## 3 Results and Discussion
### Ionic Conductivity and Dielectric Relaxation Investigations
#### 3.2.1 Broadband Ionic Conductivity Measurements
The frequency exponent \(n\) reflects the degree of connectedness of long-range ion-hopping pathways or the extent of tortuosity (a measure of the geometric complexity in porous media) for mobile charges [72]. Conductivity, being a complex phasor (\(\tilde{\sigma}\in\mathbb{C}\)) comprised of real and imaginary parts, is expressed as:  
\[\tilde{\sigma}(\omega)=\Re(\sigma(\omega))+\Im(\sigma(\omega))\equiv\sigma^{ \prime}(\omega)-i\sigma^{\prime\prime}(\omega) \tag{2}\]  
In the context of ionic conductivity studies, the metric with a complex value essentially represents a phase lag between the electric field and the ionic current density. Ultimately, primary observations are made by referring to the real part of conductivity [73]. Figure 7 shows the variation of the real and imaginary parts of conductivity, \(\sigma^{\prime}\) and \(\sigma^{\prime\prime}\), respectively, against frequency over the full temperature range for NASICON pellets assembled over different isostatic pressure levels. The double-logarithmic plot of conductivity over frequency typically displays three regions classified on the basis of frequency: (i) the low-frequency polarization region; (ii) the frequency-independent dc conductivity plateau; and (iii) the high-frequency dispersive region corresponding to ac conductivity. For all samples scanned at RT, a similar behavior was observed: the polarization region extended beyond the low frequency zone by reaching into and over the \(10^{4}\) Hz mark, followed by the dc conductivity plateau and the dispersion zone at the high frequency region. This pattern shifted when moving toward the lower temperatures near the cryogenic regime. At \(-100\)\({}^{\circ}\)C, the polarization region became steeper and shorter, establishing firmly at lower frequencies below \(1\) Hz, then gradually relocating its upper limits toward higher frequencies as the temperature rose to RT levels. The plateau that followed was confined to within an order of magnitude on the frequency range, and shifted toward higher frequencies as the temperature rose. The dispersion zone dominated at low temperatures, also following the higher-frequency shifting trend with rising temperature. At high temperatures this behavior persists, with the polarization region spanning the entire lower frequency range, albeit with a milder slope. The dc plateau contracts toward the middle frequency range (below \(10^{4}\) Hz). The ac dispersion zone dominates the higher frequencies (\(10^{4}\) Hz and above), with ionic conductivities at the typical mS cm\({}^{-1}\) benchmark for rhombohedral NASICON at temperatures above \(160\)\({}^{\circ}\)C.  
Identification of this pattern of plateau shifting toward higher frequencies with rising temperature is critical for the successful application of data fitting by the Jonscher power law. In Figure 8, the compound conductivity evolution over frequency and temperature is visualized more comprehensibly in 2D and 3D phase-space mappings, allowing for the detection of any apparent trends in the ionic dynamics that are not readily apparent in more traditional representations. Phase-space mappings for all NASICON samples revealed a frequency- and temperature-dependent regional shift manifesting as a narrow diagonal ridge preceded by the dc plateau. Juxtapsing the various representation formats allows for demarcating the narrow frequency band where the Jonscher power law is applicable (i.e., by excluding the polarization regions and post-dispersion features), so that fitting parameters can be extracted. In the above graphical representations, displays corresponding to the commercial reference sample have been omitted for economy of space, although all fitting parameters have been extracted for comparison.