## 3 Results and Discussion
### Electrochemical Analysis  
The electrochemical study of the prepared electrode was carried out in a three-electrode system. Electrochemical tests were used to determine the capacitance of MnO\({}_{2}\) (\(\alpha\), \(\beta\), and \(\gamma\)). To examine the behavior of the charge storage mechanism, CV was performed with a diverse scan rate (10 to 100 mV/s) and a voltage window of 0.8 V in 1 M Na\({}_{2}\)SO\({}_{4}\). Figure 5a-c show the CV plots corresponding to \(\alpha\)-, \(\beta\)-, and \(\gamma\)-MnO\({}_{2}\). At the lower scan rate, all the CV curves show a rectangular shape due to the formation of a double layer [21]. With the increase in the scan rate, the shape of the CV curves becomes distorted, probably due to the redox reaction occurring on the surface of the electrode. The distorted shape of the CV curves represents the pseudocapacitive behavior of the charge storage [22]. The linearity of the CV curves for all the MnO\({}_{2}\) phases is maintained even at the high scan rate, demonstrating the good charge intercalation of ions of the electrolyte or smooth charge transfer. The area under the CV curves represents the specific capacitance of the electrode [23]. It is clearly shown in Figure 5a that \(\alpha\)-MnO\({}_{2}\) exhibited the highest specific capacitance. Figure 5d-f show the GCD plots of \(\alpha\)-, \(\beta\)-, and \(\gamma\)-MnO\({}_{2}\) performed at a fixed voltage of 0.8 V V/s Ag/AgCl at various current densities (1 to 10 A/g). The shape of the GCD curves represents the mixed charge storage mechanism (pseudo and electrostatic) for the electrodes. The below equation was used to determine the specific capacitance of the material:  
\[C_{s}=\frac{I_{d}\times t}{\Delta V} \tag{1}\]  
Figure 5: (**a**–**c**) CV plots and (**d**–**f**) GCD plots of \(\alpha\)-, \(\beta\)-, and \(\gamma\)-MnO\({}_{2}\), respectively. where \(C_{s}\) is the specific capacitance, \(I_{d}\) is the current density, \(t\) is the discharging time, and \(\Delta V\) is the voltage window [24]. It can be seen from the GCD plots that the discharging time increased uniformly for each of the MnO\({}_{2}\) phases. Among all the GCD plots, \(\alpha\)-MnO\({}_{2}\) had a large discharging time and the highest specific capacitance. The specific capacitance calculated at 1 Ag\({}^{-1}\) was 138, 112, and 103 F g\({}^{-1}\) for \(\alpha\)-, \(\beta\)-, and \(\gamma\)-MnO\({}_{2}\), respectively. The \(\alpha\)-MnO\({}_{2}\) electrode maintained a specific capacitance of approximately 112, 90, 48, and 40 Fg\({}^{-1}\) at 1.5, 3, 6, and 10 Ag\({}^{-1}\), respectively. The \(\beta\)-MnO\({}_{2}\) electrode maintained a specific capacitance of approximately 88, 60, 39, and 30 Fg\({}^{-1}\) at 1.5, 3, 6, and 10 Ag\({}^{-1}\), respectively. The \(\gamma\)-MnO\({}_{2}\) electrode maintained a specific capacitance of approximately 78, 55, 36, and 27 Fg\({}^{-1}\) at 1.5, 3, 6, and 10 Ag\({}^{-1}\), respectively. The electrochemical performance of these distinct crystallographic phases was found to be different, which might be due to their atomic arrangements. \(\alpha\)-MnO\({}_{2}\) exhibited a higher electrochemical performance as compared to the other phases due to its 2 \(\times\) 2 tunnel-containing network that supports the electrolytic ion moment or electron transfer process in the charge storage mechanism. Thus, the electrochemical performance of MnO\({}_{2}\) changed with a change in the crystallographic phase. Figure 6a-d show the electrochemical analysis comparison plots for the different phases of MnO\({}_{2}\); as demonstrated in Figure 6a, \(\alpha\)-MnO\({}_{2}\) had the highest area of the CV curve of all three phases and, consequently, an excellent specific capacitance, which is due to its surface morphology [25]. Figure 6b shows the GCD curve, which indicates that \(\alpha\)-MnO\({}_{2}\) had the highest specific capacitance. Figure 6: (**a–d**) Comparison of the CV, GCD, Nyquist, and Bode plots of \(\alpha\)-, \(\beta\)-, and \(\gamma\)-MnO\({}_{2}\), respectively.