Electrodynamics of swift-electron momentum transfer to a large spherical nanoparticle

Abstract

Swift electrons from highly focused beams produced in aberration-corrected scanning transmission electron microscopes offer a powerful route for probing and manipulating matter at the nanoscale. Although linear momentum transfer from swift electrons to nanoparticles has been investigated theoretically and experimentally, subsequent analyzes revealed that several earlier predictions relied on non-causal dielectric functions or insufficient numerical convergence, leading to spurious sign reversals in the transferred momentum. Here, we derive analytical expressions and develop a numerically efficient electrodynamic framework to compute the linear momentum transferred from a swift electron to an isolated spherical nanoparticle described by a fully causal, local dielectric response. We apply our framework to large nanoparticles with 50 nm radius and explicitly resolve the spectral density of linear momentum transfer across the full frequency domain. Using causal dielectric functions for aluminum and bismuth, we analyze the role of electron velocity, impact parameter, and material-specific resonances. We find that, when causality and full multipolar convergence are enforced, the net transverse linear momentum transferred to spherical nanoparticles remains attractive toward the electron trajectory for all nanoparticles considered, despite the presence of material-dependent sign changes in individual electric and magnetic contributions. These results contrast with earlier theoretical predictions of net repulsive behavior and indicate that additional physical mechanisms beyond the present isolated, local description are required to account for experimentally observed repulsion. Our work establishes a robust reference framework for momentum transfer calculations and provides quantitative benchmarks relevant for electron-beam-based nanoscale manipulation.

Figures (9)

• Figure 1: Schematics of the system under study. A spherical nanoparticle of radius $a$, described by a frequency-dependent dielectric function $\epsilon(\omega)$, is embedded in vacuum and centered at the origin. A swift electron (blue dot) travels with constant velocity $\mathbf{v}=v\,\hat{\mathbf{z}}$ at an impact parameter $b$ relative to the NP center.
• Figure 2: Transverse spectral density of linear momentum $\mathcal{P}_{\bot}(\omega)$ transferred by a swift electron to an aluminum nanoparticle with radius $a=50$ nm. (a) Total spectral density (yellow) decomposed into electric (red) and magnetic (cyan) contributions for $b=50.5$ nm and $v=0.58c$ ($120$ keV). (b) Total spectral density (yellow) decomposed into interaction (external-scattered, red) and scattered-scattered (cyan, scaled by a factor of 50) contributions. The external-external term vanishes identically and it is not shown. (c) $\mathcal{P}_{\bot}(\omega)$ for fixed $b=50.5$ nm and electron velocities indicated in the inset. (d) $\mathcal{P}_{\bot}(\omega)$ for fixed $v=0.58c$ and impact parameters indicated in the inset. Vertical dashed lines indicate representative plasmonic resonances.
• Figure 3: Transverse linear momentum ($\Delta P_{\bot}$, yellow dots) transferred by a swift electron to an aluminum NP with $a=50$ nm, (a) as function of $v$ with $b=50.5$ nm, and (b) as function of $b$ with $v=0.58c$ ($120$ keV). The red and blue dots represent the electric and magnetic contributions to $\Delta P_{\bot}$, respectively. The red, blue, and yellow lines are guides to the eye.
• Figure 4: Momentum transfer to a bismuth nanoparticle with radius $a=50$ nm. (a) Transverse spectral density $\mathcal{P}_{\bot}(\omega)$ for fixed impact parameter $b=50.5$ nm and electron velocities indicated in the inset. (b) Total transverse momentum $\Delta P_{\bot}$ as a function of electron velocity for $b=50.5$ nm. (c) $\mathcal{P}_{\bot}(\omega)$ for fixed velocity $v=0.58c$ and impact parameters indicated in the inset. (d) $\Delta P_{\bot}$ as a function of impact parameter for $v=0.58c$. Yellow dots denote total momentum, while red and blue symbols indicate electric and magnetic contributions, respectively. The red, blue, and yellow lines are guides to the eye.
• Figure 5: Convergence of the transverse spectral momentum-transfer density $\mathcal{P}_{\bot}(\omega)$ with respect to the multipole truncation $\ell_{\max}$ for a Drude aluminum nanoparticle with $a=50~\mathrm{nm}$, $b=50.5~\mathrm{nm}$, and $v=0.7c$. As $\ell_{\max}$ increases, additional multipolar contributions modify the spectrum until convergence is reached. The spectrum is converged for $\ell_{\max}=47$; larger values do not produce visible changes.