Form factor of prismatic particles for small-angle scattering analysis

Since the morphology of nanoparticles directly influences many of their properties, accurately determining their shape is crucial for targeted applications. In this work, we focus on nanoprisms due to their widespread use and the limitations of direct imaging techniques in accurately describing their polygonal cross section. Specifically, we introduce a new tool for small-angle scattering (SAS) analysis of nanoprisms that requires minimal computation time compared with all-atom simulations and other form factor analyses. A key innovation in this work is the implementation of the Lebedev quadrature for isotropic averaging, which allows for accurate form factor calculations using few sampling points. This form factor model is developed for any n-sided prism and is compared with small-angle X-ray scattering and transmission electron microscopy experimental data for gold and/or silver nanoprisms (n = 3, 4, 5). For small sizes, the nanoprism form factor model is compared with the result obtained with the Debye equation from atomic coordinates, showing a very good agreement. We explore the effects of the aspect ratio and cross-sectional shape of the nanoprisms on the form factor curves and discuss the limitations of our approach. Overall, our method combines precise shape determination with rapid computation time, paving the way for detailed characterization of nanoprisms using SAS techniques, potentially even during their growth.