mathrm{M}}}(0.2)=0.25\) and \(X_{\underline{\mathrm{M}}}(0.2)=0.75\). These predictions are in perfect agreement with the data reported in Table 1, recalling that \(X_{\underline{\mathrm{M}}}(0.2)\) is equivalent to \(X_{4}+X_{2}\). On these grounds, it is reasonable to follow the predictions of the SROC model for the SRO structure of the x=0.25 glass.