## 3 Results and Discussion
### Dielectric Studies of Unirradiated and Irradiated PLC Nanocomposites  
The dielectric property of a material is related to the dipole polarization with respect to the applied field. It is usually frequency dependent at lower frequencies and frequency independent at higher frequencies, which is due to the better orientation of the dipoles at the applied field at lower frequencies than at high frequencies [40]. The variations in the dielectric constant (\(\varepsilon^{\prime}\)) and dielectric loss (\(\varepsilon^{\prime\prime}\)) of the unirradiated and EB-irradiated PLC nanocomposites at various temperatures, as well as the EB dose, as a function of frequencies, were studied and are presented in Figures 9 and 10, respectively. The \(\varepsilon^{\prime}\) and \(\varepsilon^{\prime\prime}\) parameters were calculated using the following equations:  
\[\varepsilon^{\prime}=C_{\mathrm{p}}d/(\varepsilon_{0}\ A) \tag{6}\]  
\[\varepsilon^{\prime\prime}=\varepsilon^{\prime}-\tan\delta \tag{7}\]  
where \(d\) is the weight of the sample, \(A\) is the area of the electrode, and \(\varepsilon_{0}\) is the dielectric permittivity in a vacuum (8.85 \(\times\) 10\({}^{-12}\) Fm\({}^{-1}\)). From Figures 9 and 10, it can be seen that \(\varepsilon^{\prime}\) and \(\varepsilon^{\prime\prime}\) were completely frequency dependent at low frequencies of up to 1 kHz, and they decreased suddenly and then remained constant afterward. There was a gradual decrease in \(\varepsilon^{\prime}\) and \(\varepsilon^{\prime\prime}\) for both the unirradiated and EB-irradiated PLC nanocomposites, with an increase in the frequency for all temperatures. This was due to the change in the direction of the dipole polarization with respect to the applied field and collection of charges between the sample (PLC nanocomposite pallet) and the electrode responsible for increasing \(\varepsilon^{\prime}\) and \(\varepsilon^{\prime\prime}\) at lower frequencies. The \(\varepsilon^{\prime}\) and \(\varepsilon^{\prime\prime}\) of the unirradiated and EB-irradiated PLC nanocomposites increased smoothly in the low-frequency range for all temperatures and EB irradiation doses of 40, 80, and 120 kGy (Figure 11a). This is attributed to an increase in the mobility of the dipoles. The dipoles are easily oriented with the direction of the applied field, and the dissociation of ion aggregates within the polymer matrix at high temperature, as well as EB energy, leads to an increase in the charge carrier density, resulting in an increase in the conductivity [41,42].  
Figure 9: _Cont._  
Figure 10: Variation in the dielectric loss (\(\epsilon^{\prime\prime}\)) with the frequency at various temperatures in the unirradiated (**a**) and EB-irradiated PLC nanocomposites, (**b**) PLC40 kGy, (**c**) PLC80 kGy, and (**d**) PLC120 kGy.  
Figure 9: Dielectric constant (\(\epsilon^{\prime}\)) variation with the frequency at various temperatures in the unirradiated (**a**) and EB-irradiated (**b**) PLC40 kGy, (**c**) PLC80 kGy, and (**d**) PLC120 kGy nanocomposites.  
It can be noticed that \(\varepsilon^{\prime}\) and \(\varepsilon^{\prime\prime}\) increased by more than one order of magnitude after irradiation with a 120 kGy EB dose (Figure 11b). This is due to the formation of defects and disorders in the band gap of the polymer matrix in the form of cross-links, chain scission processes, discontinuities, and bond breaks in the molecular chains that may produce a greater number of dipoles [35,43] and increase the delocalization of charge carriers in the polymer matrix. These results suggests that the irradiation process can modulate or improve the dielectric properties.