1：Experimental Design and Method Selection:

The Li0.5ZnFe1.5O4 nanoparticles were synthesized using a sol–gel auto-combustion method followed by annealing at 500 and 1100 °C. The phase-purity and crystal structure were examined by X-ray diffraction spectroscopy. Surface morphological characterization and chemical compositions were investigated using SEM and TEM. Thermal analysis was carried out using TGA and DTA. FTIR and Raman spectra were recorded to analyze the vibrational bands and elastic properties. Electrical characterization was conducted by impedance spectroscopy.

2：5ZnFe5O4 nanoparticles were synthesized using a sol–gel auto-combustion method followed by annealing at 500 and 1100 °C. The phase-purity and crystal structure were examined by X-ray diffraction spectroscopy. Surface morphological characterization and chemical compositions were investigated using SEM and TEM. Thermal analysis was carried out using TGA and DTA. FTIR and Raman spectra were recorded to analyze the vibrational bands and elastic properties. Electrical characterization was conducted by impedance spectroscopy. Sample Selection and Data Sources:

2. Sample Selection and Data Sources: Stoichiometric amounts of Fe(NO3)3·9H2O, Zn(NO3)2·6H2O, and LiNO3 were dissolved in distilled water with citric acid added in a 1:2 molar ratio of metal cation to citric acid.

3：List of Experimental Equipment and Materials:

BRUKER diffractometer with CuKa radiation, Merlin scanning electron microscope (SEM), FEI Tecnai F20 microscope, Q600 SDT for TGA and DTA, Shimadzu-8700 FTIR spectrometer, SENTERRA spectrometer (Bruker, Germany), Agilent 4294A impedance analyzer, Shimadzu UV-3101PC scanning spectrophotometer.

4：Experimental Procedures and Operational Workflow:

The mixed solution was magnetically stirred at 80 °C until a viscous gel was formed. The viscous gel was heated on a hot plate at 300 °C until it self-ignited, yielding dark brown ash. The as-burnt powder ferrites were then pelletized and finally annealed separately at 500 °C and 1100 °C for 8 h in an electrical muffle furnace and cooled slowly to room temperature.

5：Data Analysis Methods:

The Rietveld FULLPROF program was used for the structural refinements. The Scherrer equation and Williamson–Hall equation were used to calculate the crystallite size. The elastic parameters were calculated using the relations provided in the paper.