Abstract text (incl. figure legends and references)

The introduction of momentum-resolution in STEM, thanks to fast cameras [1], brought new motivations for the quantitative understanding of low-angle electron diffraction, which, due for instance to its richness in inelastic contributions [2], is challenging to simulate. Previous work, employing energy-filtering, led to the observation of intensity redistribution in diffraction space following the excitation of a volume plasmon mode [3]. This process could be included in calculations by use of a transition potential, following a dipolar approximation. A qualitative agreement between simulation and experiment was thus obtained, though rigorous quantitative approaches require the introduction of further contributions [4].

In this context, the influence of multiple plasmon-losses was left out. Nevertheless, whether the redistribution process is the same from each successive transition, in terms of the recorded intensity, remains an open question. In particular, by removing the propagation of the mutually incoherent wave-functions created by each distinct energy-loss, it becomes possible to express the inelastic intensity as the convolution of the elastic intensity by a Lorentzian kernel.

An energy-filtered 4D-STEM experiment was thus performed on a 74 nm-thick Al specimen, using an aberration-corrected Hitachi HF5000 instrument equipped with a CEFID energy filter leading to a Medipix3 camera. 10 eV-wide windows were positioned on the first five plasmon peaks of the EEL spectrum (PL_n) as well as on the zero-loss peak (ZL=PL₀) [5]. The resulting angle-dependent measurements are plotted in fig. 1.a and show a similar spreading of diffracted intensity as previously reported [3]. This is further verified by the ratios plotted in fig. 1.b, displaying a peak at the angle where the intensity tends to be redistributed to.

For each transition, e.g. PL₀ to PL₁, PL₁ to PL₂, etc…, the convolution approximation was tested by optimizing a Lorentzian L_n, to minimize the sum of |(L_n⊗PL_n-1)-PL_n|² in diffraction space. The resulting intensity curves are depicted in fig. 2.a-e and show a high quality of fit, thus demonstrating the accuracy of the approximation. The extracted kernels, depicted in fig. 2.f, are found to differ slightly from one another, though the characteristic spatial frequency q_δE remains in an interval of 0.15 to 0.23 nm^-1. Similar results were obtained in regions of higher thickness.

Those findings constitute experimental evidence for the validity of an empirical convolutional approach to account for multiple plasmon scattering. Aspects of simulation efficiency, reliability of Lorentzian parameters, and propagation, are also to be discussed.

Fig.1: a) Angle-dependent profiles obtained from each recording, expressed as proportions of incident intensity by Sr. b) Ratio of successive profiles. c) PACBED patterns. d) BF images.

Fig.2: a-e) Plots of successive transitions, with depiction of pre-transition and post-transition intensities normalized to incident beam, and result of convolution. f) Extracted Lorentzian kernels.

Fundings from the Initiative and Network Fund of the Helmholtz Association under contracts VH-NG-1317 and ZT-I-0025 are gratefully acknowledged.

[1] H. Ryll ; J. instr. 11, P04006 (2016)

[2] K. Müller-Caspary et al.; Sci. rep. 6, 37146 (2016)

[3] A. Beyer et. al. ; Sci. rep. 10, 17890 (2020)

[4] T. Grieb et. al. ; Ultram. 221, 113175 (2021)

[5] H. L. Robert et. al. ; App. Phys. Lett. (submitted)