## 3 Results and Discussion
### Ionic Conductivity and Dielectric Relaxation Investigations
#### 3.2.2 Broadband Dielectric Permittivity Measurements
This image was unclear for the 245 MPa sample, however, where the relaxation peak had almost vanished, careful observation of the waveform revealed that it had greatly diminished in amplitude, instead spreading out over a much wider FWHM. Aside from that, dielectric loss followed the temperature dependency typical of dielectric materials, increasing in magnitude with rising temperature due to interfacial polarization and thermal vibration losses [56]. Figure 11: Real and imaginary components of permittivity, \(\varepsilon^{\prime}\) (dielectric constant) and \(\varepsilon^{\prime\prime}\) (dielectric loss), respectively, 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 are plotted according to the common colormaps displayed on the right of figures; scans corresponding to RT (25 \({}^{\circ}\)C) are highlighted in bold. By tracking the relaxation peaks on the frequency series, the central relaxation time \(\tau\) can be determined as the inverse peak frequency (critical frequency) and correlated with temperature (Figure 12a). However, it needs to be noted that the central relaxation time determined from the associated relaxation peaks did not coincide with the inverse of the crossover frequency obtained from the Jonscher power law by nonlinear fitting [69]; this implies that dielectric relaxation does not correspond to a single Debye process [78]. The activation energy of relaxation \(E_{a,R}\) can be evaluated from the slope of relaxation time (in the same manner as activation energy of ionic transport is calculated from dc conductivity), since it follows an exponential relationship based on the formula [69]:  
\[\tau=\tau_{0}\exp(E_{a,R}/k_{B}T) \tag{7}\]  
where \(\tau_{0}\) is the pre-exponential factor of relaxation time. Activation energies of relaxation for the commercial reference and the in-lab prepared samples were evaluated at 349 meV (vs. \(E_{a}\) of 322 meV for reference), 354 meV (vs. \(E_{a}\) of 350 meV for 540 MPa), 347 meV (vs. \(E_{a}\) of 349 meV for 345 MPa), and 279 meV (vs. \(E_{a}\) of 330 meV for 245 MPa), respectively. The very close proximity of \(E_{a,R}\) and \(E_{a}\)--with the striking exception of the 245 MPa sample--revealed that Na\({}^{+}\) ions within the NASICON lattice have to overcome the same energy barrier for both transport and relaxation. In the case of the 345 MPa and 540 MPa samples, the two processes happen simultaneously--in the commercial reference, without excess zirconia inhibiting ionic transport, relaxation succeeded migration by 27 meV. On the other hand, the 245 MPa sample displayed significant hysteresis between relaxation and ionic transport (~50 meV), with the lower energy barrier for relaxation implying an early dissipation of energy and thus diminishing ionic conductivity. Figure 12: (**a**) Relaxation time \(\tau\) over scaled reciprocal temperature for all in-lab prepared NASICON samples compared with the commercial reference; Cole—Cole plots of dielectric loss against dielectric constant for temperatures ranging from 200 \({}^{\circ}\)C to \(-\)100 \({}^{\circ}\)C for NASICON pellets assembled by applying pressure levels of (**b**) 245 MPa, (**c**) 345 MPa, and (**d**) 540 MPa. Temperature profiles were plotted according to the common colormaps displayed to the right of the figures; scans corresponding to RT (25 \({}^{\circ}\)C) are highlighted in bold.