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Article
Exploiting the Acceleration Voltage Dependence of EMCD
Stefan Löffler 1,* , Michael Stöger-Pollach 1, Andreas Steiger-Thirsfeld 1, Walid Hetaba 2
and Peter Schattschneider 1,3
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 Editor: Lucia Nasi
Received: 18 December 2020
Accepted: 26 February 2021
Published: 9 March 2021
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1University Service Centre for Transmission Electron Microscopy, TU Wien,
Wiedner Hauptstraße 8-10/E057-02, 1040 Wien, Austria; michael.stoeger-pollach@tuwien.ac.at (M.S.-P.);
andreas.steiger-thirsfeld@tuwien.ac.at (A.S.-T.); peter.schattschneider@tuwien.ac.at (P.S.)
2Max Planck Institute for Chemical Energy Conversion, Stiftstraße 34–36,
45470 Mülheim an der Ruhr, Germany; hetaba@fhi-berlin.mpg.de
3Institute of Solid State Physics, TU Wien, Wiedner Hauptstraße 8-10/E138-03, 1040 Wien, Austria
*Correspondence: stefan.loeffler@tuwien.ac.at
Abstract:
Energy-loss magnetic chiral dichroism (EMCD) is a versatile method for measuring mag-
netism 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
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∗
and
s→σ∗
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
ex±iey
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
Materials 2021,14, 1314. https://doi.org/10.3390/ma14051314 https://www.mdpi.com/journal/materials
Materials 2021,14, 1314 2 of 14
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 spa-
tial resolution of the method allows the study of nanoparticles [
14
], 3d–4f coupling in su-
perlattices [
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 advan-
tages 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:
1.
We limit the derivation to an incident three-beam and outgoing two-beam case in the
zero-order Laue zone of a sample that is single-crystalline in the probed region with a
centro-symmetric crystal structure;
2.
We assume that the sample is a slab of thickness
t
with an entrance and an exit plane
essentially perpendicular to the beam propagation axis;
3.
We assume that the inelastic scattering process is at least four-fold rotationally sym-
metric around the optical axis and that the characteristic momentum transfer
qe
is
Materials 2021,14, 1314 3 of 14
much smaller than the chosen reciprocal lattice distance
|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=
2
mπ
,
m∈Z
for all atom positions
xand 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
−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 reci-
procity 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 0to G.
The probability of measuring a particular state
|ψouti
(a “click” in the detector corre-
sponding to a plane wave at the exit plane of the sample) given a certain incident state
|ψini(a plane wave incident on the entry plane of the sample) is given by Fermi’s Golden
rule [45–49]:
p=∑
I,F
pI(1−pF)hψout| hF|ˆ
V|Ii|ψinihψin|hI|ˆ
V†|Fi|ψoutiδ(EF−EI−E), (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
ˆ
Vis the transition operator. In momentum representation, ˆ
Vfor a single atom is given by
h˜
k|ˆ
V|ki=eiq·ˆ
R
q2with q=k−˜
k. (2)
With the mixed dynamic form factor (MDFF) [45,49–51],
S(q,q0,E) = ∑
I,F
pI(1−pF)h˜
k|hF|eiq·ˆ
R|Ii|ki hk0|hI|e−iq0·ˆ
R|Fi|˜
k0iδ(EF−EI−E), (3)
the probability for a “click” in the detector can be written as [8,45,48–50]
p=ZZZZ ∑
x
ei(q−q0)·xψout(˜
k)∗ψout(˜
k0)S(q,q0,E)
q2q02ψin(k)ψin(k0)∗dkdk0d˜
kd˜
k0, (4)
where the
∑xei(q−q0)·x
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 S(q,q0,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
|ψini=∑
j∈{1,2,3}
∑
g∈{−G,0,G}
C∗
j,0Cj,g|χ+γjn+gi(5)
|ψouti=∑
l∈{1,2}
∑
h∈{0,G}
˜
C∗
l,0e−i˜
γlt˜
Cl,h|˜χ+˜
γl˜n+hi, (6)
Materials 2021,14, 1314 4 of 14
where
j
,
l
are the Bloch wave indices,
g
,
h
run over the diffraction spots, the
Cj,g
are the
Bloch wave coefficients, the
γj
are the so-called anpassung,
n
is the surface normal vector,
t
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
η=Asin2(κt)−Bsin2(κ0t)
t+Csin(2κt)·=[S(q1,q2,E)]
S(q1,q1,E), (7)
where
t
is the sample thickness and the coefficients
A
,
B
,
C
,
κ
,
κ0
are defined in
Equation (A18)
(with Equations (A1) and (A3)).
Figure 1shows 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 differ-
ences (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
κ=γ1−γ2
2=q(|G|2−U2G)2+8U2
G
4χ·nand κ0=˜
γ1−˜
γ2
2=UG
2 ˜χ·˜n(8)
which are closely related to the extinction distances for the incident and outgoing beams.
As the wavevectors
χ
,
˜χ
scale with the square root of the acceleration voltage
√V
, the
frequencies of the oscillations of the EMCD effect scale with 1
/√V
. This is corroborated by
Figure 2.
Materials 2021,14, 1314 5 of 14
Figure 2.
EMCD effect
η
for various acceleration voltages
V
and thicknesses
t
for bcc
Fe
as simulated
with “bw”. The dashed lines show (arbitrary) curves with t∝√Vas guides for the eye.
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
√V
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 4and shows
excellent agreement.
Materials 2021,14, 1314 6 of 14
Figure 3.
EMCD spectra for different acceleration voltages (as indicated) after background subtrac-
tion and post-edge normalization using the
Fe
L-edge in Magnetite tilted to a
(
4 0 0
)
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 3according 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
/q2q02
term in Equation (4) (any additional beams would give
Materials 2021,14, 1314 7 of 14
rise to much longer
q
vectors). The crystal structure was assumed to be centro-symmetric,
resulting in
UG=U−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 rota-
tions around the optical axis by integer multiples of
90°
. Strong anisotropy would lead to a
distinct directional dependence of the MDFF [
48
,
59
,
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(q1
,
q2
,
E)] <[S(q1
,
q2
,
E)]
in dipole approximation [
11
,
61
]. This is fulfilled reasonably well for typical EMCD ex-
periments (for example, for
Fe (
2 0 0
)
,
|G| ≈ 7 nm−1
; for the
Fe
L-edge,
qe≈0.8 nm−1
at
200 keV and qe≈1.5 nm−1at 40 keV).
Assumption four requires the investigated atoms to be distributed homogeneously
and fulfill the condition
G·x=
2
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=
2
mπ∀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= (
1 0 0
)
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=
2
mπ
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 Gvector not fulfilling the condition is unfavorable anyway.
As can be seen from Figure 1, Equation (7) reproduces sophisticated numerical simu-
lations 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, Hills-
boro, 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, distinctly-
shaped feature nearby was found and the sample was oriented in systematic row condition
including the
(
4 0 0
)
diffraction spot (see Figure 5). At each high tension, the instrument
Materials 2021,14, 1314 8 of 14
was carefully aligned, the sample position was readjusted, the EMCD experiment was per-
formed, and a thickness measurement was taken. Both the convergence and the collection
semi-angle were approximately 3mrad [58].
200 nm
(4 0 0)
(1 1 1)
2 nm−1
(0 0 0)
(4 0 0)
(4 0 0)
I+
I−
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+
and
I−
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]
η=I+−I−
I++I−
2
. (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
√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.;
Materials 2021,14, 1314 9 of 14
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:
DDSCS Double-differential scattering cross-Section
EMCD Energy-loss magnetic chiral dichroism
EELS Electron energy-loss spectrometry
MDFF Mixed dynamic form factor
TEM Transmission electron microscopy
XAS X-ray absorption spectroscopy
XMCD X-ray magnetic circular dichroism
Appendix A. Derivation of the EMCD Effect
In the following, we will extensively use the abbreviations
α=UG
2χ·n˜
α=UG
2 ˜χ·˜n(A1)
V=U2G−|G|2
2UG
(A2)
W=q(|G|2−U2G)2+8U2
G
2UG
=pV2+2, (A3)
where the
Ug
are the Fourier coefficients of the crystal potential
V(r) = h2
2me ∑gUg
e
2πig·r
with Planck’s constant
h
, electron mass
m
and elementary charge
e
. We note in passing that
in the present case, UG=U∗
G=U−G.
With these abbreviations and the assumptions mentioned above, the Bloch wave
parameters can be calculated analytically and take the form
γ1=α(V+W)γ2=α(V−W)γ3=−α·|G|2+U2G
UG
C1,−G=1
p|V−W|2+2C2,−G=1
p|V+W|2+2C3,−G=−1
√2
C1,0=−V−W
p|V−W|2+2C2,0=−V+W
p|V+W|2+2C3,0=0
C1,G=1
p|V−W|2+2C2,G=1
p|V+W|2+2C3,G=1
√2
(A4)
for |ψiniand
˜
γ1=˜
α˜
γ2=−˜
α
˜
C1,0=1
√2˜
C2,0=1
√2
˜
C1,G=1
√2˜
C2,G=−1
√2
(A5)
Materials 2021,14, 1314 10 of 14
for |ψouti.
Inserting Equations (5) and (6) into Equation (4), evaluating the integrals, collect-
ing all terms with the same Bloch wave index, and neglecting the weak dependence of
S(q,q0,E)/(q2q02)on j,j0,l,l0[8,41,55] yields
p=∑
x
∑
g,g0,h,h0
DgD∗
g0˜
D∗
h˜
Dh0ei(g−g0−h+h0)·xS(q,q0,E)
q2q02(A6)
with
Dg=∑
j
C∗
j,0Cj,geiγjn·x˜
Dg=∑
l
˜
C∗
l,0e−i˜
γlt˜
Cl,hei˜
γj˜n·x(A7)
and
q=∆χ+g−h q0=∆χ+g0−h0∆χ=χ−˜χ. (A8)
Direct summation results in
D−G=DG=i
WeiαVn·xsin(αWn·x)
D0=eiαVn·xcos(αWn·x)−iV
Wsin(αWn·x)
˜
D0=cos(˜
α(˜n·x−t))
˜
DG=i sin(˜
α(˜n·x−t)).
(A9)
Performing the complete sums over
g
,
g0
,
h
,
h0
in Equation (A6) produces very many
terms, some of which are very small. This can be understood from the fact that
∆χ·G=
±G/
2 in the chosen setup. Therefore,
∆χ
and
∆χ−G
have the same magnitude, whereas
∆χ+G
and
∆χ−
2
G
are significantly larger. Owing to the 1
/q2q02
term, large
q
are
strongly suppressed. Hence, only the combinations
g−h=0
and
g−h=−G
are retained
(the same applies to the primed versions as well). Hence, we end up with two distinct
q
vectors, namely
q1=∆χand q2=∆χ−G. (A10)
Note that, due to the symmetry of the setup q1=|q1|=|q2|=q2.
Using S(q,q0,E) = S(q0,q,E)∗[45], Equation (A6) now takes the form
p=1
q4
1
∑
x
[
D0˜
D∗
0+DG˜
D∗
G
2S(q1,q1,E) +
D−G˜
D∗
0+D0˜
D∗
G
2S(q2,q2,E) +
2<hD0˜
D∗
0+DG˜
D∗
GD∗
−G˜
D0+D∗
0˜
DGeiG·xS(q1,q2,E)ii
=1
q4
1
[A11S(q1,q1,E) + A22S(q2,q2,E) + 2<[A12S(q1,q2,E)]]
=1
q4
1
[(A11 +A22)S(q1,q1,E) + 2<[A12 S(q1,q2,E)]].
(A11)
In the last line, the four-fold rotational symmetry was used, i.e.,
S(q1
,
q1
,
E) =
S(q2
,
q2
,
E)
since
q2=ˆ
C4[q1]
with
ˆ
C4
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
q17→ ˆ
C3
4[q1] = ˆ
C2
4[q2]
and
q27→ ˆ
C4[q2] = ˆ
C2
4[q1]
. Owing to the as-
sumed rotational symmetry of the MDFF, this replacement results in
S(ˆ
C2
4[q2]
,
ˆ
C2
4[q1]
,
E) =
S(q2,q1,E) = S(q1,q2,E)∗and hence
p0=1
q4
1
[(A11 +A22)S(q1,q1,E) + 2<[A12 S(q1,q2,E)∗]]. (A12)
Materials 2021,14, 1314 11 of 14
Thus, the quotient EMCD effect is
η=2·p−p0
p+p0=2·−2=[A12]=[S(q1,q2,E)]
(A11 +A22)S(q1,q1,E) + 2<[A12 ]<[S(q1,q2,E)] (A13)
Assuming that the scattering vectors were chosen such that
S(q1
,
q2
,
E)
is purely
imaginary (technically, (in dipole approximation) this occurs slightly inside the Thales
circle where
q2
y=G2/
4
−q2
e
; as
qeG
in typical EMCD experiments, the real part of
S(q1
,
q2
,
E)
, which is of the order
q2
e
, can be neglected compared to
S(q1
,
q1
,
E)
, which is of
the order of G2/2), this can be simplified further to
=−4·=[A12]
A11 +A22 ·=[S(q1,q2,E)]
S(q1,q1,E). (A14)
The coefficients can be calculated directly as
A11 +A22 =∑
x1−1
W2sin2(αWn·x)
=[A12] = ∑
x
1
21−3
W2sin2(αWn·x)sin(2˜
α(˜n·x−t))
−1
Wsin(2αWn·x)cos(2˜
α(˜n·x−t)).
(A15)
with the assumptions 2 and 4, the dot products can be evaluated and the sums can be
replaced by integrals over z, yielding
A11 +A22 =t1−1
2W2+sin(2tWα)
4W3α
=[A12] = 1
4(W2α2−˜
α2)−2α+3˜
α
W2sin2(αWt)
+(3−2W2)α2
˜
α+2(α+˜
α)sin2(˜
αt)
(A16)
Hence the full formula for the EMCD effect reads
η=4W3α
(W2α2−˜
α2)h2α+3˜
α
W2sin2(αWt)−(3−2W2)α2
˜
α+2(α+˜
α)sin2(˜
αt)i
2W(2W2−1)αt+sin(2tWα)·=[S(q1,q2,E)]
S(q1,q1,E)
=Asin2(κt)−Bsin2(κ0t)
t+Csin(2κt)·=[S(q1,q2,E)]
S(q1,q1,E)(A17)
with
A=C·4κκ0
κ2−κ022Wκ
κ0+3
B=C·4κκ0
κ2−κ022Wκ
κ0+3κ2
κ02+2W21−κ2
κ02
C=1
2κ(2W2−1)
κ=αW=γ1−γ2
2
κ0=˜
α=˜
γ1−˜
γ2
2.
(A18)
Materials 2021,14, 1314 12 of 14
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