Convergent-Beam EMCD: Benefits, Pitfalls, and Applications
CConvergent-Beam EMCD: Benefits, Pitfalls, andApplications
S. L¨offler a,b, ∗ , W. Hetaba a,1 a University Service Centre for Transmission Electron Microscopy, TU Wien, Vienna,Austria b Dept. for Materials Science and Engineering, McMaster University, Hamilton, Ontario,Canada
Abstract
Energy-loss magnetic chiral dichroism (EMCD) is a versatile method forstudying magnetic properties on the nanoscale. However, the classical EMCDtechnique is notorious for its low signal to noise ratio (SNR). Here, we studythe theoretical possibilities of using a convergent beam for EMCD. In partic-ular, we study the influence of detector positioning as well as convergence andcollection angles on the detectable EMCD signal. In addition, we analyzethe expected SNR and give guidelines for achieving optimal EMCD results.
Keywords:
EMCD, convergence angle, collection angle, aperture position,signal-to-noise ratio, STEM
1. Introduction
Electron magnetic chiral dichroism (EMCD), the electron microscopicequivalent to X-ray magnetic circular dichroism (XMCD), is a very versatile ∗ Corresponding author
Email address: [email protected] (S. L¨offler) Currently at Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin, Germany
Preprint submitted to Elsevier October 3, 2018 a r X i v : . [ phy s i c s . i n s - d e t ] J u l ool for investigating magnetic materials on the nanometer scale. Ever sinceits theoretical prediction [1] and subsequent realization [2], EMCD has beengaining popularity in many fields, including magnetic nano-engineering andspintronics.There are, however, two severe limitations with the classical EMCD ap-proach: spatial resolution and signal-to-noise (SNR) ratio. In the classicalEMCD approach, one sends a plane wave into a crystal that was tilted intosystematic row condition and subsequently measures the inelastically scat-tered electrons at particular points of the diffraction plane far away fromthe diffraction spots (see also fig. 1). While plane waves are well-suited foran elegant theoretical treatment, they are not so useful in practice. First ofall, from a fundamental point of view, it is impossible to actually create ormeasure true plane waves, due to the limited extent of the microscope andthe apertures, as well as the beam rotation induced by the magnetic lenses[3]. Secondly, from an experimental point of view, a (quasi) plane wave hasa very low current density at the sample. Together with the fact that thesignal has to be measured off-axis — where it can be orders of magnitudesmaller than on-axis — with (ideally infinitely) small detectors, this resultsin a notoriously low SNR. Another issue is resolution. When acquiring spec-tra in diffraction mode, the spatial resolution is usually defined by using aselected area aperture (typically of the order of 100 nm), thereby reducingthe signal even further. Alternatively, one can measure in image mode usingenergy-filtered TEM (EFTEM) [4, 5]. Due to the required energy-slit, thisagain leads to low intensity, in addition to poor energy resolution.To overcome these limitations, several approaches have been proposed,2anging from alternative measurement geometries in scanning transmissionelectron microscopy (STEM) [6–9], over vortex beams [10–12], to the useof aberration correctors to manipulate the phase of the electron beam [13].However, all these methods exhibit very low signal, are typically limitedto atomic resolution [14, 15], and may require changing components of themicroscope or operating it under non-standard conditions. Thus, these newmethods are not yet applicable for many practical applications.Here, we analyze another way to improve both the spatial resolution andthe SNR at the same time while making use of the original, straight-forwardmeasurement setup: using a convergent beam and finite collection aperturesinstead of plane waves. While this method has been used experimentallyat several occasions to boost the spatial resolution of classical EMCD (see,e.g., [16–20], and it has long been known that large collection apertures canimprove the SNR [21], it is surprising that, to our knowledge, the influence ofthe convergence angle and the interplay between convergence and collectionangle has not been studied extensively from a theoretical point of view before.In this work, we present simulations that show that convergent beamEMCD is in many ways superior to classical EMCD. In particular, we presentsimple rules of thumb for how to obtain a substantial improvement of theSNR while at the same time improving the spatial resolution to close toatomic resolution. This is expected to open new avenues for optimizingEMCD measurements in general, but particularly for the characterizationof fine grained materials, thin films, as well as the magnetic structure inthe vicinity of interfaces and defects. Thus, it is expected to lead to greatadvances in material science. 3 ( ) α β ABC D α AB,CD β a) c)d) I + I - b) Figure 1: Sketch of the convergent beam setup. (a) The incident beam with convergencesemi-angle α is centered on a crystal plane. (b) Sketch of the general positions of theareas with “positive” (i.e., higher than non-magnetic) signal I + and “negative” (i.e., lowerthan non-magnetic) signal I − . (c) Schematic elastic diffraction pattern for large α . (d)Schematic elastic diffraction pattern for small α . The diffraction spots are labeled 0, G,-G. Diffraction disks are depicted as black dashed lines, the Thales circles are depicted asgray dotted lines. α is the convergence semi-angle, β is the collection semi-angle. Thefour detector positions A–D are described in the text.
2. Methods
In this work, we present extensive simulations for the model system ofa 10 nm thick bcc Fe crystal, tilted 10 ◦ from the [0 0 1] zone axis (ZA) toproduce a systematic row case including the (2 0 0) diffraction spot. Allsimulations were performed using an acceleration voltage of 300 kV withoutspherical aberration . The beam was focused (with varying convergence semi-angle α ) onto the entry surface of the sample and positioned on an atomicplane. The complete measurement setup is depicted in fig. 1.The inelastic scattering was performed using the mixed dynamic form fac-tor (MDFF) approach [2, 22, 23]. The MDFF was modeled with an idealized The spherical aberration is not expected to play a major role here, though, as we areworking mostly in the diffraction pattern. p → d .The elastic scattering both before and after the inelastic scattering weretaken into account using the multislice algorithm [25]. A 2048 × ≈ .
09 ˚A / px was used together with a slice thickness of 1 ˚A and theelectrostatic potentials given by Kirkland [25].For extracting the relative EMCD effect, one needs to measure the signalstrengths at two different positions I + , I − and then divide the difference ofthe two by their average [1, 7, 26] S = 2 · I + − I − I + + I − = ∆ II , I = I + + I − . (1)In some cases, only the difference signal ∆ I is used instead of the relativeEMCD effect, especially in low-signal/high-noise situations. Therefore, wewill also study how to obtain the difference signal in a convergent beamgeometry and what SNR can really be achieved that way.To that end, two different schemes were used. On the one hand, a point-wise comparison of corresponding points on the upper/lower or left/righthalves of the diffraction plane was performed to obtain a visual indicationof the distribution of the EMCD effect. On the other hand, circular col-lection apertures (of varying collection semi-angle β ) were centered at fourdifferent points of the diffraction plane: (A) on the Thales circle, (B) atthe intersection of adjacent elastic diffraction disks , (C) just outside theelastic diffraction disks such that the collection aperture touched adjacent In case the elastic diffraction disks did not overlap, the apertures were centered on thesystematic row , (D) in an “optimal position”, i.e. at a convergence andcollection angle dependent point determined by a downhill simplex optimiza-tion algorithm [27] where the maximal EMCD effect can be obtained. Allfour positions are also depicted in fig. 1.
3. Results
In order to check the applicability of convergent beam EMCD, it is firstnecessary to determine where an EMCD effect can be expected in the diffrac-tion plane (if at all). To that end, fig. 2a–d show simulated energy filtereddiffraction patterns for the Fe L edge for different convergence angles. Forclassical EMCD (i.e., fig. 2a), it is well known that there are four areas ex-hibiting magnetic information, one in each quadrant of the diffraction plane.Therefore, in fig. 2e–h, we plotted the EMCD effect calculated pixel by pixelfrom the difference of the upper and the lower half-plane. Likewise, fig. 2i–l,show the EMCD effect calculated pixel by pixel from the difference of theright and the left half-plane.The first main result from those maps is that with increasing convergenceangle, the areas where the EMCD is strong is “pushed out” such that it cangenerally be found close to the rim of the elastic diffraction disks. Thiscan be explained by considering the relative contributions of the differentscattering vectors. Assuming ideal conditions, a point-like detector, andusing the dipole approximation [7, 28, 29], the EMCD difference signal is In case such a touching configuration was not possible, the aperture was positionedon the systematic row .51.00.50.0-0.5-1.0-1.5 q y [ Å - ] q y [ Å - ] l o g . I n t e n s i t y [ a r b . u . ] E M C D [ % ] a) b) c) d)e) f) g) h) q y [ Å - ] x [Å -1 ] 1.51.00.50.0-0.5-1.0-1.5 q x [Å -1 ] 1.51.00.50.0-0.5-1.0-1.5 q x [Å -1 ] 1.51.00.50.0-0.5-1.0-1.5 q [Å -1 ] E M C D [ % ] i) j) k) l) Figure 2: Energy-filtered diffraction patterns (a–d), pointwise EMCD maps based onupper/lower halfplane subtraction (e–h) and pointwise EMCD maps based on left/righthalfplane subtraction (i–l) for convergence semi-angles of 0 mrad (a, e, i), 7 mrad (b,f, j), 14 mrad (c, g, k), and 20 mrad (d, h, l). The black dotted circles indicate thethree most intense diffraction disks, whereas the white dashed circles indicate the classicalThales circles. The energy-filtered diffraction patterns are shown in contrast-optimizedlogarithmic scale. (cid:90) (cid:126)q × (cid:126)q (cid:48) q q (cid:48) d qd q (cid:48) , (2)where one has to integrate over all combinations of scattering vectors con-necting points inside the convergence disks (with radii α , see Fig. 1) withthe point-like detector. Due to the 1 / ( q q (cid:48) ) dependence, contributions fromshort scattering vectors are dominant and due to the (cid:126)q × (cid:126)q (cid:48) dependence,contributions are strongest for perpendicular scattering vectors.In the limit of small convergence angles, only one pair of scattering vectorsis possible and the situation reduces to the case of classical EMCD: theperpendicularity requirement suggests that the signal is strongest close tothe Thales circle. For large convergence angles, this explanation no longerholds as then, many combinations of scattering vectors can contribute.First, we consider detector positions inside the diffraction disks. Withoutloss of generality, we will assume a detector position inside the 0 diffractiondisk. As stated above, the dominant contributions stem from short scatteringvectors. For any sufficiently short scattering vector (cid:126)q from a point inside thediffraction disk to the detector, the scattering vector − (cid:126)q also connects a pointinside the diffraction disk to the detector. As the contributions of ( (cid:126)q, (cid:126)q (cid:48) ) and( − (cid:126)q, (cid:126)q (cid:48) ) are equal in magnitude but opposite in sign for any scattering vector (cid:126)q (cid:48) , all these contributions will average out. This implies that inside the elasticdiffraction disks, the EMCD effect will be small.Secondly, if the detector is positioned far away from large diffraction disks,neither the perpendicularity constraint nor the shortness requirement can be The exact position depends on the characteristic momentum transfer q z = q (cid:48) z , as wellas the details of the elastic scattering. . Due to the asymmetric Ewald sphere andthe HOLZ contributions, the intensity in the upper half-plane is slightly lowerthan the corresponding intensity in the lower half-plane. While this intensitydifference is not caused by the spin-polarization of the sample, it can easilybe misinterpreted as a “fake” EMCD effect. As the setup is symmetric withrespect to a right/left mirror operation, the right/left difference maps do notsuffer from this effect. Note that the figures show only a subset of the total simulated area, so the “artifacts”close to the edge are not calculation artifacts but actually coincide with HOLZ reflectionsconsistent for the chosen scattering geometry. .51.00.50.0-0.5-1.0-1.5 q y [ Å - ] q y [ Å - ] l o g . I n t e n s i t y [ a r b . u . ] E M C D [ % ] a) b) c) d)e) f) g) h) q y [ Å - ] x [Å -1 ] 1.51.00.50.0-0.5-1.0-1.5 q x [Å -1 ] 1.51.00.50.0-0.5-1.0-1.5 q x [Å -1 ] 1.51.00.50.0-0.5-1.0-1.5 q [Å -1 ] E M C D [ % ] i) j) k) l) Figure 3: Same as Fig. 2 but for a hypothetical non-magnetic case.
10o confirm this interpretation, fig. 3 shows the same maps, but calcu-lated for a hypothetical “non-magnetic” iron where the spin-polarization wasforced to zero. Again, the upper/lower difference maps fig. 3e–h show a “fake”EMCD effect, whereas the right/left difference maps fig. 3i–l correctly showno magnetic signal.Therefore, in the remainder of this work, we use the right/left differencemethod to extract EMCD signals.
In this section, we will analyze both the achievable signal strengths S and ∆ I as well as the SNR S/δS and ∆
I/δ ∆ I associated with them as afunction of convergence and collection angles for the four detector positionsA–D defined above. This is conceptionally similar to previous studies thatincluded estimations for the SNR for plane wave illumination [21] and foraberrated probes [32]. To calculate the SNR, we will include the pre-edgebackground intensity B which does not contribute to the signal but doesincrease the noise. We will also use the jump ratio defined by r = I + BB (3)to simplify the equations.Note that while we will give general formulas that should be applicableto all cases at the beginning of each section, further derivations will be basedon the assumption of pure Poissonian shot noise to derive simplified formulasand actual numbers. This neglects other noise sources such as read-out noiseand electronic noise (which will be low compared to the shot noise as derivedbelow), or uncertainties introduced by the background subtraction process11 c o ll . s e m i - ang l e [ m r ad ] a) b) c) E M CD e ff e c t [ % ] d) c o ll . s e m i - ang l e [ m r ad ] . . . e) . . . f) . . . . . g) . . . . r e l . S NR h) Thales circle (pos. A) Intersection (pos. B) Adjacent (pos. C) Optimum (pos. D)
Figure 4: EMCD effect S (a–d) and SNR S/δS (e–h) for the four sets of detector positionsA–D as a function of convergence and collection semi-angles. The SNR is given for a jumpratio of r = 2 in fractions of the maximum SNR. [33]. Nevertheless, the numbers calculated below will give a good rule ofthumb for the intensity necessary to obtain a statistically significant EMCDsignal. Fig. 4a–d show the dependence of the EMCD effect on the convergenceand collection angles for the four different sets of detector positions A–Ddefined above. As was already noted in sec. 3.1, placing the detectors onthe Thales circle (position A) only gives a large EMCD signal for smallconvergence and collection angles. For angles larger than the Bragg angle of θ B ≈ . α < θ B and then gives a relatively12mall EMCD signal for small collection angles. This can be understood fromthe fact that for large collection angles, a significant portion of the collectedintensity stems from the areas inside the diffraction disk which does not showa significant EMCD asymmetry.Putting the detectors adjacent to the elastic diffraction disks (positionC) gives medium EMCD effects, but over a large range of convergence andcollection angles (apart from the area of α + β < θ B , where the notion of“adjacent” does not make sense). In fact, this case is mostly complementaryto the Thales circle case.The final case — putting the detectors at optimal positions D — natu-rally gives the largest EMCD effects, basically combining the “best of bothworlds”: positions A and C. While this case gives the highest EMCD signalby design, it is likely difficult to implement in many applications as it requiresextensive simulations with conditions that vary from situation to situation(e.g., with crystal thickness and orientation).While fig. 4a–d shows that convergent beam EMCD works to produce adichroic signal, it also indicates that the achievable EMCD effect is decreasingsomewhat with increasing convergence and collection angle. What was nottaken into account so far, however, is the influence of the SNR. If shot noisedominates over other noise sources (such as read out noise), I ± follows aPoisson distribution. By the central limit theorem, this can be approximatedwell by a Gaussian distribution with a standard deviation of δI ± = √ I ± + B for sufficiently large signal, where B is the background intensity. Then thevariance ( δS ) of the signal S can be calculated by error propagation to read( δS ) = 16 · ( δI + ) I − + I ( δI − ) ( I + + I − ) . (4)13ith a SNR of SδS = I − I − (cid:112) ( δI + ) I − + I ( δI − ) . (5)The former can be simplified to( δS ) = 16 I + I − ( I + + I − ) + 16 B · I + I − ( I + + I − ) (6)By virtue of I + − I − = SI I + + I − = 2 I I + I − = I (4 − S )2( I + I − ) = I (4 + S ) (7)this can also be written as( δS ) = I (4 − S ) + B · (4 + S )2 I . (8)Thus, the SNR becomes SδS = √ SI (cid:112) I (4 − S ) + B · (4 + S ) . (9)Not surprisingly, the SNR increases with total intensity and EMCD effectand decreases with pre-edge background.Fig. 4e–h depicts the SNR associated with the four sets of detector po-sitions. It clearly confirms the experimental evidence that for very smallconvergence and/or collection angles, the SNR drops dramatically due tothe greatly reduced recorded intensity for a given exposure time. When em-ploying convergent beam EMCD, however, the SNR can easily be increaseddramatically. Note that the SNR naturally takes into account the decreasingEMCD effect with larger convergence/collection angles, but the increase in14ecorded intensity and correspondingly decreasing noise more than compen-sate for it, giving more reliable and statistically significant results under oth-erwise identical recording conditions. As a good rule of thumb, one can usea convergence semi-angle slightly larger than θ B — which results in slightlyintersecting elastic diffraction disks —, a collection angle of about θ B , andposition the collection aperture adjacent to the diffraction disks (position C).For the present case studied here, this results in approximately 80 % of themaximal achievable SNR and an EMCD effect of 15 %.To answer the question of how many counts need to be recorded to achievea certain statistical significance, one naturally needs to consider the ratiobetween the elemental edge and the pre-edge background (which increasesthe noise level but not the signal). Assuming a jump ratio r of r = I + BB ⇔ B = I r − ⇔ I + B = I · rr − , (10)the SNR can be rewritten as SδS = √ I S (cid:113) − S + S r − = S (cid:112) r − I (cid:112) r (4 − S ) + 2 S . (11)If B = I , i.e. for a jump ratio of r = 2, the SNR takes the particularlysimple form of SδS = S √ I . (12)To reach a SNR of at least k , I must be chosen such that I ≥ k S (cid:18) − S + 4 + S r − (cid:19) (13)or, equivalently, that the total intensity fulfills I + B ≥ k r S ( r − (cid:18) − S + 4 + S r − (cid:19) (14)15 c o ll . s e m i - ang l e [ m r ad ] . . . a) . . . . b) . . c) . . . D i ff e r en c e [ a r b . u .] d) c o ll . s e m i - ang l e [ m r ad ] . . . e) . . . f) . . . . . g) . . . . r e l . S NR h) Thales circle (pos. A) Intersection (pos. B) Adjacent (pos. C) Optimum (pos. D)
Figure 5: Difference signal ∆ I (a–d) and SNR ∆ I/δ ∆ I (e–h) for the four sets of detectorpositions A–D as a function of convergence and collection semi-angles. The SNR is givenfor a jump ratio of r = 2 in fractions of the maximum SNR. For the special case of k = 3 and r = 2, this gives I + B ≥ S (15)i.e., for an expected EMCD effect of 10 %, an intensity of at least 7200 countsneeds to be achieved. In low signal/large noise situations, it is often assumed that the divisionby I renders the relative EMCD effect statistically unstable. In such cases,one might consider looking only at the difference signal ∆ I which, thoughnot being quantifiable in absolute numbers, can still give a qualitative indi-cation of whether a magnetic signal is non-zero and how it changes acrossthe sample. Therefore, here we will also look at the signal strength and theSNR properties of the difference signal ∆ I .16ig. 5a–d shows the difference signal dependence on the convergence andcollection semi-angles for the four sets of detector positions. Three featuresimmediately catch the eye: first of all, unlike the relative EMCD effect infig. 4, the signal strength tends to increase with increasing convergence andcollection angles. Secondly, position B (on the intersection of the diffractiondisks) yields higher signal strengths, in stark contrast to the case described insec. 3.2.1, where position B exhibited by far the lowest intensity. Thirdly, theregion with appreciable signal for position A (on the Thales circle) movedfrom low convergence/collection angles to medium ones where the Thalescircle position is close to the intersection of the diffraction disks. All theseeffects can easily be understood from the method of signal extraction. Whilethe calculation of the EMCD effect S includes the division by the averagesignal, the calculation of the difference signal ∆ I does not. Consequently,adding the intensity from points that show little or no asymmetry does notalter the difference signal, while it decreases the EMCD effect. Of course, forboth methods, adding points that contribute little to nothing to the signalinherently decreases the SNR, though, and as such should be avoided.For calculating the SNR, we follow the same steps as in sec. 3.2.1. Underthe same assumptions as above, the variance of the difference signal ∆ I isgiven by ( δ ∆ I ) = ( δI + ) + ( δI − ) = I + + I − + 2 B = 2( I + B ) (16)and the SNR reads∆ Iδ ∆ I = I + − I − (cid:112) ( δI + ) + ( δI − ) = SI (cid:112) I + B ) = S (cid:112) I ( r − √ r (17)From the third expression, it is obvious that the SNR increases with the17MCD effect and the total intensity while it decreases with increasing back-ground intensity as expected. If B = I , i.e. for a jump ratio of r = 2, theSNR takes the same form as S/δS , i.e.,∆ Iδ ∆ I = ∆ I √ I = S √ I . (18)To reach an SNR of k , one needs to achieve a total intensity of I + B ≥ r k S ( r − (19)counts.
4. Discussion
As mentioned above, it is often assumed that the use of the differencesignal is beneficial particularly when the signal is weak. This notion pre-sumably comes from the fact that an increase of the exposure time or theincident beam current actually increases the difference signal but does notchange the relative EMCD effect. While it can be argued that this com-plicates post-processing by requiring additional normalizing by the incidentdose — which is inherently included in a way in the relative EMCD signal—, such a discussion misses the most important point: the SNR. No mat-ter how large or small the EMCD signal itself is, the real question is if itis detectable under given conditions. If a condition A yields a SNR that isgood enough to detect a small signal, it is still preferable over a condition Bwhich gives a larger signal but also a bad SNR resulting in an EMCD signalindistinguishable from the noise. Therefore, the crucial aspect is really SNR,and not total signal strength. 18 comparison of eq. 9 and eq. 17 gives √ SI (cid:112) I (4 − S ) + B · (4 + S ) < SI (cid:112) I + B ) (20)4( I + B ) < I (4 − S ) + B · (4 + S ) (21)0 < S · ( B − I ) (22)Therefore, only for B > I O ⇔ r <
2, i.e. for thick specimens, using the differ-ence signal is actually better than using the relative EMCD effect. However,thick specimens typically yield a low overall EMCD effect owing to oscil-lations and sign reversal caused by the elastic scattering and pendell¨osung[29, 34]. Therefore, the relative EMCD signal should generally preferred overthe difference signal.
In this section, we investigate the dependence of the convergent beamEMCD signal on the beam position. For small convergence and collectionangles, one can expect that the EMCD signal is largely independent of thebeam position due to the large illuminated area and, consequently, the lowspatial resolution. For convergence and collection semi-angles significantlylarger than the Bragg angle, however, one can expect a position-dependence.To study this effect, we also performed calculations with the beam displacedby half a lattice plane distance so that it was positioned directly in-betweenadjacent lattice planes.Fig. 6 compares the energy-filtered diffraction patterns and point-wiseEMCD effects for on-plane and off-plane beam positions for a large conver-gence angle. While there are obvious differences, it is remarkable that an19 .51.00.50.0-0.5-1.0-1.5 q y [ Å - ] q y [ Å - ] l o g . I n t e n s i t y [ a r b . u . ] E M C D [ % ] a) b)d) x [Å -1 ] 1.51.00.50.0-0.5-1.0-1.5 q [Å -1 ] c) Figure 6: Energy-filtered diffraction patterns (a, b) and pointwise EMCD maps basedon left/right halfplane subtraction (c, d) for on-plane (a, c) and off-plane (b, d) beampositions. The convergence semi-angle is 20 mrad. The black dotted circles indicate thethree most intense diffraction disks, whereas the white dashed circles indicate the classicalThales circles. The energy-filtered diffraction patterns are shown in contrast-optimizedlogarithmic scale. Of course, the question of how muchwhich atom contributes to the EMCD effect depends crucially on how theincident and outgoing electron beams channel through the crystal [37, 38].However, a full quantitative description of the resulting thickness dependenceis beyond the scope of this work.Fig. 7 shows the convergence and collection semi-angle dependence of theEMCD signal for a probe beam positioned between atomic planes, togetherwith the corresponding SNR. It is not surprising that, qualitatively, it lookssimilar to the on-plane case depicted in fig. 4. In particular for small conver-gence and collection semi-angles, the maps are identical, as is to be expected.Perhaps the most noticeable difference is the different large-angle behaviorof the SNR for positions A and B. This can be understood from the fact thatboth positions pick up intensity from inside the area of the elastic diffractiondisks. This intensity is strongly influenced by the local potential the beamtraverses [39], and, hence, strongly dependent on the beam position, as is This can also be understood from the fact that the initial p-states contributing to theL-edge have vanishing probability density at the position of the nucleus. E M CD [ % ] c o ll . s e m i - ang l e [ m r ad ] c o ll . s e m i - ang l e [ m r ad ] . . . . . . . . . r e l . S NR . . . . . . Thales circle (pos. A) Intersection (pos. B) Adjacent (pos. C) Optimum (pos. D)
Figure 7: EMCD effect S (a–d) and SNR S/δS (e–h) for the four sets of detector positionsA–D as a function of convergence and collection semi-angles for a beam position in-betweenatomic planes. The SNR is given for a jump ratio of r = 2 in fractions of the maximumSNR. shown in figs. 6a, b. If the collection aperture is placed outside the area ofthe elastic diffraction disks, however, as is the case for positions C and D,the off-plane signal becomes remarkably similar to the on-plane signal, withthe above-mentioned enhancement for large convergence angles.
5. Concluding Remarks
In this work, we have explored the possibility of convergent-beam EMCD.We found that this method should not only give a similar EMCD signal asthe classical, parallel beam EMCD method, but in fact is expected to havesuperior SNR characteristics. As a rule of thumb, choosing a convergencesemi-angle slightly larger than the Bragg angle, a collection angle close tothe Bragg angle, positioning the collection aperture just outside the elasticdiffraction disks, and using an exposure time giving more than 7200 counts22t the edge under investigation should give close to optimal results. Of course, further work is necessary, e.g., to adapt the EMCD sumrules [40] to the convergent beam case and to characterize the thickness-dependence of convergent-beam EMCD. However, especially the improve-ments in SNR, as well as in spatial resolution, open exciting new possibilitiesfor EMCD that may soon lead to an even broader applicability of this excitingtechnique for material science.
Funding
This work was supported by the Austrian Science Fund (FWF) [grantnumber J3732-N27].
Acknowledgements
The authors gratefully acknowledge access to the USTEM computationalfacilities, as well as fruitful discussions with Peter Schattschneider.
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