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Bridging nano-optics and condensed matter formalisms in a unified description of inelastic scattering of relativistic electron beams

by Hugo Lourenço-Martins, Axel Lubk, Mathieu Kociak

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Submission summary

As Contributors: Hugo Lourenço-Martins · Axel Lubk
Arxiv Link: (pdf)
Date submitted: 2020-07-20 02:00
Submitted by: Lourenço-Martins, Hugo
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approach: Theoretical


In the last decades, the blossoming of experimental breakthroughs in the domain of electron energy loss spectroscopy (EELS) has triggered a variety of theoretical developments. Those have to deal with completely different situations, from atomically resolved phonon mapping to electron circular dichroism passing by surface plasmon mapping. All of them rely on very different physical approximations and have not yet been reconciled, despite early attempts to do so. As an effort in that direction, we report on the development of a scalar relativistic quantum electrodynamic (QED) approach of the inelastic scattering of fast electrons. This theory can be adapted to describe all modern EELS experiments, and under the relevant approximations, can be reduced to any of the last EELS theories. In that aim, we present in this paper the state of the art and the basics of scalar relativistic QED relevant to the electron inelastic scattering. We then give a clear relation between the two once antagonist descriptions of the EELS, the retarded green Dyadic, usually applied to describe photonic excitations and the quasi-static mixed dynamic form factor (MDFF), more adapted to describe core electronic excitations of material. We then use this theory to establish two important EELS-related equations. The first one relates the spatially resolved EELS to the imaginary part of the photon propagator and the incoming and outgoing electron beam wavefunction, synthesizing the most common theories developed for analyzing spatially resolved EELS experiments. The second one shows that the evolution of the electron beam density matrix is proportional to the mutual coherence tensor, proving that quite universally, the electromagnetic correlations in the target are imprinted in the coherence properties of the probing electron beam.

Current status:
Has been resubmitted

Submission & Refereeing History

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Submission 2007.02773v2 on 20 July 2020

Reports on this Submission

Report 1 by Jo Verbeeck on 2020-9-17 (Invited Report)

  • Cite as: Jo Verbeeck, Report on arXiv:2007.02773v2, delivered 2020-09-17, doi: 10.21468/SciPost.Report.2003


1. Develops common foundation for EELS and nanoptical experiments
2. Builds on many earlier theoretical developments but puts them under one unifying umbrella
3. Will form the basis for interpreting many new EELS related experiments involving phase manipulation, phonon scattering, time modulated beams and many more.
4. Very thorough development of the theory, giving attention to all assumptions applied


1. Can be heavy to digest for the experimental reader


This manuscript describes a theoretical derivation of the foundation of inelastic energy loss spectroscopy in a transmission electron microscope and its link to nano optics.
I would describe the manuscript as nothing less than a master piece, bringing together many of the concepts and theories that underpin all experimental work in this domain. The paper develops in a pedagogical way, a common description on how accelerated electron beams inelastically interact with a sample taking into account both quantum and relativistic aspects, augmenting and clearing up the work of previous derivations.
The authors succeed in giving a fair overview of the existing state of the art, while pointing out how these fit together in a more generalised scheme.
The findings are very important in a time where more control over the quantum state of the electron in such experiments is becoming available and where confusion exists in the community as to how to interpret its results in terms of the materials properties.
The results give hope that much more is to be expected from novel electron spectroscopic setups, also going beyond the assumptions made in this manuscript (time modulated beams, phonons, ...).

Requested changes

Minor comments:

1. p1: "The overlap of the electron beam with the sample e.g ...". It is not clear what is meant with 'overlap' here. I think it refers to aloof experiments vs. bulk? Otherwise both suface and bulk effects play a role and seem to be independent on the overlap of the beam with the sample?
2. fig1: an ending bracket is missing in the description of the illumination system
3. p2: the magic angle is defined as the angle at which electrons are most likely deflected. This is confusing to me, but may be right. I would define the magic angle as the collection angle at which there is no dependence of the recorded spectrum on the orientation of an anisotropic sample.
4. p2: "X or optical ...", I would write "X-ray or optical photon"
5. p2: "in a STEM has yet been provided, which includes...", I would write "in a STEM has been provided, which implies..."
6. p3: "the later to the retarded case,...", I think this needs to be "latter". I suggest to search for the word "later" and replace with "latter" wherever appropriate as this occurs several times in the manuscript.
7. ref 18: check the spelling of Hebert (accents)
8. fig2: the 'grey' part of the figure is not easy to distinguish
9. p5: "The zz indices hind at", I think this needs to be "hint"
10. p7: "The are therefore not sufficient to model such experiments". I would argue that even for 'simple' EELS experiments, the partial coherence of the outgoing electron wave does make an important difference as even going to diffraction space will make such coherence visible in the recorded intensities. In this sense, I would argue that we always need to keep the formalism developed in this manuscript in mind, even for the seemingly simpler setups.
11. p7: "In other words, non-zero out of diagonal terms entail electron interferences in the image plane". Well, this depends on which plane you are looking at and in which plane the density matrix was given. E.g. if the image plane would be in r-space, the intensity would only depend on the diagonal (assuming no lens effects etc.).
12. p7: "Upon comparision"...should be "comparison"
13. p9: "Thanks to the Schwarz's inequality...", no need for the "'s"
14. p9: "For the sake of clearness", could be "for the sake of clarity"
15. p11: Around e.q. 54, the temperature is assumed zero. I wonder if the equations are linear whether the effect of the ignored temperature can later be added as an additive effect on top of the beam driven effect described here?
16. p15: "at it generalize the Kubo approach...", should be "generalises"
17. p15: after caption on concluding remarks "one usually calculate the Green dyadic" should be "calculates"
18. p16: after eq. 105 "This term no encompass elastic...", should be "encompasses"
19. p18: after eq. 119 "for a set of Gaussian random variable", should be "variables"
20. p18: above eq. 123 "simulataneous" should be "simultaneous"
21. p22: after caption VII, "We will therefore make furthers approximations", should be "further"
22. p23: after eq. 169: "one respectively obtain equations (10)...", should be "obtains"

  • validity: top
  • significance: top
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: good

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