## 3 Results and Discussion
### Ionic Conductivity and Dielectric Relaxation Investigations
#### 3.2.1 Broadband Ionic Conductivity Measurements
As a general observation, the NASICON pellet produced by applying 345 MPa of pressure displayed the closest approximation to the reference (e.g., 0.34 mS cm\({}^{-1}\) vs. 0.35 mS cm\({}^{-1}\) of conductivity at the highest measured frequency of 10\({}^{7}\) Hz, respectively) within the range of typical values for NASICON at RT [27]. The parameters \(\sigma_{dc}\), \(A\), and \(n\) of the Jonscher power law (Equation (1)) were extracted from the frequency series after conversion into the angular frequency format by using the nonlinear Levenberg-Marquardt iteration algorithm. The crossover frequency \(\omega_{p}\) [rad s\({}^{-1}\)] marks the transition from dc conductivity to ac dispersion (its inverse provides an assessment of relaxation time), and is also derived from the power law [69,74]:  
\[\omega_{p}=(\sigma_{dc}/A)^{1/n} \tag{3}\]  
Figure 7: Real and imaginary components, \(\sigma^{\prime}\) and \(\sigma^{\prime\prime}\) respectively, of conductivity scanned over the broadband frequency range of 10\({}^{-1}\) to 10\({}^{11}\) Hz and for temperatures ranging from 200 \({}^{\circ}\)C to \(-\)100 \({}^{\circ}\)C. Scans for NASICON pellets assembled by applying isostatic pressure levels of (**a**) 245 MPa, (**b**) 345 MPa, and (**c**) 540 MPa are displayed. Temperature profiles were plotted according to the common colormaps displayed on the right of figures; scans corresponding to RT (25 \({}^{\circ}\)C) are highlighted in bold, along with ac conductivity values recorded at the highest frequency levels of the frequency-dependent dispersion zones for RT. In Figure 9, the results of the fitting procedure are displayed. Bulk dc conductivity, \(\sigma_{dc}\), clearly displays temperature dependence by rising proportionally with temperature for all  
Figure 8: Parametric 2D and 3D phase-space mappings of the real component \(\sigma^{\prime}\) of conductivity scanned over a broadband frequency range of \(10^{-1}\) to \(10^{11}\) Hz and for temperatures ranging from 200 \({}^{\circ}\)C to \(-100\)\({}^{\circ}\)C. Scans for NASICON pellets assembled by applying pressure levels of (**a**) 245 MPa, (**b**) 345 MPa, and (**c**) 540 MPa are displayed. 3D mappings portray the \(XY\) plane reflected across the \(Y\)-axis bisector, revealing the frequency-dependent dispersion region as a narrow diagonal ridge preceded by the dc plateau. Conductivity scales adhere to the common colormap to the right of the figures. samples, from \(10^{-9}\) to \(10^{-8}\) S cm\({}^{-1}\) at \(-100\)\({}^{\circ}\)C and reaching the \(10^{-4}\) to \(10^{-3}\) S cm\({}^{-1}\) scale at \(100\)\({}^{\circ}\)C. The in-lab prepared sample at 345 MPa displayed the closest approximation to the commercial one, with dc conductivity reaching 2.49 mS cm\({}^{-1}\) vs. 2.68 mS cm\({}^{-1}\), respectively at 100 \({}^{\circ}\)C, but diverging by an order of magnitude at the low-temperature zone. This can be attributed to the higher availability of charge carriers becoming thermally excited and overcoming the minimum energy barriers at the higher temperatures [56]. The activation energy \(E_{a}\) associated with ionic transport can be evaluated from the variation of dc conductivity expressed by the empirical Arrhenius equation [75]:  
\[\sigma=\sigma_{0}\exp(-E_{a}/k_{B}T) \tag{4}\]  
where \(\sigma_{0}\) [S cm\({}^{-1}\)] is the pre-exponential factor; \(E_{a}\) [eV] is the activation energy for the Na\({}^{+}\) ion transfer; \(k_{B}\) is the Boltzmann constant (8.617\(\cdot 10^{-5}\) eV K\({}^{-1}\)); and \(T\) [K] is the temperature. Activation energies for the commercial reference and the in-lab prepared samples were evaluated at 322 meV (reference), 350 meV (540 MPa), 349 meV (345 MPa), and 330 meV (245 MPa), respectively. The measured conductivities and derived activation energies were within the reported limits for NASICON [27; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55].