## 3 The Material
More specifically, the equation governing the mixture is given by  
\[G=\sum_{i}g_{i}x_{i} \tag{15}\]  
where \(g\) is the bulk parameter of constituent \(i\), \(x\) is the additivity parameter (volume fraction in this case), and \(G\) is the property of the aggregate glass. For purposes of discussion, assume a binary glass mixture having constituents A and B, each possessing a bulk material value of g (which are refractive index [65], mass density [66], and the inverse of the acoustic velocity [67]; a simple derivation can be found in [68]). The constituents are assumed to be well-mixed and that the glass structure does not change significantly across a desired compositional range. In other words, the binary system acts as a simple mixture across that compositional range. The volume fraction can be expressed in terms of the mass density, molar mass (\(M\)), and the molar concentration \([C]_{m00\%}\) of each constituent as \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dopant** & \(\Delta n\) **(\%/wt\%)** & \(\Delta V_{S}\) **(\%/wt\%)** & \(\Delta V_{L}\) **(\%/wt\%)** \\ \hline GeO\({}_{2}\) & +0.056 & \(-0.49\) & \(-0.47\) \\ P\({}_{2}\)O\({}_{5}\) & +0.020 & \(-0.41\) & \(-0.31\) \\ F & \(-0.31\) & \(-3.1\) & \(-3.6\) \\ TiO\({}_{2}\) & +0.23 & \(-0.45\) & \(-0.59\) \\ Al\({}_{2}\)O\({}_{3}\) & +0.063 & +0.21 & +0.42 \\ B\({}_{2}\)O\({}_{3}\) & \(-0.033\) & \(-1.18\) & \(-1.23\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Empirical effect of some common dopants on the optical index and shear and longitudinal acoustic velocities of silica. Data from [55]. \[x_{A}=\frac{\frac{M_{A}}{P_{A}}\big{[}C_{A}\big{]}_{mol\%}}{\frac{M_{A}}{P_{A}} \big{[}C_{A}\big{]}_{mol\%}+\frac{M_{B}}{P_{B}}\big{[}C_{B}\big{]}_{mol\%}}, \tag{16}\]  
with \(x_{B}=1-x_{A}\). The resulting calculation gives the property of the binary glass, ideally across the whole compositional range of 100% of constituent A to 100% of constituent B. In the cases where the bulk values are not known, they can become empirical fitting parameters that, once found, can serve as a basis for extrapolation to a more optimized composition. As previously discussed, a summary of material properties obtained through these means can be found in [35]. Above-and-beyond what has been identified in [55], the work in [35] includes Y\({}_{2}\)O\({}_{3}\)[69], MgO [33], SrO [70], BaO [71], Yb\({}_{2}\)O\({}_{3}\)[72], La\({}_{2}\)O\({}_{3}\)[73], Lu\({}_{2}\)O\({}_{3}\)[74], and Li\({}_{2}\)O [34], providing a broad palette of dopant selection for glass design. Interestingly, it is possible to deduce both the density and index of the bulk materials, so long as both are assumed for one of the constituents (usually for pure silica as the base material). To illustrate this, Figure 4 provides an example calculation for binary germanosilicate glass utilizing the parameters found in [75] (blue curve). When the mass density is modified for the same bulk refractive index value (these can be read from the endpoints on the graph), the curvatures of the graphs change, offering a means to obtain fittings for it with some degree of confidence. To clarify, Equation (15) does not necessarily give insight into the glass structure. Instead, it is a simple _empirical_ model that can be used in the design of the glass comprising an optical fiber. Most importantly, it provides sufficient degrees-of-freedom to obtain very good fits to experimental data across wide compositional ranges. With caution, this enables simple extrapolations to compositions more suitable for a given application. The methodology has successfully been applied to BaO [71] and Al\({}_{2}\)O\({}_{3}\)[76] silicates, but to the best of the authors' knowledge has not yet been used in germanosilicate glasses of varying mass density for a fixed composition (or, in other words, a varying bulk GeO\({}_{2}\) mass density).