## 2 Theoretical Background: The Lever Rule and the Short-Range Order Configuration Model
### The Original SROC Model  
According to the lever rule, the SRO structure of binary borate glasses with the same nominal composition should be identical and independent of the cation type. Unfortunately, this is not the case, and the SRO structure not only exhibits a strong dependence on the cation type, but in addition, for a particular glass composition, there is clear evidence for the existence of structural units that are not predicted from the lever rule; see, for example, [15; 16; 19; 20; 21; 44]. In order to resolve the previous issues, the lever rule approach was extended by assuming that the functions of Figure 1 represent the normalized contributions of the particular short-range order configurations (SROC) in binary borate glasses, namely xMO-(1-x)B\({}_{2}\)O\({}_{3}\), with x = 0 (N), x = 0.5 (M), x = 0.67 (P), and x = 0.75 (Q). Then, these functions, denoted as \(X_{\underline{N}}(x),X_{\underline{M}}(x),X_{\underline{P}}(x)\), and \(X_{\underline{O}}(x)\), can be used to derive the SRO structure of any intermediate glass composition. The same extension is valid also in the case of xM\({}_{2}\)O-(1-x)B\({}_{2}\)O\({}_{3}\) glasses, where M is a monovalent metal ion. To further formulate the SROC model, a mathematical formalism was adopted starting from a 5-dimensional vector space, where the fundamental basis is the set of the short-range order building units, denoted as \(B_{j}\), where \(j\) is the number of bridging oxygen atoms per boron atom (\(j=0,1,2,3,4\)). In addition, the components of each vector in this representation are denoted by \(f_{\underline{X}j}\), where the first index \(\underline{X}\) denotes a fundamental SROCnode, namely \(\underline{\rm N}\), \(\underline{\rm M}\), \(\underline{\rm P}\) or \(\underline{\rm Q}\), and \(j\) is again the number of bridging oxygen atoms per boron atom (\(j=0,1,2,3,4\)). For example, within this notation, \(B_{4}\) represents the metaborate tetrahedron \({\rm B}{\cal O}_{4}\)-, and \(B_{0}\) represents the orthoborate triangle \({\rm B}{\cal O}_{3}\)\({}^{3-}\). This formalism does not account for the presence of orthoborate tetrahedra \({\rm B}{\cal O}_{2}{\rm O}_{2}{}^{3-}\) since in the original study there was no experimental evidence for their presence in the structure of the binary zinc borate glasses investigated. Nevertheless, an extension to a 6-dimensional vector space, including the possible presence of orthoborate tetrahedra, is straightforward. With the above formalism, for a particular composition of a fundamental node, i.e., the metaborate one, \({\rm M}\), with \({\rm x}=0.50\) and \(X_{\underline{\rm M}}(0.50)=1\), the SROC, namely \(\underline{\rm M}\), is generally a linear combination of the borate species \(B_{j}\):  
\[\underline{\rm M}=\sum_{j=0}^{4}f_{\underline{\rm M},j}B_{j}=f_{\underline{ \rm M},0}B_{0}\ +\ f_{\underline{\rm M},1}B_{1}+f_{\underline{\rm M},2}B_{2}+f_{\underline{ \rm M},3}B_{3}+f_{\underline{\rm M},4}B_{4} \tag{6}\]  
where \(f_{\underline{\rm M},j}\) denotes the components of the \(\underline{\rm M}\) vector in the \(B_{j}\) basis. Equivalently, we can represent the vector \(\underline{\rm M}\) with its components, \(\underline{\rm M}=\big{(}f_{\underline{\rm M},0},f_{\underline{\rm M},1},f_{ \underline{\rm M},2},f_{\underline{\rm M},3},f_{\underline{\rm M},4}\big{)}\). We observe that \(\sum_{j=0}^{4}f_{\underline{\rm M},j}=1\) for any fundamental node \(\underline{\rm X}\) and that the components \(f_{\underline{\rm M},j}\) coincide with the molar fractions of the borate units for glass with the metaborate composition. As a starting point, it is clear that for \({\rm x}=0\), the corresponding SROC, \(\underline{\rm N}\), consists exclusively of neutral borate triangles (\({\rm B}{\cal O}_{3}\)) or in vector notation \(\underline{\rm N}=(0,0,0,1,0)\).