1：Experimental Design and Method Selection:

The study utilized small-angle X-ray scattering (SAXS) to investigate the structural properties of colloidal crystals formed by NaYF4 nanocrystals. The methodology included the preparation of nearly monodisperse particles of the hexagonal β-phase through a synthesis process involving heating a solution of sodium oleate and rare-earth oleate in a mixture of oleic acid and octadecene at 300 °C in the presence of solid NaHF2 as a fluoride source.

2：Sample Selection and Data Sources:

Two samples of β-NaYF4 particles with mean radii of approximately 4.5 nm (sample A) and 12.5 nm (sample B) were prepared. Sample A was dispersed in hexane, and sample B in toluene.

3：5 nm (sample A) and 5 nm (sample B) were prepared. Sample A was dispersed in hexane, and sample B in toluene. List of Experimental Equipment and Materials:

3. List of Experimental Equipment and Materials: The synthesis involved oleic acid, 1-octadecene, sodium hydrogen difluoride, and sodium oleate. SAXS/WAXS measurements were performed on a goniometer-based X-ray scattering platform (Empyrean Nano Edition, Malvern Panalytical, The Netherlands) configured in a sample transmission geometry with a focusing X-ray mirror and an area detector.

4：Experimental Procedures and Operational Workflow:

Colloidal crystals were obtained by slow drying of colloidal dispersions in a capillary. SAXS data were collected using a pencil beam in a point collimation setup, with additional measurements done with an elongated beam in a line collimation setup for higher scattering intensities.

5：Data Analysis Methods:

The SAXS data were analyzed to determine the particle size, size distribution, and the structure of the colloidal crystals. The pair distance distribution functions p(r) were calculated using an indirect Fourier transformation procedure, and the experimental data were fitted to a model for homogeneous noninteracting spheres with an assumed Gaussian particle size distribution.