## 2 Experimental Part
The reliability of the measurements is confirmed by the excellent agreement between experimental and calculated values of the parameter 10\({}^{4}\)/d\({}^{2}\), as well as the consistency between theoretical and experimental density values (both pycnometric and X-ray) and the unit cell parameters. This agreement demonstrates the high accuracy of the performed studies and validates the correctness of the methodology (Table 1). \begin{table}
\begin{tabular}{c c c c c c c} \hline
**h, mm** & **VI\({}^{0}\)** & **d, A** & **hkl** & **10\({}^{4}\)/d\({}^{2}\)ep. ** & **10\({}^{4}\)/d\({}^{2}\)**calc. ** & **Difference 10\({}^{4}\)/d\({}^{2}\)ep. and 10\({}^{4}\)/d\({}^{2}\)calc. \%** \\ \hline  & & & & NdNa\({}_{2}\)NiMnO\({}_{5}\) & & \\  
The analysis of X-ray diffraction data for NdNa\({}_{2}\)NiMnO\({}_{5}\) and NdK\({}_{2}\)NiMnO\({}_{5}\) shows that the accuracy of the experimental data lies in the fact that the difference between the calculated and experimental values of 10\({}^{4}\)/d\({}^{2}\)\({}_{\text{exp. }}\) and 10\({}^{4}\)/d\({}^{2}\)\({}_{\text{calc. }}\) is minimal in most cases, indicating high data precision. The maximum discrepancies do not exceed 1.49% for NdNa\({}_{2}\)NiMnO\({}_{5}\) and 1.05% for NdK\({}_{2}\)NiMnO\({}_{5}\), which is acceptable in crystallographic studies. The most intense peaks (I/I\({}^{0}\) = 100) are observed for NdNa\({}_{2}\)NiMnO\({}_{5}\) at d = 2.723 A (hkl = 731) and NdK\({}_{2}\)NiMnO\({}_{5}\) at d = 2.723 A (hkl = 821). This indicates a similarity in their structures and the preferential orientation of atomic planes. A comparison of the parameters for the two compounds shows similar d-values and Miller indices (hkl), confirming structural relatedness. However, slight discrepancies in d-values and intensities are noted, possibly due to the difference in atomic radii of Na and K, which affects the lattice parameters. These findings confirm the high consistency between experimental and calculated values, demonstrating the accuracy of the applied data processing methodology. To evaluate the quality of the experimental data, parameter refinement was performed using the Rietveld method. The calculated quality factors indicate agreement between the model and the experimental data. Since the DRON-2.0 diffractometer does not support automatic calculation of these factors, the calculations were conducted manually based on experimental intensities. For NdNa\({}_{2}\)NiMnO\({}_{5}\), the weighted quality factor is approximately 0.892 [16]. The tolerance factor (tt) plays a key role in determining the stability of the crystal structure of perovskite-like compounds [17]. The ionic radii were taken from [18; 19]. The calculation is carried out using [PERSON]'s formula:  
\[t=\frac{r_{A}+r_{O}}{\sqrt{2(r_{B}+r_{O})}}, \tag{1}\]  
where \(r_{A}\) is the effective radius of cations in the A-site position (Nd\({}^{3+}\), Na\({}^{+}\), or K\({}^{+}\)), \(r_{B}\) is the effective radius of cations in the B-site position (Ni\({}^{2+}\) and Mn\({}^{3+}\)), \(r_{O}\) is the radius of oxygen ions (O\({}^{2-}\)), and \(\sqrt{2}\) is the geometric coefficient for perovskite structures. The calculation of the tolerance factor according to [PERSON] and [PERSON] is presented in Table 2.  
In both cases, NdNa\({}_{2}\)NiMnO\({}_{5}\) and NdK\({}_{2}\)NiMnO\({}_{5}\), the tolerance factor values indicate a cubic structure. The tolerance factor is a criterion for structural stability in perovskites and can range from 0.8 to 1.11 [20]. Infrared spectroscopic analysis of the compounds was carried out using the ALPHA Fourier spectrometer (Bruker, Berlin, Germany). IR spectroscopy revealed bond vibrations characteristic of Ni-O, Mn-O, Nd-O, and Na-O in NdNa\({}_{2}\)NiMnO\({}_{5}\), as well as Ni-O, Mn-O, Nd-O, and K-O in NdK\({}_{2}\)NiMnO\({}_{5}\), confirming the features of their crystal structures. The IR spectrum of the NdNa\({}_{2}\)NiMnO\({}_{5}\) compound: at 634.70 cm\({}^{-1}\), the peak may indicate vibrations of nickel-oxygen (Ni-O) or manganese-oxygen (Mn-O) bonds.