## 3 Results and Discussion
### Analysis of Dose\(-\)Response Graphs
#### 3.2.2 Dose\(-\)Response  
The samples were exposed to several doses of radiation ranging from 20 to 70 Gy at a constant linear heating rate of 5 K s\({}^{-1}\) to examine the impact of dose-dependency on the reported peak positions. The TL response curve for ZnZrO\({}_{3}\) (3% Mn) nanoparticles at various beta-irradiation doses ranging from 20 to 70 Gy is displayed in Figure 6A. A phosphor must have a linear dose\(-\)response to serve as a TL dosimeter. Because beta-irradiated phosphors cannot go deeper and cannot induce lattice defects, the TL response is mainly generated by the surface traps. As the irradiation dose increases, the density of surface defects also rises, increasing the peak intensity. Competition between radiative and nonradiative centers, or between various types of trapping centers, results in the TL intensity [34]. It has been found that the concentration (\(n_{\mathrm{o}}\)) of the trapped charge carriers is proportional to the glow curve intensity (I) during heating [35; 36]. The primary glow curve at 349 K and a secondary one at 473 K are displayed in Figure 6A. It has been noted that as the dosage increases, so does the peak intensity. This is because an increase in dose causes an increase in the number of luminescence centers, which raises the peak intensity. In other words, the higher the energy density of the ionizing radiation, the more electron\(-\)hole pairs are trapped, and as a result, the TL intensity peak rises as the dose of beta irradiation increases. It has been noted that as the dosage increases, so does the peak intensity. This is because an increase in dose causes an increase in the number of luminescence centers, which raises the peak intensity. In other words, the higher the energy density of the ionizing radiation is increased. Figure 5: TL glow curve of ZnZrO\({}_{3}\)-Mn phosphor with repeated TL measurements, measured at a heating rate of 5.0 K s\({}^{-1}\) following irradiation to 70 Gy. radiation, the more electron\(-\)hole pairs are trapped, and as a result, the TL intensity peak rises as the dose of beta irradiation increases. The dose\(-\)response behavior of Mn-doped ZnZrO\({}_{3}\) phosphor shows a systematic shift in TL glow-curve peaks toward higher temperatures with increasing radiation doses. At lower doses, electrons occupy shallow traps with lower activation energies, releasing at lower temperatures. With higher doses, deeper traps with higher activation energies are filled, leading to thermal stimulation at higher temperatures and shifting the TL peaks, revealing dose-dependent trapping and thermal-release mechanisms [18]. The underlying premise of the first-rise method for determining trap depths from six stability measurements in Figure 6B is that the concentration of trapped charge carriers is comparatively stable on the low-temperature side of a TL glow curve [37]. Only a very small portion of the charge carriers can escape with the limited thermal energy available [18]. \[I(T)=Cexp\left(-\frac{E}{KT}\right) \tag{3}\]  
where \(E\) is the activation energy, \(k\) is Boltzmann's constant, and \(C\) is a constant of proportionality, which comprises the frequency factor s, which is assumed to be independent of temperature, allowing us to approximate the equations for first-, second-, and general-order kinetics. The low temperature is shown by an Arrhenius plot, which plots the glow curve as ln(I) versus 1/T. The activation energy for TL was determined at a heating rate of 5 \({}^{\circ}\)C s\({}^{-1}\) after irradiation to 70 Gy; the first glow curve in Figure 6B illustrates the impact of repeated measurements on the same sample. Therefore, using the following equation, \(E_{v}\) can be evaluated without knowing the frequency factor s, as follows [18]:  
\[E_{v}=\frac{d(lnI)}{d\left(\frac{1}{T}\right)} \tag{4}\]  
Figure 7A highlights the effects of the Variable Heating Rate (VHR) on the thermoluminescence (TL) glow curve.