## 2 Experimental Part
Such vibrations are typically observed in the frequency range of 400 to 600 cm\({}^{-1}\) and might suggest deformation movements within the crystal structure. At 821.84 cm\({}^{-1}\), the peak could be associated with neodymium-oxygen (Nd-O) bond vibrations. Vibrations in this range often point to interactions between rare-earth elements and oxygen, potentially resulting from symmetric or asymmetric stretching within the crystal lattice. At 1427.70 cm\({}^{-1}\), the peak might arise from sodium-oxygen (Na-O) bond vibrations. Na-O vibrations are commonly \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **NdNa\({}_{2}\)NiMnO\({}_{5}\)** & **NdK\({}_{2}\)NiMnO\({}_{5}\)** \\ \hline [PERSON] & 0.92 & 1.02 \\ [PERSON] & 0.96 & 1.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the results of the tolerance factor according to [PERSON] and [PERSON]. found in higher frequency ranges and could indicate movements of light sodium ions within the crystal structure (Figure 2a). For the IR spectrum of the NdK\({}_{2}\)NiMnO\({}_{5}\) compound, at 638.68 cm\({}^{-1}\), this peak may be associated with vibrations of nickel-oxygen (Ni-O) or manganese-oxygen (Mn-O) bonds. Such vibrations are typically observed in the range from 600 to 800 cm\({}^{-1}\) and may indicate deformation movements in the crystal structure. At 705.79 cm\({}^{-1}\), the peak could correspond to neodymium-oxygen (Nd-O) bond vibrations. Vibrations in this range often suggest interactions between rare-earth elements and oxygen and may involve  
Figure 2: The infrared absorption spectra of the compounds NdNa\({}_{2}\)NiMnO\({}_{5}\) (**a**) and NdK\({}_{2}\)NiMnO\({}_{5}\) (**b**). symmetric or asymmetric stretching within the crystal lattice. At 837.85 cm\({}^{-1}\), this peak may also relate to Nd-O bond vibrations, potentially due to symmetric or asymmetric stretching. At 1370.25 cm\({}^{-1}\), the peak might indicate potassium-oxygen (K-O) bond vibrations. Such vibrations often appear in the higher frequency range, pointing to the movement of potassium ions within the crystal structure (Figure 2b) [21]. Thermal analysis (DTA/TGA) of NdNa\({}_{2}\)NiMnO\({}_{5}\) and NdK\({}_{2}\)NiMnO\({}_{5}\) was carried out using the Setaram Labsys(tm) Evo TG-DTA/DSC (Manufacturer: SETARAM Instrumentation, Caluire, France, 2022). This device enables simultaneous measurement of thermogravimetric (TG) and differential thermal analysis (DTA/DSC), which helps accurately determine the temperatures of phase transitions, decomposition, and other thermal effects for NdNa\({}_{2}\)NiMnO\({}_{5}\) (Figure 3). For NdNa\({}_{2}\)NiMnO\({}_{5}\), Figure 3 presents the DTA curve of the mixture Na\({}_{2}\)CO\({}_{3}\) + Nd\({}_{2}\)O\({}_{5}\) + NiO + Mn\({}_{2}\)O\({}_{3}\) during the formation of NdNa\({}_{2}\)NiMnO\({}_{5}\). As shown in Figure 3, the thermogravimetric curve reveals six characteristic mass-loss regions, each associated with specific processes (carbonate decomposition and target compound formation): at 30-220 \({}^{\circ}\)C, a gradual moisture loss (-4%) is observed, attributed to the removal of adsorbed moisture from the sample surface. At 220-400 \({}^{\circ}\)C, this stage involves more pronounced moisture loss (-4%). At 400-540 \({}^{\circ}\)C, interaction between Na\({}_{2}\)CO\({}_{3}\) and oxides occurs, releasing CO\({}_{2}\), with a mass loss of approximately 5%. This stage also marks the initial interaction of Na\({}_{2}\)CO\({}_{3}\) with oxides and possibly the formation of intermediate phases. At 540-760 \({}^{\circ}\)C, interaction continues with CO\({}_{2}\) release, leading to a 4.5% mass loss. The oxides begin reacting, initiating the formation of NdNa\({}_{2}\)NiMnO\({}_{5}\). At 760-900 \({}^{\circ}\)C, the formation of NdNa\({}_{2}\)NiMnO\({}_{5}\) progresses, accompanied by further CO\({}_{2}\) release and a mass loss of 5.5%. At 900-1100 \({}^{\circ}\)C, a sharp mass decrease is evident, corresponding to the main phase reaction for synthesizing NdNa\({}_{2}\)NiMnO\({}_{5}\).