Excitation of confined bulk plasmons in metallic nanoparticles by penetrating electron beams within a nonlocal analytical approach

Using a linear hydrodynamic model, we theoretically investigate the interaction between penetrating electron beams and sub-5-nm metallic spherical nanoparticles. We derive an analytical expression of the electron energy loss probability that includes nonlocal effects in the response of the confined electron gas. Our study focuses on the longitudinal plasmon excitations—also known as confined bulk plasmons (CBPs)—which are inaccessible within local dielectric frameworks, and it shows that their excitation is highly sensitive to the impact parameter, the kinetic energy of the incident electron, and the nanoparticle size. In contrast to local approaches, our model predicts the emergence of several peaks associated with CBPs above the plasma frequency and a blueshift of their spectral envelope as the nanoparticle size decreases and the impact parameter of the electron beam increases from its center toward the surface. Moreover, it predicts a threshold impact parameter, corresponding to the minimum length of the electron's trajectory inside the nanoparticle required for efficient excitation of specific CBP modes. By exploiting a multipolar description of the CBPs, we identify the underlying selection rules governing their excitation by electron beams and correlate the observed blueshift at larger impact parameters with the preferential excitation of higher-order CBPs. The size-dependent dispersion of the CBPs further enhances this impact-parameter-dependent blueshift and also explains the decrease in the impact parameter threshold for smaller nanoparticles. Compared with the local response approximation, the hydrodynamic model captures these trends without momentum cutoffs and with rapid multipolar convergence. Applied to sodium nanoparticles as a canonical free-electron system, the theory reproduces qualitative behaviors observed in experiments and calculations, providing practical guidance for the design and interpretation of CBP-sensitive electron energy loss spectroscopy measurements.