## 3 Results and Discussion
### The Augmented SROC Model, ASROC  
As previously evidenced, the SROC model cannot describe the SRO structure in bismuth borate glasses. Moreover, this weakness is probably attributed to the onset of disproportionation reaction (14) that affects directly the SRO structure of the glass and is not considered in the SROC model with the four fundamental nodes. In this section, we enhance the SROC model by inserting an additional node in order to incorporate the available experimental data of ref. [43]. The obvious choice is the node with x = 0.30, denoted as \(X_{0.3}\), since the SRO structure is already known from Table 1.  
First, we observe that x = 0.30 lies in region II (0.25 \(\leq\) x \(\leq\) 0.40), so there is a partition of this range into two new subintervals, namely region IIa, 0.25 \(\leq\) x \(\leq\) 0.30, and region IIb, 0.30 \(\leq\) x \(\leq\) 0.40. In order to proceed, we observe that for x = 0.30, \(A(0.3)\)\(\approx\) 2.143. Thus, we have the following set of equations:  
**Region IIa**  
\[X_{\underline{M}}+X_{0.3}=1\]  
\[2X_{\underline{M}}+2.143X_{0.3}=A(x)\]It is more instructive to express the coefficients appearing in the second equation with the values of the \(A(x)\) function, that is, \(A(0.25)=2\) and \(A(0.3)=2.143\). Solving the above system, we find  
\[X_{\underline{M}}=\frac{A(0.3)-A(x)}{A(0.3)-A(0.25)} \tag{15a}\] \[X_{0.3}=\frac{A(x)-A(0.25)}{A(0.3)-A(0.25)} \tag{15b}\]  
**Region IIb**: \[X_{0.3}+X_{\underline{P}}=1\] \[2.143X_{0.3}+2.5X_{\underline{P}}=A(x)\]  
Recalling that \(A(0.4)=2.5\), we obtain  
\[X_{\underline{P}}=\frac{A(x)-A(0.3)}{A(0.4)-A(0.3)} \tag{16a}\] \[X_{0.3}=\frac{A(0.4)-A(x)}{A(0.4)-A(0.3)} \tag{16b}\]  
Equations (15) and (16) are quite symmetric and can be easily generalized in the case of inserting a node q between the fundamental nodes \(\underline{M}\) and \(\underline{P}\). This is accomplished by replacing the term \(A(0.3)\) in the above equations with the appropriate \(A(q)\) value. It is easy to see that the additional node fulfills the necessary conditions II and III, as described in Section II. The SROC functions calculated by Equations (10), (12), (15) and (16) of the augmented SROC model (ASROC) are represented in Figure 4. \(X_{\underline{N}}(x),X_{\underline{M}}(x),X_{\underline{P}}(x),X_{\underline{Q }}(x)\), and \(X_{0.3}(x)\) denote the corresponding SROC functions of the four fundamental nodes \(\underline{N}\), (\(\kappa=0\)); \(\underline{M}\) (\(\kappa=0.25\)); \(\underline{P}\) (\(\kappa=0.4\)); and \(\underline{O}\) (\(\kappa=0.5\)) and of the new node at x = 0.3, respectively. \[X_{\underline{M}}+X_{0.3}=1\]0 SROC functions in xBi\({}_{2}\)O\({}_{3}\)-(1-x)B\({}_{2}\)O\({}_{3}\) glasses calculated by Equations (10), (12), (15) and (16). \(X_{\underline{N}}(x),X_{\underline{M}}(x),X_{\underline{P}}(x),X_{\underline{Q }}(x)\), and \(X_{0.3}(x)\) denote the corresponding SROC functions of the four fundamental nodes \(\underline{N}\), (\(\kappa=0\)); \(\underline{M}\) (\(\kappa=0.25\)); \(\underline{P}\) (\(\kappa=0.4\)); and \(\underline{O}\) (\(\kappa=0.5\)) and of the new node at x = 0.3, respectively. For details, see text. In the framework of the ASROC model, the molar fractions of the borate entities for the SRO structure of bismuth borate glasses can be calculated by extending Equation (13a,b).