## 3 Results and Discussion
### Ionic Conductivity and Dielectric Relaxation Investigations
#### 3.2.2 Broadband Dielectric Permittivity Measurements  
Following the formalism for conductivity, dielectric permittivity is expressed as a complex phasor (\(\hat{\epsilon}\in\mathbb{C}\)) comprised of real and imaginary parts:  
\[\hat{\epsilon}(\omega)=\Re(\epsilon(\omega))+\Im(\epsilon(\omega))\equiv \epsilon^{\prime}(\omega)-i\epsilon^{\prime\prime}(\omega)=|D_{0}/E_{0}|( \cos\delta-i\sin\delta) \tag{5}\]  
where \(\epsilon^{\prime}\) and \(\epsilon^{\prime\prime}\) are the dielectric constant and dielectric loss terms, respectively; \(\omega\) is the angular frequency; \(D_{0}\) and \(E_{0}\) are the amplitudes of the charge displacement and electric fields respectively, and \(\delta\) is the loss angle, which quantifies the dielectric's inherent dissipation of electromagnetic energy. The loss tangent (dissipation factor) is then defined by the complex-plane angle between the dissipative and non-dissipative terms:  
\[\tan\delta=\epsilon^{\prime\prime}/\epsilon^{\prime} \tag{6}\]  
Frequency dependence reflects the material's causal response to an oscillating electric field, represented by a phase difference; dielectric relaxation is the momentary delay in polarization, analogous to hysteresis in oscillating magnetic fields. Figure 11 shows the variation of the real and imaginary components of permittivity, \(\epsilon^{\prime}\) and \(\epsilon^{\prime\prime}\), respectively, against frequency over the full temperature scanning range (from 200 \({}^{\circ}\)C to \(-\)100 \({}^{\circ}\)C) for NASICON pellets assembled over different pressure levels. Initial observations were made for both components regarding the dissipation and relaxation profiles of the material. The real component of dielectric permittivity, \(\epsilon^{\prime}\), remained constant at RT throughout the lowest frequencies and up to 10 kHz--this zone of stability was considerably diminished for the 245 MPa sample, which started deteriorating at a much lower frequency of 100 Hz. At higher temperatures, the dielectric constant rose steeply while its stable zone became transposed toward the higher frequencies. At high temperatures and low frequencies, this rise can be attributed to temperature-dependent ionic transport--at high frequencies, thermal vibration instead enhances the loss of dispersion [56]. The reverse holds true for the lower temperature regime, with the dielectric constant receding toward the lower frequencies and diminishing in value, its stable high zone effectively vanishing by \(-\)100 \({}^{\circ}\)C. The above occur as direct effects of free charge carriers at low frequencies and interfacial polarization at the higher end of the spectrum, where the rapid periodic reversals of the electric field at the interface counteracts and negates the ionic contribution to the dielectric constant [67]. Essentially, over high frequencies, the decrease in orientation polarization and the stabilization of the dielectric constant occurred due to the interruption of long-range ionic migration [70]. This steady-state dielectric constant constitutes the high-frequency permittivity limit and is denoted as \(\varepsilon_{\infty}\); its value can be assessed by scans near the cryogenic regime without having to access prohibitively high frequencies (above \(10^{7}\) Hz, which is the limit of the broadband scanner) to negate dipole contributions to polarization. The imaginary component of permittivity, the dielectric loss \(\varepsilon^{\prime\prime}\), provides information on the dissipative processes within the material under the influence of an applied electric field. In the frequency series of Figure 11, the onset of relaxation peaks was clearly observed over the mid-frequency range (\(10^{4}\) to \(10^{5}\) Hz) at RT, the waveform moved toward lower frequencies with decreasing temperature and the reverse for increasing--the amplitude of the relaxation peak remained constant throughout these transposals.