## 3 Results and Discussions  
### The Effect of Different Mn and Zn Contents on the Crystal Phase  
Figure 2a shows the XRD diffractogram of [Mn\({}_{x}\)Zn\({}_{1-x}\)]Fe\({}_{2}\)O\({}_{4}\) for different proportions of Mn and Zn. The diffraction peak shows that a change in the proportions affects the crystal phase. The crystal phase for a composition of Mn\({}_{0.54}\)Zn\({}_{0.46}\)Fe\({}_{2}\)O\({}_{4}\) is the most integrated and the strongest. The crystal phase for Mn\({}_{0.8}\)Zn\({}_{0.2}\)Fe\({}_{2}\)O\({}_{4}\) is the least integrated and the weakest. The nano-Mn-Zn ferrite ([Mn\({}_{x}\)Zn\({}_{1-x}\)]Fe\({}_{2}\)O\({}_{4}\)) is a spinel structure. The average particulate diameter is calculated using the diffraction peaks for different Mn-Zn proportions and the Scherrer equation. It is seen that the particulate diameter is different for different crystal phases. For the composition [Mn\({}_{0.54}\)Zn\({}_{0.46}\)]Fe\({}_{2}\)O\({}_{4}\) with the strongest crystal phase, the largest average particulate diameter is 11.50 nm. For the composition [Mn\({}_{0.8}\)Zn\({}_{0.2}\)]Fe\({}_{2}\)O\({}_{4}\) with the weakest crystal phase, the average particulate diameter is the smallest, at about 4.28 nm.  
Figure 2b shows the effect of the pH value of the waste battery on the crystal phase of [Mn\({}_{\rm x}\)Zn\({}_{\rm 1-x}\)]Fe\({}_{2}\)O\({}_{4}\). It is seen that for pH values between 6 and 7, there are many phases in the diffraction peaks. For a pH value of 8, the strength of the crystal lattice of the miscellaneous phase is greatly reduced. For a pH value of 10, the diffraction peak for the miscellaneous phase is totally absent. The miscellaneous peak is the diffraction peak for Zn\({}^{2+}\)\({}_{3}\)O\({}_{4}\), Fe\({}_{3}\)(SO\({}_{4}\))O\({}_{3}\) and MnSO\({}_{4}\), and when the pH value is between 6 and 7, there is insufficient alkaline content so the Fe ions and Mn ions are not fully precipitated, and the precipitated Zn\({}^{2+}\)\({}_{3}\)O\({}_{4}\) forms Zn ions. The solution must be neutralized to a pH value of greater than 9.  
The average particulate diameter for [Mn\({}_{\rm x}\)Zn\({}_{\rm 1-x}\)]Fe\({}_{2}\)O\({}_{4}\) at different pH values is calculated using the diffraction peak and the Scherrer equation. Normally, crystallite size decreases as the particle size decreases, but not at the same rate. Nitrogen adsorption (BET) is a good technique for measuring nano-ferrite particles [23]. A pH value of 6 produces the smallest average particulate diameter of about 31.10 nm because the alkaline content is not sufficient and the sulfate in the ionic solution is  
Figure 2: XRD patterns for (**a**) [Mn\({}_{\rm x}\)Zn\({}_{\rm 1-x}\)]Fe\({}_{2}\)O\({}_{4}\) sintered at 600 \({}^{\circ}\)C and (**b**) the effect of the pH value of the waste dry battery.  
not neutralized, so the metallic ions are not precipitated and a complete spinel structure cannot grow. For higher pH values, the average particulate diameter is greater. A pH value of 10 gives the maximum average particulate diameter of about 59.61 nm. Further increases in the pH value result in a decrease in the average particulate diameter because when the pH value is too large, Zn(OH)\({}_{2}\) reacts with NaOH to form Na\({}_{2}\)ZnO\({}_{2}\) and the Zn atoms in the spinel structure are removed. A pH value of 12 gives an average particulate diameter of only 35.03 nm. The particulate diameter varies between 31.10 and 59.61 nm for the range of pH values that is used in this study.