**Correlative Light and Transmission Electron Microscopy Showed Details of Mitophagy by Mitochondria Quality Control in Propionic Acid Treated SH-SY5Y Cell**

#### **Minkyo Jung 1,**†**, Hyosun Choi 1,2,**†**, Jaekwang Kim <sup>3</sup> and Ji Young Mun 1,\***


Received: 25 August 2020; Accepted: 27 September 2020; Published: 29 September 2020

**Abstract:** Propionic acid is a metabolite of the microbiome and can be transported to the brain. Previous data show that propionic acid changes mitochondrial biogenesis in SH-SY5Y cells and induces abnormal autophagy in primary hippocampal neurons. Maintaining mitochondrial function is key to homeostasis in neuronal cells, and mitophagy is the selective autophagy involved in regulating mitochondrial quality. Monitoring mitophagy though light microscopy or conventional transmission electron microscopy separately is insufficient because phases of mitophagy, including autophagosome and autolysosome in nano-resolution, are critical for studies of function. Therefore, we used correlative light and electron microscopy to investigate mitochondrial quality in SH-SY5Y cells after propionic acid treatment to use the advantages of both techniques. We showed, with this approach, that propionic acid induces mitophagy associated with mitochondrial quality.

**Keywords:** propionic acid; autophagy; mitophagy; correlative light and electron microscopy (CLEM)

#### **1. Introduction**

Short-chain fatty acids (SCFAs) such as acetic, propionic, and butyric acid are by-products of fermentation of dietary fiber by the gut microbiome [1]. Microbe-derived metabolites can cross the blood–brain barrier and affect the neurons. As the relationship between gut microbiome and the brain, gut–brain axis, has become interesting, SCFAs have attracted increasing attention. Propionic acid (PPA) is increased in stools from patients with autistic spectrum disorder, and prenatal exposure to PPA causes significant impairment of the social behavior of neonatal rat offspring [2]. Further, PPA administration to rodents alters expression of genes associated with neurotransmitters, neuronal cell adhesion molecules, inflammation, oxidative stress, lipid metabolism, and mitochondrial function [3–5]. Conversely, decreases in PPA are reported in patients with multiple sclerosis, an autoimmune and neurodegenerative disease [6]. One important cellular process negatively affected by PPA is mitochondrial function [7]. Rats exposed to PPA show mitochondrial dysfunction and an increase in free acyl-carnitine, a factor for the transport of long-chain fatty acids into mitochondria [8]. PPA and butyric acid also induce autophagy in human colon cancer cells that limits apoptosis, and inhibition of autophagy potentiates SCFA-induced apoptosis [9]. As our previous data indicated that PPA induces abnormal autophagy in PPA-treated hippocampal neuron [10], we

investigated the relationship between mitochondrial defects and the regulation of mitochondrial quality though mitophagy.

Mitophagy is a selective degradative process responsible for removing damaged mitochondria to maintain cytoplasmic homeostasis [11]. Mitochondrial dysfunction is involved in various neurodegenerative or neurodevelopmental diseases [12]. Once mitophagy is initiated, a balance between autophagosome formation and autophagic degradation is necessary. Thus, accumulation of autophagosomes and disruption of the autophagic process in neurons is associated with disease [13]. Until now, conventional techniques for analysis of mitophagy have been based on immunofluorescence staining and immunoblotting of several specific mitochondrial proteins, qPCR for mitochondrial DNA copy number, and nano-resolution imaging using transmission electron microscopy (TEM). TEM is a direct imaging method for the early stage of mitophagy, which is the starting point of engulfing mitochondria and early autophagosome showing specific mitochondrial structures such as cristae [14,15]. However, the assessment of the late phases of mitophagy requires specific imaging techniques.

Recently, the engineering of two fluorescent proteins (mCherry-GFP-mito, and mt-Keima) has permitted monitoring of the status of mitophagy in live cells. These reporters change the fluorescence profile in response to pH changes. For example, the excitation wavelength for mt-Keima is 488 at neutral pH and 561 at acidic pH for late mitophagy observed in the lysosome [16,17]. mCherry-GFP-mito protein, fused to a mitochondrial targeting sequence of a mitochondrial protein, such as the outer mitochondrial membrane (OMM) protein FIS1 (comprising amino acids 101–152) [18] can be used to detect mitophagy. The mitochondrial network can be seen as a green fluorescence, and mitochondria delivered to lysosomes show as a red color after mitophagy. This is because mCherry fluorescence is stable, but GFP fluorescence is quenched in the acidic condition. However, these tools do not allow the monitoring of all phases of mitophagy. To analyze the entire dynamic phase of autophagy regarding mitophagy, TEM is employed to classify the specific type of autophagy including phagophore, autophagosome, and autolysosome in high resolution [19,20]. Therefore, we analyzed structural changes by TEM to study specific stages of autophagy that are more tightly linked with the mechanisms of mitophagy dysfunction. Thus, correlative light and electron microscopy (CLEM) is an effective method to analyze mitophagy or autophagic pathways [21]. Because the Keima protein is incompatible with fixation [22], we used GFP and mCherry conversion depending on the pH level of lysosomes to investigate details of various steps of mitophagy in nano-resolution though electron microscopy (EM). The CLEM technique of overlaying two images from fluorescence and EM makes the investigation of all phases of mitophagy possible. Evans et al. suggested that CLEM can open new avenues using light-up through (fluorescent) dyes in the dark by EM observation [23]. Thus, we applied CLEM to study mitophagy after PPA treatment.

#### **2. Materials and Methods**

#### *2.1. Cell Culture*

SH-SY5Y control cells, obtained from Dr. Kim H.J (KBRI), and the tandem mCherry–GFP tag fused to FIS1 stable SH-SY5Y cells, a kind gift from Dr. Ian G. Ganley (University of Dundee, Dundee, UK) [18], were grown in normal culture conditions with DMEM/F12 (ThermoFisher, Waltham, MA, USA) supplemented with 15% fetal bovine serum (ThermoFisher, USA), 100 units/mL of penicillin, and 100 ug/mL of streptomycin (ThermoFisher, USA) at 37 ◦C in a humidified 5% CO2 atmosphere. The tandem mCherry–GFP tag fused to FIS1 stable SH-SY5Y cells were selected with 500 μg/mL of hygromycin (Sigma, St. Louis, MO, USA) and a stable pool was used for experiments.

#### *2.2. Viability Assay*

SH-SY5Y cells were plated and treated with PPA (0.1, 1, 2, 6, and 12 mM, Sigma, St. Louis, MO, USA) in 96-well plates for 48 and 72 h Approximately 10 μL of CCK-8 (Dojindo, Kumamoto, Japan) was added to cells, and the optical density (OD) value was measured at 450 nm.

#### *2.3. Immunocytochemistry*

SH-SY5Y cells were grown on coverslips and treated with 1 mM PPA (Sigma, USA) for 72 h Cells were fixed with 1% paraformaldehyde (EMS, Hatfield, PA, USA) in phosphate-buffered saline (PBS, Welgene, Gyeongsangbuk-do, Gyeongsan-si, Korea) containing 4% sucrose for 5 min at room temperature. Primary antibodies against LC3A/B (#12741, Cell Signaling, Danvers, MA, USA) were added with blocking solution containing 0.1% gelatin, 0.3% Triton X-100, 16 mM sodium phosphate, and 450 mM NaCl, and cells were incubated overnight at 4 ◦C. After being washed with PBS, coverslips were incubated with Alexa Fluor488 (#4412, Cell Signaling, USA)-conjugated secondary antibodies for 1 h at room temperature and then again washed with PBS. Next, coverslips were mounted with a mounting medium (H-1000, Vector Laboratories, Burlingame, CA, USA) and were imaged with fluorescence microscopy (Nikon, Tokyo, Japan) using a 488 nm fluorescence filter.

#### *2.4. Transmission Electron Microscopy for Quantifying Autophagic Elements*

SH-SY5Y cells were treated with 1 mM of PPA for 72 h and then fixed with 2.5% glutaraldehyde/2% paraformaldehyde solution for 2 h Fixed cells were then post-fixed with 2% osmium tetroxide (EMS, USA) for 2 h at 4 ◦C, and the block was stained in 0.1 mg thiocarbohydrazide (TCH, TCI, Tokyo, Japan) in 10 mL distilled water and 1% uranyl acetate (EMS, USA) and dehydrated with a graded ethanol series. The samples were then embedded with an EMBed-812 embedding kit (EMS, USA). The embedded samples were sectioned (60 nm) with an ultramicrotome (Leica, Wetzlar, Germany), and the sections were then viewed on a Tecnai G2 transmission electron microscope (Thermofisher) at 120 kV. The numbers of autophagosomes and autolysosomes per cell were assessed.

#### *2.5. Correlative Light and Electron Microscopy*

CLEM was performed as previously described [23]. The mCherry–GFP tag fused to FIS1 stable SH-SY5Y cells were grown in 35 mm glass grid-bottomed culture dishes to 50–60% confluency. Cells with or without 1 mM or 2 mM PPA treatment were stained with 100 nM LysoTracker (LysoTracker Blue DND-22, Thermofisher, USA) for 15 min and then imaged under a confocal light microscope (Ti-RCP, Nikon, Japan), and after 24 h treatment of PPA, cells were fixed with 1% glutaraldehyde and 1% paraformaldehyde in 0.1 M cacodylate solution (pH 7.0). After being washed, cells were dehydrated with a graded ethanol series and infiltrated with an embedding medium. After embedment, 60 nm sections were cut horizontally to the plane of the block (UC7; Leica Microsystems, Germany) and were mounted on copper slot grids with a specimen support film. Sections were stained with uranyl acetate and lead citrate. The cells were observed at 120 kV in a Tecnai G2 microscope (ThermoFisher, USA). Confocal micrographs were produced as high-quality large images using PhotoZoom Pro 8 software (Benvista Ltd., Houston, TX, USA). Enlarged fluorescence images were fitted to the electron micrographs using the Image J BigWarp program.

#### *2.6. Measurement of Mitophagy*

The mCherry–GFP tag fused to FIS1 stable SH-SY5Y cells (5 <sup>×</sup> 104 cells/well) was grown in 35 mm glass-bottomed culture dishes (MatTEK, Ashland, MA, USA) and treated with 1 and 2 mM PPA. Parallel incubation of cells without PPA was used for control. Measurement of mitophagy has been described previously [16]. Briefly, quantitation was performed for five fields of view for each group. Red-alone puncta were defined as round structures found only in the red channel with no corresponding structure in the green channel. Quantitative data were collected by manually counting all red-only puncta within each cell for each field of view.

#### **3. Results**

The viability of SH-SY5Y cells after treatment with PPA showed that optimal concentrations of PPA were less than 2 mM. Viability was assessed with CCK-8 assays after treatment with concentrations of 0, 0.1, 1, 2, 6, and 12 mM. Viability was significantly decreased in response to 2 mM after 48 h incubation with PPA (Figure 1A). After 72 h, treatment with 1 mM PPA a significant change was also shown (Figure 1B). Therefore, we used 1 mM PPA for 72 h to assess autophagy in SH-SY5Y cells. The number of LC3 puncta in PPA-treated cells was increased, as shown by immunofluorescence (Figure 1C). The number of LC3 puncta was 2.9 ± 0.28, compared with 1.8 ± 0.3293 for untreated cells (Figure 1D). No difference in the intensity of LC3 puncta was observed.

**Figure 1.** Viability assay and increase of of LC3 level in SH-SY5Y cells after propionic acid (PPA) treatment. The cells were treated with PPA for (**A**) 48 h and (**B**) 72 h Statistical analysis used a two- way ANOVA. Results are presented as mean ± SEM. When the concentration reached 2–12 mM, cell viability was significantly reduced. (**C**) Representative immunofluorescence images showing LC3 puncta in SH-SY5Y control cells and PPA-treated cells. The white scale bar is 50 μm. (**D**) Numbers of LC3 puncta and LC3 puncta intensity from images in (**A**) (n = 5), illustrating a significant increase of number of LC3 puncta following treatment with 1 mM PPA. Statistical analysis used a one-way ANOVA. Results are presented as mean ± SEM. \* *p* < 0.05, \*\*\*\* *p* < 0.0001.

We further analyzed the number of autophagosomes and autolysosomes in control and PPA-treated cells using TEM images. The numbers of both organelles were increased in PPA-treated cells. The numbers of autophagosomes per cell were 6.1 ± 0.7371 and 1.9 ± 0.4333 for treated and control cells, respectively (Figure 2A). Similar results for autolysosomes were found: 6.6 ± 1.067 for PPA-treated cells and 1.9 ± 0.5667 for untreated cells (Figure 2B).

**Figure 2.** Theincreasein autophagy following PPA treatmentin SH-SY5Y cells using TEM. (**A**) Representative TEM images showing autophagy in control SH-SY5Y cells and PPA-treated cells. The black arrow indicates autophagosomes, and red arrows indicate autolysosomes. (**B** and **C**) Numbers of autophagosomes and autolysosomes from images in (**A**) (n = 10). TEM analysis shows that numbers of both autophagosomes and autolysosomes increase in cells treated with 1 mM PPA. Statistical analysis used a one-way ANOVA. Results are presented as mean ± SEM. \*\* *p* < 0.01, \*\*\* *p* < 0.0005.

PPA was reported as a small molecule leading to mitochondrial dysfunction [7]. Therefore, we used the tandem mCherry–GFP tag fused to FIS1-stable SH-SY5Y cells to confirm the induction of mitophagy by PPA treatment. GFP and mCherry show green and red fluorescence, respectively, with the former specific for mitochondria and the latter for mitophagy (Figure 3A). After treatment with 1 or 2 mM PPA, the number of mCherry red puncta increased 4.6 times, indicating the induction of mitophagy in treated cells (Figure 3B).

**Figure 3.** Mitophagy assays in SH-SY5Y cells following treatment with 1 and 2 mM PPA for 24 h. (**A**) mCherry–GFP-FIS1101–152 stably expressing cells were subjected to (1) control, (2) 1 mM PPA, and (3) 2 mM PPA for 24 h. (**B**) Numbers of red puncta (mitophagy) per cell (n > 40). Statistical analysis used a one-way ANOVA. Results are presented as mean ± SEM. \*\*\*\* *p* < 0.0001.

We employed CLEM to confirm the ultrastructure of red puncta (Figure 4). Live cells were imaged using confocal microscopy treatment with PPA after 24 h. Images were then aligned with stitched TEM images of the same cells. Almost healthy mitochondria showed the green fluorescence of GFP, and some red puncta were co-localized with LysoTracker (Figure 4A). In control cells, LysoTracker-positive structures (blue) are seen contacting red puncta (Figure 4A, white dot line box, and enlarged images). In the PPA treatment cells, damaged round mitochondria show high electron density in the electron micrographs and are co-localized with LysoTracker, suggesting mitophagy (Figure 4B).

**Figure 4.** Correlative light and electron microscopy. Correlative confocal and electron microscope images of (**A**) co-localized mCherry–GFP-FIS1101–152 in SH-SY5Y cells or (**B**) cells treated with 2 mM PPA. Yellow indicates co-localization of GFP and mCherry signals. Magenta indicates co-localization of mCherry and LysoTracker. Multiple TEM images were taken at 1700× magnification. Images were stitched for a large field of view at higher resolution. The black dot line box indicates structures corresponding to magenta fluorescent puncta on fluorescence images. The structures showed the morphology of mitophagy, as demonstrated by the black dot line box shown at higher magnification in the inserted images (A1–A6 and B1–B9). L, lysosome; M, mitochondria. Size bar in A and B = 5 μm, A1~A6 = 1 μm, B1~B9 = 1 μm.

#### **4. Discussion**

Mitochondria are continuously replenished. As new mitochondria are produced, dysfunctional organelles are removed by autophagy-mediated degradation though mitophagy [24]. There are several control or repair systems for mitochondrial structure and function maintaining essential energy metabolism. Oxidatively damaged proteins in the mitochondrial outer membrane can be degraded by the ubiquitin–proteasome system. When damage is more extensive, e.g., through exposure to elevated reactive oxygen species (ROS) or aging, mitochondria are sequestered by autophagosome and fused with lysosome for degradation. It is called mitophagy. The extent of mitophagy in neurites is influenced by various factors related to mitochondrial function, and the contribution of mitophagy to mitochondrial function in soma or neurites is critical to understanding the regulation of energy metabolism in these cells [25]. Our previous work shows that PPA induces defects in mitochondria [6] and autophagy [10]. In this study, we investigated the relationship between mitochondrial dysfunction and increased autophagy.

Some mitochondrial toxins, including rotenone, concomitantly activate autophagy, including mitophagy. Like rotenone, PPA-treated cells showed elevated autophagic sequestration of mitochondria. More prominent LC3 signals (Figure 1C) and LC3-II/β-actin ratios (data not shown) indicate change of autophagy in PPA-treated cells. Several reports that focus on mitochondrial function in PPA-treated cells are available. Kim et al. showed an increase of mitochondrial copy number and expression of PGC-1a, COX4, SIRT3, and, TFAM (mitochondrial biogenesis-related proteins) after PPA treatment of SH-SY5Y cells [7]. Dysfunction of mitochondria caused by RNA interference-mediated knockdown of peroxisome proliferator-activated receptor-γ coactivator 1α (PGC-1α) in neurons showed abnormal synapse formation in developing neural circuits and failure to maintain synapses in the hippocampus of adults [26]. Induction of mitochondrial biogenesis following expression of PGC-1α is stimulated by brain-derived neurotrophic factor, which can be modulated by changes of SCFAs in the brain [27]. PPA-induced mitochondrial dysfunction suggests a mechanism for neurotoxicity. El-Ansary et

al. showed PPA-induced neurotoxicity in rat pups though depletion of gamma-aminobutyric acid and serotonin. [28] Frye et al. showed oxidative stress after PPA exposure, and Alfawaz et al. showed that factors related to mitochondria, such as carnosine, N-acetylcysteine, and vitamin D, can rescue neurotoxicity caused by PPA in rat pups [29]. The protective effect of carnosine (β-alanyl-L-histidine) is related to autophagy and causes a decrease in Drp-1 expression. Further, treatment with N-acetylcysteine shows inhibition of Atg32-dependent mitophagy [29]. In PPA-treated SH-SY5Y cells in the present study, TEM analysis shows an increase in numbers of both autophagosomes and autolysosomes, which reflects properly functioning autophagy flux (Figure 2).

Several technical challenges using fluorescence and biochemical techniques to analyze autophagic processes, including mitophagy, are recognized [17,30]. Such challenges were met in this study using CLEM techniques to monitor autophagy for cellular homeostasis due to mitochondrial dysfunction in PPA-treated SH-SY5Y cells. The tandem mCherry–GFP tag fused to FIS1 was used in our approach. Red fluorescence of mCherry increased in PPA-treated cells, suggesting increased mitophagy, and the green fluorescent of GFP in mitochondria did not change significantly (Figure 3). CLEM confirmed the ultrastructure associated with these colors as mitochondria and mitophagy (Figure 4). There is a mismatch between some healthy mitochondrial fluorescence signal (yellow color) and EM images due to technical limitations of the TEM based CLEM method (Supplementary Figure S1 A1 and B1). It is difficult to accurately match the Z axis of the optical section (LM) and the physical section (EM), since LM and EM image thicknesses are different. Due to the difference, there is more information in the fluorescence micrograph (LM: 300 nm optical thickness, EM: 60 nm physical thickness). Although there is some technical limitation, the damaged mitochondria are well correlated with the lysotracker (Supplementary Figure S1, black arrow). Thus, observations depicted in Figures 3 and 4 indicate that increased autophagy shown in Figures 1 and 2 is mitophagy.

#### **5. Conclusions**

Changes in mitophagy under stress condition is associated with pathological conditions, including neurodegenerative diseases and myopathies. Therefore, identifying mitophagy modulators and understanding their mechanisms of action will provide critical insight into neurodegenerative diseases. We confirmed that mitophagy was induced by PPA treatment in SH-SY5Y cells. CLEM is a useful technique for monitoring mitophagy in cells under stress. Various stages of mitophagy, including initiation of autophagy, vesicle completion, lysosome fusion, and degradation of mitochondria in lysosomes, can be monitored in CLEM, if the time points in such studies are adequately controlled. CLEM might also be applied to study structural changes of other cellular organelles.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/1996-1944/13/19/4336/s1, Figure S1: Correlative confocal and electron microscope images of the control (A) and 2 mM PPA treated cell (B).

**Author Contributions:** Conceptualization, M.J., H.C., and J.Y.M.; methodology, M.J., H.C., and; formal analysis, M.J., H.C.; investigation, M.J., H.C., and J.Y.M.; resources, J.K.; data curation, M.J., H.C.; writing—original draft preparation, J.Y.M.; writing—review and editing, M.J., H.C., J.K., and J.Y.M.; visualization, M.J., H.C.; supervision, J.Y.M.; project administration, J.Y.M.; funding acquisition, J.K. and J.Y.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the National Research Foundation of Korea (NRF) grant funded by the government of Korea (MSIP) (No. 2019R1A2C1010634), the Organelle Network Research Center (NRF-2017R1A5A1015366), and KBRI basic research program though the Korea Brain Research Institute funded by the Ministry of Science and ICT (20-BR-02-09).

**Acknowledgments:** Instrument (transmission electron microscopy and confocal microscopy) data were acquired at the Brain Research Core Facilities in KBRI.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Exploiting the Acceleration Voltage Dependence of EMCD**

**Stefan Löffler 1,\*, Michael Stöger-Pollach 1, Andreas Steiger-Thirsfeld 1, Walid Hetaba <sup>2</sup> and Peter Schattschneider 1,3**


**Abstract:** Energy-loss magnetic chiral dichroism (EMCD) is a versatile method for measuring magnetism down to the atomic scale in transmission electron microscopy (TEM). As the magnetic signal is encoded in the phase of the electron wave, any process distorting this characteristic phase is detrimental for EMCD. For example, elastic scattering gives rise to a complex thickness dependence of the signal. Since the details of elastic scattering depend on the electron's energy, EMCD strongly depends on the acceleration voltage. Here, we quantitatively investigate this dependence in detail, using a combination of theory, numerical simulations, and experimental data. Our formulas enable scientists to optimize the acceleration voltage when performing EMCD experiments.

**Keywords:** EMCD; TEM; EELS; magnetism; acceleration voltage

**Citation:** Löffler, S.; Stöger-Pollach, M.; Steiger-Thirsfeld, A.; Hetaba, W.; Schattschneider, P. Exploiting the Acceleration Voltage Dependence of EMCD. *Materials* **2021**, *14*, 1314. https://doi.org/10.3390/ma14051314

Academic Editors: Lucia Nasi and Matteo Ferroni

Received: 18 December 2020 Accepted: 26 February 2021 Published: 9 March 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

#### **1. Introduction**

Circular dichroism in X-ray Absorption Spectroscopy (XAS) probes the chirality of the scatterer, related either to a helical arrangement of atoms or to spin polarized transitions as studied in X-ray Magnetic Circular Dichroism (XMCD). Before the new millenium, it was considered impossible to see such chirality in electron energy-loss spectrometry (EELS). On the other hand, the formal equivalence between the polarization vector in XAS and the scattering vector in EELS tells us that any effect observable in XAS should have its counterpart in EELS. For instance, anisotropy in XAS corresponds to anisotropy of the double differential scattering cross section (DDSCS) in EELS. A well known example is the directional prevalence of either *s* → *π*<sup>∗</sup> and *s* → *σ*<sup>∗</sup> transitions in the carbon K-edge of graphite, depending on the direction of the scattering vector [1,2].

In XMCD, the polarization vector is helical—a superposition of two linear polarization vectors **e***<sup>x</sup>* ± *i***e***<sup>y</sup>* orthogonal to each other—resembling a left- and right-handed helical photon, respectively. However, what is the counterpart of photon helicity in EELS?

In 2002, one of the authors and their postdoc speculated about what the counterpart of photon helicity could be in EELS—an arcane issue at the time. This led to a keen proposal to study spin polarized transitions in the electron microscope [3]. Closer inspection revealed that in EELS, a superposition of two scattering vectors orthogonal to each other with a relative phase shift of ±*π*/2 is needed, exactly as the formal similarity with XMCD dictated. This, in turn, called for a scattering geometry that exploits the coherence terms in the DDSCS [4,5]. These insights led to the CHIRALTEM project [6].

The multidisciplinary team elaborated the appropriate geometry for the analysis of ionization edges in the spirit of XMCD. The first EELS spectrum was published in 2006 [7]. In that paper, the new method was baptized EMCD—Electron (Energy Loss) Magnetic Chiral Dichroism—in analogy to XMCD. The term "chiral" was deliberately chosen instead of "circular" because the chirality of electronic transitions was to be detected, and because there is

no circular polarization in EELS. The experiment confirmed that the physics behind EMCD is very similar to the physics of XMCD. Rapid progress followed: consolidation of the theory [8,9], optimization of experimental parameters [10], dedicated simulation software [11,12], and spatial resolution approaching the nm [13,14] and the atomic scale [15–23].

A genuine feature of EMCD is the ability to probe selected crystallographic sites [18,24], e.g., in Heusler alloys [25], ferrimagnetic spinels [26], or perovskites [27,28]. The high spatial resolution of the method allows the study of nanoparticles [14], 3d–4f coupling in superlattices [29], specimens with stochastically oriented crystallites and even of amorphous materials [30]. EMCD has also been used to investigate spin polarization of non-magnetic atoms in dilute magnetic semiconductors [31], magnetic order breakdown in MnAs [32], GMR of mixed phases [33] and magnetotactic bacteria [34]. A key experiment on magnetite, exploiting the combination of atomic resolution in STEM with the site specificity showed the antiferromagnetic coupling of adjacent Fe atoms directly in real space [16]. An overview of EMCD treating many aspects of anisotropy and chirality in EELS can be found in [35].

To date, EMCD measurements have predominantly been performed at the highest available acceleration voltages—typically 200 keV to 300 keV—which has several advantages such as better resolution, a larger inelastic mean free path, and optimal detector performance resulting in a reasonable signal-to-noise ratio. However, by limiting oneself to a specific acceleration voltage and hence electron energy, EMCD cannot be used to its full potential.

One example where choosing a lower acceleration voltage can be tremendously helpful is the reduction or avoidance of beam damage [36–39]. Another is the investigation of the magnetization dependence: in a TEM, the sample is placed inside the objective lens with a typical field strength of the order of 2 T for 200 keV electrons. By changing the acceleration voltage, the objective lens field applied at the sample position is changed as well [40], thereby enabling magnetization-dependent investigations. This can even be used to drive magnetic field induced phase transitions [27]. Moreover, EMCD is strongly affected by elastic scattering, and, hence, thickness and sample orientation [8,11,25,41]. Therefore, changing the electron energy and therefore the details of the elastic scattering processes enables EMCD measurements even at a thickness and orientation where no significant EMCD effect is observable at a high acceleration voltage. This proposition is corroborated by early numerical simulations [42], which to our knowledge have not been followed up on or widely adopted by the community.

#### **2. Results**

#### *2.1. Theory*

The general formula governing EMCD has already been outlined in the original publications theoretically predicting the effect and demonstrating it experimentally [3,7]. Detailed ab initio studies soon followed [8]. However, those formulations all aimed at very high accuracy; none of them gave a simple, closed form to quickly calculate the EMCD effect and easily see the influence parameters such as, e.g., the acceleration voltage have on the outcome. Recently, Schneider et al. [41] published such a formula; however, they neglected any elastic scattering the beam can undergo after an inelastic scattering event by approximating the outgoing wave by a simple plane wave.

Here, we present a derivation of a simple formula taking into account elastic scattering both before and after the inelastic scattering event. In the process, we will make four major assumptions:


much smaller than the chosen reciprocal lattice distance <sup>|</sup>*G*|. This implies that the inelastic scattering in the chosen geometry is only dependent on the scattering atom's spin-state, but not influenced significantly by any anisotropic crystal field;

4. We assume that the atoms of the investigated species are homogeneously distributed along the beam propagation axis and that *G* · *x* <sup>=</sup> <sup>2</sup>*mπ*, *<sup>m</sup>* <sup>∈</sup> <sup>Z</sup> for all atom positions *x* and the chosen lattice vector *G*.

Assumption 1 comes from the conventional EMCD setup: the (crystalline) sample is tilted into systematic row condition and the detector is placed on (or close to) the Thales circle between neighboring diffraction spots. In a symmetric systematic row condition, the strongest diffraction spots are the central one (**0**) and the two diffraction spots at <sup>−</sup>*G*, *G*, which have the same intensity. Any higher-order diffraction spots are comparatively weak and will therefore be neglected.

To understand the reason behind the outgoing two-beam case, we follow the reciprocity theorem [43,44]. A (point-like) detector in reciprocal space detects exact plane-wave components. If we trace those back to the exit plane of the sample, we can expand them into Bloch waves. For the typical EMCD detector positions, they correspond exactly to the Bloch waves we get in a two-beam case (where the Laue circle center is positioned somewhere along the bisector of the line from **<sup>0</sup>** to *G*.

The probability of measuring a particular state |*ψ*out (a "click" in the detector corresponding to a plane wave at the exit plane of the sample) given a certain incident state |*ψ*in (a plane wave incident on the entry plane of the sample) is given by Fermi's Golden rule [45–49]:

$$p = \sum\_{I, \sf F} p\_I \left(1 - p\_F\right) \left<\psi\_{\sf out} \right| \left \left|\psi\_{\sf in}\right> \left<\psi\_{\sf in}\right| \left *\left|\psi\_{\sf out}\right> \delta\left(E\_F - E\_I - E\right), \tag{1}*$$

where *I*, *F* run over all initial and final states of the sample, *pI*, *pF* are their respective occupation probabilities, *EI*, *EF* are their respective energies, *E* is the EELS energy loss, and *V*ˆ is the transition operator. In momentum representation, *V*ˆ for a single atom is given by

$$
\langle \vec{k} | \hat{V} | k \rangle = \frac{\mathbf{e}^{i\mathbf{q} \cdot \hat{R}}}{q^2} \quad \text{with} \quad q = k - \tilde{k}. \tag{2}
$$

With the mixed dynamic form factor (MDFF) [45,49–51],

$$S(\boldsymbol{\mathfrak{q}}, \boldsymbol{\mathfrak{q}}', \boldsymbol{E}) = \sum\_{I, \boldsymbol{F}} p\_I (1 - p\_{\boldsymbol{F}}) \left< \mathbf{k} \right| \left< \mathbf{F} \middle| \mathbf{e}^{i \boldsymbol{q} \cdot \hat{\mathbf{R}}} \middle| I \right> \left| \mathbf{k} \right> \left< \mathbf{k}' \middle| \left< I \middle| \mathbf{e}^{-i \boldsymbol{q}' \cdot \hat{\mathbf{R}}} \middle| F \right> \left< \mathbf{k}' \right> \delta \left( E\_{\boldsymbol{F}} - E\_{\boldsymbol{I}} - E \right), \tag{3}$$

the probability for a "click" in the detector can be written as [8,45,48–50]

$$p = \iiint \int \sum\_{\mathbf{x}} \mathbf{e}^{\mathbf{i}(q - q') \cdot \mathbf{x}} \psi\_{\text{out}}(\ddot{\mathbf{k}})^{\*} \psi\_{\text{out}}(\ddot{\mathbf{k}}') \frac{S(q, q', \mathbf{E})}{q^{2} q'^{2}} \psi\_{\text{in}}(\mathbf{k}) \psi\_{\text{in}}(\mathbf{k}')^{\*} \, \text{d} \mathbf{k} \, \text{d} \mathbf{k}' \, \text{d} \mathbf{k} \, \text{k} \mathbf{k}', \qquad (4)$$

where the <sup>∑</sup>*x <sup>e</sup>*i(*q*−*<sup>q</sup>* )·*<sup>x</sup>* stems from the summation over all atoms (of the investigated species) in the sample and the MDFF is taken to be the MDFF of a single such atom located at the origin.

Specific expressions for the MDFF for various models under different conditions and approximations are well known (see, e.g., [7,49,52]), but their details will be irrelevant for the majority of our derivation for which we will keep the general expression *<sup>S</sup>*(*q*, *q* , *E*).

Using the Bloch wave formalism [8,36,53–55], the three-beam incident wavefunction and the two-beam outgoing wave function can be written as

$$|\psi\_{\rm in}\rangle = \sum\_{j \in \{1,2,3\}} \sum\_{\substack{\mathfrak{g} \in \{-G, \mathfrak{g}, G\} \\ \text{....}}} \mathbb{C}\_{j, \mathfrak{g}}^{\*} \mathbb{C}\_{j, \mathfrak{g}} |\mathfrak{g} + \gamma\_{j} \mathfrak{n} + \mathfrak{g}\rangle \tag{5}$$

$$\left|\psi\_{\rm out}\right\rangle = \sum\_{l \in \{1, 2\}} \sum\_{h \in \{0, G\}} \mathcal{C}\_{l,0}^\* \mathbf{e}^{-i\gamma\_l t} \mathcal{C}\_{l,h} \left|\tilde{\chi} + \gamma\_l \bar{n} + h\right\rangle,\tag{6}$$

where *<sup>j</sup>*, *<sup>l</sup>* are the Bloch wave indices, *<sup>g</sup>*, *<sup>h</sup>* run over the diffraction spots, the *Cj*,*g* are the Bloch wave coefficients, the *<sup>γ</sup><sup>j</sup>* are the so-called anpassung, *<sup>n</sup>* is the surface normal vector, *<sup>t</sup>* is the sample thickness, and *χ*, *χ*˜ are the wave vectors of the incident and outgoing plane waves, respectively.

The derivation of the EMCD effect can be found in Appendix A. The final expression is

$$\eta = \frac{A\sin^2(\kappa t) - B\sin^2(\kappa' t)}{t + C\sin(2\kappa t)} \cdot \frac{\odot[S(\mathbf{q}\_1, \mathbf{q}\_2, E)]}{S(\mathbf{q}\_{1'}, \mathbf{q}\_{1'}, E)},\tag{7}$$

where *t* is the sample thickness and the coefficients *A*, *B*, *C*, *κ*, *κ* are defined in Equation (A18) (with Equations (A1) and (A3)).

Figure 1 shows a comparison of the thickness dependence predicted by Equation (7) and a full simulation based on Equation (4) for some typical, simple magnetic samples. Owing to the approximations made in the derivation, there naturally are some small differences (which are more pronounced at small thicknesses), but they are well within typical experimental uncertainties.

**Figure 1.** Comparison of the thickness dependence of the EMCD effect *η* predicted by Equation (7) (solid lines) and by the "bw" software using Equation (4) (dotted lines) for different acceleration voltages for bcc Fe and hcp Co.

Two main conclusions about the thickness-variation of the EMCD effect can be drawn from Equation (7). On the one hand, the numerator nicely shows the oscillatory nature of the effect. On the other hand, the denominantor clearly implies that the strength of the EMCD effect decreases approximately as 1/*t*.

The numerator is composed of two oscillations with different amplitudes (*A*, *B*) and the frequencies

$$\kappa = \frac{\gamma\_1 - \gamma\_2}{2} = \frac{\sqrt{(|\mathbf{G}|^2 - \ell I\_{2G})^2 + 8\ell I\_G^2}}{4\chi \cdot \mathfrak{n}} \quad \text{and} \quad \kappa' = \frac{\tilde{\gamma}\_1 - \tilde{\gamma}\_2}{2} = \frac{\ell I\_G}{2\tilde{\chi} \cdot \tilde{\mathfrak{n}}} \tag{8}$$

which are closely related to the extinction distances for the incident and outgoing beams. As the wavevectors *<sup>χ</sup>*, *<sup>χ</sup>*˜ scale with the square root of the acceleration voltage <sup>√</sup>*V*, the frequencies of the oscillations of the EMCD effect scale with 1/ <sup>√</sup>*V*. This is corroborated by Figure 2.

Both the oscillations and the 1/*t* decay can be understood from the fact that EMCD is essentially an interferometry experiment. As such, it crucially depends on the relative phases of the different density matrix components after traversing the sample from the scattering center to the exit plane. Some scattering centers are positioned in a way that the resulting components contribute positively to the EMCD effect, other scattering centers are positioned such that their contribution to the EMCD effect is negative. As a result, there are alternating "bands" of atoms contributing positively and negatively [11], where the size of the bands is related to the extinction length. With increasing thickness, more and more alternating bands appear—the non-magnetic signal increases linearly with *t*, but the magnetic EMCD signal of all but one band averages out, ultimately resulting in a 1/*t* behavior of the relative EMCD effect.

Our theoretical results have several important implications. First, the EMCD effect can indeed be recorded at a wide variety of acceleration voltages as already proposed on numerical grounds in [42], thereby enabling magnetization-dependent measurements. Second, the thickness dependence scales with 1/*t*, thus necessitating thin samples. Third, for a given sample thickness in the region of interest, a candidate for the optimal high tension yielding the maximal EMCD effect can easily be identified based on any existing simulation and the <sup>√</sup>*<sup>V</sup>* scaling behavior (note, however, that other effects such as multiple plasmon scattering can put further constraints on the useful range of sample thicknesses, particularly at very low voltages).

#### *2.2. Experiments*

To corroborate our theoretical finding, we performed experiments at various high tensions to compare to the simulations. The experiments were performed on a ferrimagnetic magnetite (Fe3O4) sample [56], which has the advantage over pure Fe that it is unaffected by oxidation (it may, however, be partially reduced to Wüstite by prolonged ion or electron irradiation). The individual recorded spectra are shown in Figure 3. It is clearly visible that the EMCD effect changes with the high tension as predicted in Section 2.1. A quantitative comparison between the calculations and the experiments is shown in Figure 4 and shows excellent agreement.

**Figure 3.** EMCD spectra for different acceleration voltages (as indicated) after background subtraction and post-edge normalization using the Fe L-edge in Magnetite tilted to a (400) systematic row condition. The sample-thickness was determined to be *t* ≈ 35 nm for the 40 kV and 60 kV measurement positions and *t* ≈ 45 nm for the 200 kV measurement position.

**Figure 4.** Comparison between numerical EMCD simulations ("bw", solid curves) and experiments (points) for Magnetite for three different acceleration voltages. For the experimental points, *η* was calculated from the data in Figure 3 according to Equation (9), the measured thickness values are given in the caption of Figure 3, and the error bars were determined as described in [57,58].

#### **3. Discussion**

Although Equation (7) is—to our knowledge—the first complete, analytical, closed form predicting the EMCD effect, several assumptions and approximations were made in its derivation. As such it is no replacement for full simulations with sophisticated software packages if ultimate accuracy is vital. Nevertheless, it can be a good starting point for EMCD investigations, and it helps elucidating the underlying physical principles and understanding the effects the experimental parameters have on EMCD. In this section, we will discuss the limits of the theoretical derivation based on the approximations made.

Assumption one deals with the scattering geometry and the crystal structure. The incident three-beam and outgoing two-beam case is the simplest approximation taking into account elastic scattering both before and after an inelastic scattering event. Adding more beams to the calculation can, of course, improve the results somewhat. However, the effect was found to be very small and well within typical experimental uncertainties [11], owing primarily to the 1/*q*2*q* <sup>2</sup> term in Equation (4) (any additional beams would give

rise to much longer *q* vectors). The crystal structure was assumed to be centro-symmetric, resulting in *<sup>U</sup>G* <sup>=</sup> *<sup>U</sup>*−*G*. While this limits the applicability of the formula to relatively simple crystals, very complex, non-symmetric crystals will likely violate some of the other assumptions as well. In addition, the constraints implied by centro-symmetry are necessary in the first place to arrive at a reasonably simple final formula.

Assumption two requires the sample's surface to be essentially perpendicular to the beam direction. This requirement is necessary to avoid complex phase factors down the line. A small tilt of up to a few degrees is not expected to cause any major issues, and larger tilts of -45 ◦C are not recommended (and often not even possible) in practice anyway.

Assumption three requires the inelastic scattering process to be invariant under rotations around the optical axis by integer multiples of 90°. Strong anisotropy would lead to a distinct directional dependence of the MDFF [48,59,60], thereby making it impossible to reason about the intensities at the various detector positions. In such cases, however, the classical EMCD setup would fail to properly measure the magnetic properties anyway. In addition, assumption three states *qe* |*G*|, which implies [*S*(*q*1, *<sup>q</sup>*2, *<sup>E</sup>*)] [*S*(*q*1, *<sup>q</sup>*2, *<sup>E</sup>*)] in dipole approximation [11,61]. This is fulfilled reasonably well for typical EMCD experiments (for example, for Fe (200), <sup>|</sup>*G*| ≈ 7 nm<sup>−</sup>1; for the Fe L-edge, *qe* <sup>≈</sup> 0.8 nm−<sup>1</sup> at 200 keV and *qe* ≈ 1.5 nm−<sup>1</sup> at 40 keV).

Assumption four requires the investigated atoms to be distributed homogeneously and fulfill the condition *G* · *x* <sup>=</sup> <sup>2</sup>*mπ*. The homogeneity requirement excludes involved situations such as multi-layer systems and ultimately allows to replace the sum over all atoms by an integral over the sample thickness. In practice, homogeneity is facilitated by tilting into a systematic row condition and probing a large area of the sample, as a large probed volume and a (small) tilt mean that some atoms can be found in each of the investigated lattice planes at any depth *z*.

The condition *G* · *x* <sup>=</sup> <sup>2</sup>*m<sup>π</sup>* <sup>∀</sup>*x* is perhaps the most severe limitation as it implies that all atoms fall exactly onto one of the probed set of lattice planes. This excludes, e.g., *G* = (100) for Fe (which is forbidden anyway), or *G* = (100) for Co, as for these, only some (for Fe) or none (for Co) of the atoms fulfill the condition. The reason for requiring *G* · *x* <sup>=</sup> <sup>2</sup>*m<sup>π</sup>* is that it implies that phase factors of the form exp(i*G* · *x*) are all 1. If that is not the case, different phases have to be applied to different components, thereby reducing the EMCD effect [41]. Hence, choosing a *G* vector not fulfilling the condition is unfavorable anyway.

As can be seen from Figure 1, Equation (7) reproduces sophisticated numerical simulations quite well for reasonably simple samples despite all approximations. The strongest deviations can be found for small *t*, as can be expected. For larger sample thicknesses and, consequently, many atoms, small differences that might arise for individual atoms tend to average out.

#### **4. Materials and Methods**

The numerical simulations were performed using the "bw" code [11], a software package for calculating EELS data based on Bloch waves and the MDFF. The crystal structure data for magnetite was taken from [62], all other crystallographic data was taken from the EMS program (version 4.5430U2017) [63].

The wedge-shaped magnetite sample was prepared by a FEI Quanta 200 3D DBFIB (FEI Company, Hillsboro, OR, USA) from a high-quality, natural single crystal purchased from SurfaceNet GmbH (Rheine, Germany) [64] and subsequently thinned and cleaned using a Technoorg Linda Gentlemill.

The EMCD measurements were performed on a FEI Tecnai T20 (FEI Company, Hillsboro, OR, USA) equipped with a LaB6 gun and a Gatan GIF 2001 spectrometer (Gatan Inc., Pleasanton, CA, USA). The system has an energy resolution (full width at half maximum) of 1.1 eV at 200 kV which improves down to 0.3 eV at 20 kV [65]. First, a suitable sample position with a sample thickness around 40 nm and an easily recognizable, distinctlyshaped feature nearby was found and the sample was oriented in systematic row condition including the (400) diffraction spot (see Figure 5). At each high tension, the instrument

was carefully aligned, the sample position was readjusted, the EMCD experiment was performed, and a thickness measurement was taken. Both the convergence and the collection semi-angle were approximately 3 mrad [58].

**Figure 5.** TEM bright-field overview image (**left**), corresponding diffraction pattern in (0 1 1) zone axis (**middle**) and schematic of the EMCD measurement positions in systematic row condition (**right**). The sample position used for the EMCD experiments is marked by a yellow circle in the bright-field image, the positions for *I*<sup>+</sup> and *I*<sup>−</sup> are marked by the orange and blue circles. Both the image and the diffraction pattern were recorded at 200 kV. Note that the weak, kinematically forbidden (200) reflections can be attributed to double diffraction [36] in the thicker part of the sample visible at the bottom of the bright-field image; they are negligible in the thin part of the sample used for the EMCD measurements.

> For data analysis, all spectra were background-subtracted using a pre-edge power-law fit and normalized in the post-edge region. The EMCD effect was calculated based on the L3-edge maxima according to the formula [9,58]

$$
\eta = \frac{I\_+ - I\_-}{\frac{I\_+ + I\_-}{2}}.\tag{9}
$$

The errors were estimated as described in [57,58].

#### **5. Conclusions**

In this work, we have derived an analytical formula for predicting the EMCD effect, taking into account elastic scattering both before and after inelastic scattering events. This formula not only helps elucidate the physics underlying EMCD, it also allows to directly predict the influence of various parameters on the EMCD effect. In particular, we have focused on the acceleration voltage *V* and on the thickness *t*. We showed that the periodicity of the EMCD effect scales with <sup>√</sup>*V*, while its total intensity decreases as 1/*t*. In addition, we have performed experiments at different acceleration voltages to corroborate these predictions. Our results will not only help to optimize the EMCD effect for a given sample thickness by tuning the high tension accordingly, it will also pave the way for magnetization-dependent measurements by employing different magnetic fields in the objective lens at different acceleration voltages.

**Author Contributions:** Conceptualization, S.L., P.S.; methodology, S.L., M.S.-P., W.H., P.S.; software, S.L.; formal analysis, S.L.; investigation, S.L., M.S.-P.; resources, A.S.-T., W.H.; data curation, S.L.; writing—original draft preparation, S.L., P.S.; writing—review and editing, M.S.-P., A.S.-T., W.H.; visualization, S.L.; supervision, P.S.; project administration, S.L., P.S.; funding acquisition, S.L., P.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Austrian Science Fund (FWF) under grant numbers I4309-N36 and P29687-N36.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data is contained within the article.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


XMCD X-ray magnetic circular dichroism

#### **Appendix A. Derivation of the EMCD Effect**

In the following, we will extensively use the abbreviations

$$
\boldsymbol{\alpha} = \frac{\boldsymbol{\mathcal{U}}\_{\mathcal{G}}}{2\boldsymbol{\mathcal{X}} \cdot \boldsymbol{\mathfrak{n}}} \qquad \boldsymbol{\bar{\mathfrak{n}}} = \frac{\boldsymbol{\mathcal{U}}\_{\mathcal{G}}}{2\boldsymbol{\tilde{\chi}} \cdot \boldsymbol{\mathfrak{n}}} \tag{A1}
$$

$$V = \frac{\mathcal{U}\_{2G} - |\mathcal{G}|^2}{2\mathcal{U}\_G} \tag{A2}$$

$$W = \frac{\sqrt{(|\mathcal{G}|^2 - \mathcal{U}\_{2G})^2 + 8\mathcal{U}\_G^2}}{2\mathcal{U}\_G} = \sqrt{V^2 + 2},\tag{A3}$$

where the *<sup>U</sup>g* are the Fourier coefficients of the crystal potential *<sup>V</sup>*(*r*) = *<sup>h</sup>*<sup>2</sup> <sup>2</sup>*me* <sup>∑</sup>*<sup>g</sup> <sup>U</sup>g*e2*<sup>π</sup>*i*g*·*<sup>r</sup>* with Planck's constant *h*, electron mass *m* and elementary charge *e*. We note in passing that in the present case, *<sup>U</sup>G* <sup>=</sup> *<sup>U</sup>*<sup>∗</sup> *G* <sup>=</sup> *<sup>U</sup>*−*G*.

With these abbreviations and the assumptions mentioned above, the Bloch wave parameters can be calculated analytically and take the form

$$\begin{aligned} \gamma\_1 &= a(V+W) & \gamma\_2 &= a(V-W) & \gamma\_3 &= -a \cdot \frac{|G|^2 + U\_{2G}}{U\_G} \\ \text{C}\_{1,-G} &= \frac{1}{\sqrt{|V-W|^2 + 2}} & \text{C}\_{2,-G} &= \frac{1}{\sqrt{|V+W|^2 + 2}} & \text{C}\_{3,-G} &= -\frac{1}{\sqrt{2}} \\ \text{C}\_{1,0} &= -\frac{V-W}{\sqrt{|V-W|^2 + 2}} & \text{C}\_{2,0} &= -\frac{V+W}{\sqrt{|V+W|^2 + 2}} & \text{C}\_{3,0} &= 0 \\ \text{C}\_{1,G} &= \frac{1}{\sqrt{|V-W|^2 + 2}} & \text{C}\_{2,G} &= \frac{1}{\sqrt{|V+W|^2 + 2}} & \text{C}\_{3,G} &= \frac{1}{\sqrt{2}} \end{aligned} \tag{A4}$$

for |*ψ*in and

*γ*˜1 = *α*˜ *γ*˜2 = −*α*˜ *C*˜ 1,**<sup>0</sup>** <sup>=</sup> <sup>1</sup> √2 *C*˜ 2,**<sup>0</sup>** <sup>=</sup> <sup>1</sup> √2 *C*˜ 1,*G* <sup>=</sup> <sup>1</sup> √2 *C*˜ 2,*G* <sup>=</sup> <sup>−</sup> <sup>1</sup> √2 (A5) for |*ψ*out.

Inserting Equations (5) and (6) into Equation (4), evaluating the integrals, collecting all terms with the same Bloch wave index, and neglecting the weak dependence of *<sup>S</sup>*(*q*, *q* , *E*)/(*q*2*q* <sup>2</sup> ) on *j*, *j* , *l*, *l* [8,41,55] yields

$$p = \sum\_{\mathbf{x}} \sum\_{\mathbf{g}, \mathbf{g}', h, h'} D\_{\mathbf{3}} D\_{\mathbf{3}'}^\* \bar{D}\_{h}^\* \bar{D}\_{h'} \mathbf{e}^{i(\mathbf{g} - \mathbf{g}' - h + h') \cdot \mathbf{x}} \frac{\mathbf{S}(\mathbf{q}, \mathbf{q}', \mathbf{E})}{q^2 q'^2} \tag{A6}$$

with

$$D\_{\mathcal{R}} = \sum\_{j} \mathbf{C}\_{j,\mathbf{0}}^{\*} \mathbf{C}\_{j,\mathbf{0}} \mathbf{e}^{i\gamma\_{j} \cdot \mathbf{n} \cdot \mathbf{x}} \qquad \vec{D}\_{\mathcal{R}} = \sum\_{l} \vec{\mathcal{C}}\_{l,\mathbf{0}}^{\*} \mathbf{e}^{-i\gamma\_{l}t} \vec{\mathcal{C}}\_{l,\mathbf{h}} \mathbf{e}^{i\gamma\_{j} \cdot \mathbf{n} \cdot \mathbf{x}} \tag{A7}$$

and

$$q = \Delta \chi + \mathbf{g} - h \qquad q' = \Delta \chi + \mathbf{g'} - h' \qquad \Delta \chi = \chi - \tilde{\chi}.\tag{A8}$$

Direct summation results in

$$\begin{split} D\_{-G} = D\_G &= \frac{\mathbf{i}}{W} \mathbf{e}^{\|\mathbf{v}\|\_{W} \cdot \mathbf{x}} \sin(aW\mathbf{n} \cdot \mathbf{x}) \\ D\_0 &= \mathbf{e}^{\|\mathbf{v}\|\_{W} \cdot \mathbf{x}} \left[ \cos(aW\mathbf{n} \cdot \mathbf{x}) - \frac{\mathbf{i}V}{W} \sin(aW\mathbf{n} \cdot \mathbf{x}) \right] \\ D\_0 &= \cos(\bar{\mathbf{a}}(\bar{\mathbf{n}} \cdot \mathbf{x} - t)) \\ \bar{D}\_G &= \mathbf{i} \sin(\bar{\mathbf{a}}(\bar{\mathbf{n}} \cdot \mathbf{x} - t)) .\end{split} \tag{A9}$$

Performing the complete sums over *g*, *g* , *h*, *h* in Equation (A6) produces very many terms, some of which are very small. This can be understood from the fact that <sup>Δ</sup>*<sup>χ</sup>* · *G* <sup>=</sup> <sup>±</sup>*G*/2 in the chosen setup. Therefore, <sup>Δ</sup>*<sup>χ</sup>* and <sup>Δ</sup>*<sup>χ</sup>* <sup>−</sup> *G* have the same magnitude, whereas <sup>Δ</sup>*<sup>χ</sup>* <sup>+</sup> *G* and <sup>Δ</sup>*<sup>χ</sup>* <sup>−</sup> <sup>2</sup>*G* are significantly larger. Owing to the 1/*q*2*<sup>q</sup>* <sup>2</sup> term, large *q* are strongly suppressed. Hence, only the combinations *g* <sup>−</sup> *h* <sup>=</sup> **<sup>0</sup>** and *g* <sup>−</sup> *h* <sup>=</sup> <sup>−</sup>*G* are retained (the same applies to the primed versions as well). Hence, we end up with two distinct *q* vectors, namely

$$q\_1 = \Delta \chi \quad \text{and} \quad q\_2 = \Delta \chi - G. \tag{A10}$$

Note that, due to the symmetry of the setup *<sup>q</sup>*<sup>1</sup> <sup>=</sup> <sup>|</sup>*q*1<sup>|</sup> <sup>=</sup> <sup>|</sup>*q*2<sup>|</sup> <sup>=</sup> *<sup>q</sup>*2. Using *<sup>S</sup>*(*q*, *q* , *<sup>E</sup>*) = *<sup>S</sup>*(*q* , *q*, *<sup>E</sup>*)<sup>∗</sup> [45], Equation (A6) now takes the form

*<sup>p</sup>* <sup>=</sup> <sup>1</sup> *q*4 1 ∑*x* [ *<sup>D</sup>***0***D*˜ <sup>∗</sup> **<sup>0</sup>** <sup>+</sup> *<sup>D</sup>GD*˜ <sup>∗</sup> *G* 2 *<sup>S</sup>*(*q*1, *q*1, *<sup>E</sup>*) + *<sup>D</sup>−GD*˜ <sup>∗</sup> **<sup>0</sup>** + *D***0***D*˜ <sup>∗</sup> *G* 2 *<sup>S</sup>*(*q*2, *q*2, *<sup>E</sup>*) + 2 - *D***0***D*˜ <sup>∗</sup> **<sup>0</sup>** <sup>+</sup> *<sup>D</sup>GD*˜ <sup>∗</sup> *G D*∗ *<sup>−</sup>GD*˜ **<sup>0</sup>** <sup>+</sup> *<sup>D</sup>*<sup>∗</sup> **<sup>0</sup>***D*˜ *G* <sup>e</sup>i*G*·*xS*(*q*1, *q*2, *<sup>E</sup>*) <sup>=</sup> <sup>1</sup> *q*4 1 [ *<sup>A</sup>*11*S*(*q*1, *<sup>q</sup>*1, *<sup>E</sup>*) + *<sup>A</sup>*22*S*(*q*2, *<sup>q</sup>*2, *<sup>E</sup>*) + <sup>2</sup>[*A*12*S*(*q*1, *<sup>q</sup>*2, *<sup>E</sup>*)]] <sup>=</sup> <sup>1</sup> *q*4 1 [ (*A*<sup>11</sup> <sup>+</sup> *<sup>A</sup>*22)*S*(*q*1, *<sup>q</sup>*1, *<sup>E</sup>*) + <sup>2</sup>[*A*12*S*(*q*1, *<sup>q</sup>*2, *<sup>E</sup>*)]]. (A11)

In the last line, the four-fold rotational symmetry was used, i.e., *<sup>S</sup>*(*q*1, *q*1, *<sup>E</sup>*) = *<sup>S</sup>*(*q*2, *<sup>q</sup>*2, *<sup>E</sup>*) since *<sup>q</sup>***<sup>2</sup>** <sup>=</sup> *<sup>C</sup>*<sup>ˆ</sup> <sup>4</sup>[*q***1**] with *<sup>C</sup>*<sup>ˆ</sup> <sup>4</sup> as the operator performing a 90° rotation around the optical axis.

To calculate the probability for a "click" in the detector at the second EMCD position, we have to replace *<sup>q</sup>*<sup>1</sup> → *<sup>C</sup>*ˆ3 <sup>4</sup> [*q*1] = *<sup>C</sup>*ˆ2 <sup>4</sup> [*q*2] and *<sup>q</sup>*<sup>2</sup> → *<sup>C</sup>*<sup>ˆ</sup> <sup>4</sup>[*q*2] = *<sup>C</sup>*ˆ2 <sup>4</sup> [*q*1]. Owing to the assumed rotational symmetry of the MDFF, this replacement results in *S*(*C*ˆ2 <sup>4</sup> [*q*2], *<sup>C</sup>*ˆ2 <sup>4</sup> [*q*1], *<sup>E</sup>*) = *<sup>S</sup>*(*q*2, *q*1, *<sup>E</sup>*) = *<sup>S</sup>*(*q*1, *q*2, *<sup>E</sup>*)<sup>∗</sup> and hence

$$p' = \frac{1}{q\_1^4} [(A\_{11} + A\_{22})S(q\_1, q\_{1'}E) + 2\Re[A\_{12}S(q\_1, q\_{2'}E)^\*]].\tag{A12}$$

Thus, the quotient EMCD effect is

$$\eta = 2 \cdot \frac{p - p'}{p + p'} = 2 \cdot \frac{-2 \Im[A\_{12}] \Im[S(q\_1, q\_2, E)]}{(A\_{11} + A\_{22}) S(q\_1, q\_1, E) + 2 \Re[A\_{12}] \Re[S(q\_1, q\_2, E)]} \tag{A13}$$

Assuming that the scattering vectors were chosen such that *<sup>S</sup>*(*q*1, *q*2, *<sup>E</sup>*) is purely imaginary (technically, (in dipole approximation) this occurs slightly inside the Thales circle where *q*<sup>2</sup> *<sup>y</sup>* = *<sup>G</sup>*2/4 − *<sup>q</sup>*<sup>2</sup> *<sup>e</sup>*; as *qe G* in typical EMCD experiments, the real part of *<sup>S</sup>*(*q*1, *q*2, *<sup>E</sup>*), which is of the order *<sup>q</sup>*<sup>2</sup> *<sup>e</sup>*, can be neglected compared to *<sup>S</sup>*(*q*1, *q*1, *<sup>E</sup>*), which is of the order of *G*2/2), this can be simplified further to

$$=-4\cdot \frac{\mathbb{G}[A\_{12}]}{A\_{11}+A\_{22}} \cdot \frac{\mathbb{G}[S(q\_{1},q\_{2},E)]}{S(q\_{1},q\_{1},E)}.\tag{A14}$$

The coefficients can be calculated directly as

$$\begin{split} A\_{11} + A\_{22} &= \sum\_{\mathbf{x}} \left[ 1 - \frac{1}{W^2} \sin^2(\boldsymbol{\alpha} \mathcal{W} \boldsymbol{\pi} \cdot \mathbf{x}) \right] \\ \odot [A\_{12}] &= \sum\_{\mathbf{x}} \frac{1}{2} \left[ \left( 1 - \frac{3}{W^2} \sin^2(\boldsymbol{\alpha} \mathcal{W} \boldsymbol{\pi} \cdot \mathbf{x}) \right) \sin(2\bar{\boldsymbol{\pi}}(\bar{\boldsymbol{\pi}} \cdot \mathbf{x} - t)) \\ &\quad - \frac{1}{W} \sin(2\boldsymbol{\alpha} \mathcal{W} \boldsymbol{\pi} \cdot \mathbf{x}) \cos(2\bar{\boldsymbol{\pi}}(\bar{\boldsymbol{\pi}} \cdot \mathbf{x} - t)) \right]. \end{split} \tag{A15}$$

with the assumptions 2 and 4, the dot products can be evaluated and the sums can be replaced by integrals over *z*, yielding

$$\begin{split} A\_{11} + A\_{22} &= t \left( 1 - \frac{1}{2W^2} \right) + \frac{\sin(2tWa)}{4W^3a} \\ \odot [A\_{12}] &= \frac{1}{4(\mathcal{W}^2a^2 - \bar{\mathfrak{a}}^2)} \left[ -\left( 2a + \frac{3\bar{\mathfrak{a}}}{W^2} \right) \sin^2(aWt) \\ &+ \left( \frac{(3 - 2\mathcal{W}^2)a^2}{\bar{\mathfrak{a}}} + 2(a + \bar{\mathfrak{a}}) \right) \sin^2(\bar{\mathfrak{a}}t) \right] \end{split} \tag{A16}$$

Hence the full formula for the EMCD effect reads

$$\eta = \frac{4W^3 a}{(W^2 a^2 - \tilde{a}^2)} \frac{\left[ \left( 2a + \frac{3\tilde{a}}{W^2} \right) \sin^2(aWt) - \left( \frac{(3 - 2W^2)a^2}{4} + 2(a + \tilde{a}) \right) \sin^2(\tilde{a}t) \right]}{2W(2W^2 - 1)at + \sin(2tWa)} \cdot \frac{\odot[S(\boldsymbol{q}\_1, \boldsymbol{q}\_2, E)]}{S(\boldsymbol{q}\_1, \boldsymbol{q}\_1, E)}$$
 
$$= \frac{A \sin^2(\pi t) - B \sin^2(\pi' t)}{t + C \sin(2\pi t)} \cdot \frac{\odot[S(\boldsymbol{q}\_1, \boldsymbol{q}\_2, E)]}{S(\boldsymbol{q}\_1, \boldsymbol{q}\_1, E)} \tag{A17}$$

with

$$\begin{split} A &= \mathbb{C} \cdot \frac{4\pi\kappa'}{\kappa^2 - \kappa'^2} \left( 2W \frac{\kappa}{\kappa'} + 3 \right) \\ B &= \mathbb{C} \cdot \frac{4\pi\kappa'}{\kappa^2 - \kappa'^2} \left( 2W \frac{\kappa}{\kappa'} + \frac{3\kappa^2}{\kappa'^2} + 2W^2 \left( 1 - \frac{\kappa^2}{\kappa'^2} \right) \right) \\ \mathbb{C} &= \frac{1}{2\kappa(2W^2 - 1)} \\ \kappa &= \kappa W = \frac{\gamma\_1 - \gamma\_2}{2} \\ \kappa' &= \bar{\alpha} = \frac{\bar{\gamma}\_1 - \bar{\gamma}\_2}{2} . \end{split} \tag{A18}$$

#### **References**


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