## 3 Results and Discussion
### Ionic Conductivity and Dielectric Relaxation Investigations  
#### 3.2.1 Broadband Ionic Conductivity Measurements  
As stated earlier, there exists a considerable body of work reporting on the conductivity of NASICON-type materials based exclusively on impedance analysis decoupled from any form of dielectric relaxation studies--the latter having the potential of offering valuable insight on transport phenomena in solid-state ionic conductors. Seeking to remedy this issue, we chose to forego the typical EIS-based investigative procedure, and shifted instead to coupled dielectric and conductivity measurements over a broadband frequency range (10\({}^{-1}\) to 10\({}^{11}\) Hz), with temperatures starting from 200 \({}^{\circ}\)C and approaching the cryogenic regime at \(-\)100 \({}^{\circ}\)C. The formalism applied here followed similar procedures encountered in the general dielectric/ionic conductivity literature [67; 68; 69; 70; 71; 72; 73], with the addition of parametric phase-space mapping adapted from dynamical systems theory. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Peak Signature** & \multicolumn{3}{c}{**Untreated (RT \(\mid\)80 \({}^{\circ}\)C)**} & \multicolumn{3}{c}{**Calcinated (600 \({}^{\circ}\)C)**} & \multicolumn{3}{c}{**Sintered (1250 \({}^{\circ}\)C)**} \\ \hline
**2\(\theta\)** & **Miller** & \(d\) & **FWHM** & **Size** & \(d\) & **FWHM** & **Size** & \(d\) & **FWHM** & **Size** \\ \hline
**[deg.]** & **(h k l)** & **[Å]** & **[deg.]** & **[Å]** & **[Å]** & **[deg.]** & **[Å]** & **[Å]** & **[deg.]** & **[Å]** \\ \hline
13.8 & 012 & 6.341 & 0.226 & 369 & 6.414 & 0.216 & 387 & 6.453 & 0.186 & 448 \\
19.2 & 104 & 4.564 & 0.337 & 250 & 4.572 & 0.144 & 585 & 4.634 & 0.185 & 454 \\
19.8 & 110 & 4.448 & 0.194 & 435 & 4.481 & 0.161 & 524 & 4.492 & 0.121 & 695 \\
23.0 & 113 & 3.840 & 0.316 & 268 & 3.864 & 0.264 & 320 & 3.872 & 0.215 & 393 \\
28.2 & 024 & 3.134 & 0.253 & 338 & 3.147 & 0.135 & 635 & 3.159 & 0.105 & 813 \\
30.7 & 116 & 2.896 & 0.442 & 195 & 2.923 & 0.310 & 277 & 2.925 & 0.300 & 287 \\
34.1 & 241 & 2.684 & 0.357 & 243 & 2.626 & 0.234 & 371 & 2.627 & 0.250 & 347 \\
34.6 & 300 & 2.584 & 0.452 & 193 & 2.597 & 0.204 & 426 & 2.595 & 0.160 & 544 \\
38.7 & 208 & 2.314 & 0.357 & 246 & – & – & – & 2.334 & 0.092 & 951 \\ \hline \end{tabular}
\end{table} 
Table 2: XRD parameters of primary NASICON peak signatures for the untreated, calicinated, and sintered powdered materials. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Peak Signature** & \multicolumn{3}{c}{**Low Pressure (245 MPa)**} & \multicolumn{3}{c}{**Medium Pressure (345 MPa)**} & \multicolumn{3}{c}{**High Pressure (540 MPa)**} \\ \hline
**2\(\theta\)** & **Miller** & \(d\) & **FWHM** & **Size** & \(d\) & **FWHM** & **Size** & \(d\) & **FWHM** & **Size** \\ \hline
**[deg. ]** & **(h k l)** & **[Å]** & **[deg. ]** & **[Å]** & **[Å]** & **[deg. ]** & **[Å]** & **[deg. ]** & **[Å]** & **[deg. ]** & **[Å]** \\ \hline
20.1 & – & 4.425 & 0.092 & 920 & 4.413 & 0.109 & 774 & 4.441 & 0.101 & 834 \\
27.0 & – & 3.299 & 0.078 & 1095 & 3.292 & 0.078 & 859 & 3.290 & 0.088 & 974 \\
28.3 & – & 3.159 & 0.105 & 813 & 3.154 & 0.1 The total conductivity, \(\sigma(\omega)\) [S cm\({}^{-1}\)], comprises the sum of ac and dc conductivities, expressed according to the Jonscher/Almond-West power law [70; 71; 72; 73; 74]:  
\[\sigma(\omega)=\sigma_{ac}+\sigma_{dc}\equiv A\omega^{n}+\sigma_{dc} \tag{1}\]  
where \(\sigma_{ac}\) is the frequency-dependent ac conductivity (reflective of information on the chemical environment within the material and used in the analysis of ionic dynamics) and \(\sigma_{dc}\) the temperature-dependent bulk dc electrical conductivity; \(\omega\) is the angular frequency based on the formula \(\omega=2\pi f\)[rad s\({}^{-1}\)], with \(f\) [Hz] being the ac frequency; \(A\) and \(n\) are the semi-empirical temperature-dependent coefficient and frequency exponent of the mobile ions (\(0\leq n\leq 1\)), respectively [56; 67; 68; 69; 70; 71; 72; 73; 74].