## 3 Results and Discussion
### Analysis of Dose\(-\)Response Graphs
#### 3.2.2 Dose\(-\)Response
At lower heating rates, charge carriers have sufficient time to escape at lower temperatures, resulting in a lower peak temperature. Conversely, at higher heating rates, the material's temperature increases rapidly, causing the charge carriers to be released only at higher thermal energies, which shifts the peak temperature upward. In summary, the TL intensity increases and the peak temperature shifts to higher values with increasing heating rates. These trends reflect the combined effects of thermal lag, activation energy, and the kinetic nature of the TL process, which are critical for understanding the material's luminescent behavior under varying thermal conditions. The shift of the peak temperature \(T_{m}\) with increasing heating rate \(\beta\) (K\(\cdot\)s\({}^{-1}\)) can be modelled using Equation (5) for first-order kinetics, as follows:  
\[\mathrm{T_{m}}=\frac{\mathrm{E}}{\mathrm{K_{B}}}\bigg{[}1-\frac{\mathrm{K_{B} }\mathrm{T_{m}}}{\mathrm{E}}\mathrm{ln}\bigg{(}\frac{\mathrm{s}}{\beta}\bigg{)} \bigg{]} \tag{5}\]  
where \(E\)--Activation energy (eV), \(K_{B}\)--Boltzmann constant (8.617 \(\times\) 10\({}^{-5}\) eV/K\({}^{-1}\), \(T_{m}\)--the temperature of the TL peak (K), s--Frequency factor(s\({}^{-1}\)), and \(\beta\)--heating rate (Ks\({}^{-1}\)). The TL intensity (I) at a given heating rate \(\beta\) can be expressed using the general glow curve equation, as in Equation (6), which is derived from the kinetic order. For first-order kinetics, the TL intensity is calculated as follows:  
\[I(T)=sn_{o}\ exp\ \bigg{(}\bigg{(}-\frac{E}{K_{B}T}\bigg{)}\ exp\ \ \bigg{(}-\frac{s}{\beta}\int_{T_{o}}^{T}exp\ \bigg{(}-\frac{E}{K_{B}T^{\prime}}\bigg{)}dT^{\prime}\bigg{)} \tag{6}\]  
where \(I(T)\)--TL intensity at a temperature \(T\), \(n_{o}\)--Initial concentration of trapped electrons, and \(T_{o}\)--Initial temperature (K). Therefore, TL peak temperature shifts logarithmically with heating rate, as seen in Equation (7), below:  
\[T_{m}\ \sim\ \frac{E}{K_{B}}ln\bigg{(}\frac{s}{\beta}\bigg{)} \tag{7}\]  
The provided Figure 8A,B illustrates TL glow curves and their behavior, specifically, their fading over time (0-90 s) and the effects of thermal cleaning at different temperatures. TL fading refers to the loss of TL intensity over time due to the gradual detrapping of charge carriers at ambient or elevated temperatures. The figure likely represents TL measurements conducted at different time intervals after irradiation. The TL signal arises from trapped charge carriers in defect centers that escape when thermally stimulated. At room temperature (or slightly elevated temperatures), shallow traps (low activation energy) can release carriers spontaneously over time, causing significant fading. Deeper traps (higher activation energy) are more thermally stable and retain charge carriers for longer durations, resulting in minimal fading. Thermal cleaning (seen in Figure 8B) is a process whereby the TL signal is intentionally erased by heating the sample to specific temperatures. This selectively removes carriers from shallow traps while preserving those in deeper traps. The TL glow curves show reduced intensity for peaks at lower temperatures after thermal cleaning at progressively higher temperatures. Higher-temperature peaks remain unaffected until the sample is heated beyond its corresponding activation-energy threshold. When a sample is heated to a certain temperature, \(T_{c}\) (cleaning temperature), charge carriers in traps with activation energies \(E\) below a certain threshold are released. Traps with activation energies higher than \(E\) remain occupied and contribute to TL intensity at higher temperatures. Thermal cleaning was applied to the thermoluminescence (TL) glow curve of ZnZrO\({}_{3}\):Mn nanopowders to refine the material's luminescence properties and eliminate impurities. This process significantly enhances the clarity and efficiency of the glow-curve analysis by removing unwanted elements, ensuring a more pristine and precise assessment of the sample's performance.