## 3 Results and Discussion  
### XRD and FT-IR Analysis  
The XRD of the geopolymers show a similarity in the intensity of the peaks produced by the present crystalline phases, in relation to each type of geopolymer, highlighting quartz as the most intense peak, present at 20 = 27\({}^{\circ}\), attributed to the natural structure of the FA. The main crystalline phases of FA depend on the combustion process carried out in the coal plants; however, the crystal structures present in FA (Class F) coincide with the results of other investigations [35; 36]. However, the reduction of particle size through mechanical grinding is the variable that causes a change in the XRD of the geopolymers, which is attributed, on the one hand, to the increase of the specific surface area, expanding the contact area with the alkaline activator, which promotes an increase in the reactivity of the FA. It was also observed that the mechanical grinding affected the microstructure of the FA and mainly causes a weakening in the vitreous chemical bonds of Si-O or Al-O, besides the fact that it accelerates the dissolution of these bonds, shortening the equilibrium time, gelation, and structuring of the new crystalline phases and of the different reaction products, specifically the hydrated sodium aluminosilicate gel (N-A-S-H), which is known as the reaction product that gives the mechanical properties to the geopolymer. In Figure 3, the characteristic crystalline phases that were already present in the FA did not suffer any change with respect to the activated samples, because sodium hydroxide did not cause alterations in these phases due to the strong binding energies, which oscillate between 360 and 380 kcal/mol (five times stronger than the vitreous phase) [37], through which the alkalinity of the solution is capable of dissolving the bonds of the amorphous phase of this starting material. Figure 3: X-ray diffraction pattern of geopolymers (Q; quartz, M; mullite, f; faujasite, z; type Y-Zeolite, c; sodium carbonate). A semicrystalline gel is produced in the geopolymers pastes, which is characteristic in these materials and is displayed in the range of 20 between 20\({}^{\circ}\) and 40\({}^{\circ}\). In addition, scattered peaks appear at different intensities, located between 5\({}^{\circ}\) and 50\({}^{\circ}\) in 20 [38]. These results are consistent with those shown in the infrared spectra (Figure 4) that identify small bands located between 560 cm\({}^{-1}\) and 792 cm\({}^{-1}\), where the first can be attributed to the vibration of Al-O-Si bonds contained in the mullite and the second to Si-O-Si bonds included in the quartz [39; 40]. Considering that these bands are maintained for each of the spectra of the activated materials (the sets G1, G2, and G3), it is confirmed that these crystalline phases are not modified and, therefore, the signals come from the crystals of the FA. Regarding the curing age at early ages, the sets G1 and G2 remain constant. However, set G3 presents an increase in its intensity, and this behavior remains for 14 and 28 days, indicating the development of polymerization of the material. This process reveals evolutions in the crystalline phases, determining the influence that the particle sizes of the FA have on the reactivity of the geopolymers. As curing time passes, there is a decrease in the halo of the amorphous phase (Figure 3) located between 20\({}^{\circ}\) and 40\({}^{\circ}\) of 20 due to its dissolution, which promotes the formation of new products and allows for the occurrence of the specific phases of the geopolymers. In Figure 5, it is shown how faujasite and type Y-zeolite acquire intensities with similar tendencies.