Time-dependent forces between a swift electron and a small nanoparticle within the dipole approximation

Abstract

In this paper we calculate the time-dependent forces between a swift electron traveling at constant velocity and a metallic nanoparticle made of either aluminum or gold. We consider that the nanoparticle responds as an electric point dipole and we use classical electrodynamics to calculate the force on both the nanoparticle and the electron. The values for the velocity of the electron and the radius of the nanoparticle were chosen in accordance with electron microscopy observations, and the impact parameter was selected to fulfill the constraints imposed by the dipole approximation. We find that there are times when the force on the nanoparticle is attractive and others when it is repulsive, and show that this is due to the delayed electromagnetic response of the nanoparticle. To establish the limits of validity of our approach, we calculate the total linear momentum transfer to the nanoparticle, and compare it with results obtained, in frequency space, using the full multipole expansion of the fields induced on the nanoparticle, considering the effects of electromagnetic radiation.

Effects of a non-causal electromagnetic response on the linear momentum transfer from a swift electron to a metallic nanoparticle

“…As has been argued in Ref. [40], the electromagnetic response of small spherical NPs is satisfactorily charac-terized by the quasistatic polarizability:…”

“…It is noteworthy that the experimental observation of repulsion of nanoparticles from the electron beam does not contradict the fact that the linear momentum transfer is always attractive. In fact, using the fully retarded wave solution approach, it has been theoretically shown that there are indeed repulsive forces in this interaction [29,40], and that they are present even if ∆P ⊥ is always attractive [40].…”

“…From these scattered fields, whose detailed expressions can be found in Ref. [40], together with the electromagnetic fields produced by the bare swift electron (external fields) [41], one can obtain the linear momentum transferred by the swift electron to the NP [40]:…”

“…6(b) we show ∆P ⊥ transferred to the NP as function of the pre-echo, choosing b = 1.5 nm and v = 0.5c. Previous studies have reported that most of the linear momentum is transferred from the electron to the NP in a time scale on the order of tens of attoseconds, where the main interaction between the swift electron and the NP occurs [29,40]. A remarkable effect can be observed when the non-causal pre-echo is on the order of the interaction time between the NP and the swift electron: the transverse linear momentum transferred is negative, meaning a repulsive interaction between the swift electron and the NP [red dots in Fig.…”