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Ultramicroscopy 110 (2010) 952–957

Contents lists available at ScienceDirect

Ultramicroscopy

0304-39

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/ultramic

Measurement of effective source distribution and its importance forquantitative interpretation of STEM images

C. Dwyer a,�, R. Erni b, J. Etheridge a

a Monash Centre for Electron Microscopy, and Department of Materials Engineering, Monash University, Victoria 3800, Australiab Electron Microscopy Center, Swiss Federal Laboratories for Materials Testing and Research, CH-8600 Dubendorf, Switzerland

a r t i c l e i n f o

Keywords:

STEM

Coherence

Convergent beam electron diffraction

91/$ - see front matter & 2010 Elsevier B.V. A

016/j.ultramic.2010.01.007

esponding author.

ail address: christian.dwyer@mcem.monash.e

a b s t r a c t

We review the manner in which lens aberrations, partial spatial coherence, and partial temporal

coherence affect the formation of a sub-A electron probe in an aberration-corrected transmission

electron microscope. Simulations are used to examine the effect of each of these factors on a STEM

image. It is found that the effects of partial spatial coherence (resulting from finite effective source size)

are dominant, while the effects of residual lens aberrations and partial temporal coherence produce

only subtle changes from an ideal image. We also review the way in which partial spatial and temporal

coherence effects are manifest in a Ronchigram. Finally, we provide a demonstration of the Ronchigram

method for measuring the effective source distribution in a probe aberration-corrected 300 kV field-

emission gun transmission electron microscope.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

In the latest scanning transmission electron microscopes(STEMs) equipped with small electron sources (e.g. a Schottkyor cold field emission gun, FEG), aberration-correctors, andimproved electronic and mechanical stability, the incidentelectron wave field can be brought to a focus that is smaller than1 A in diameter. The main goal of using such electron probes is togenerate data from electron scattering within a small volume ofspecimen, and hence determine the local atomic and electronicstructure specific to that volume. In order to interpret such datasets quantitatively it is necessary to know the characteristics ofthe incident electron wave field. In particular, as emphasizedrecently [1–3], and as will be further demonstrated in the presentwork, it is important to know the degree of spatial coherence asthis has a significant effect on the achievable probe size in thelatest instruments, and it determines the extent to whichinterference effects can occur and hence be detected.

Previously, we have described a method for the measurementof the spatial coherence [1], and hence the effective sourcedistribution, in systems for which the probe size is not aberration-limited. When this measurement is combined with knowledge ofthe degree of temporal coherence and lens aberrations, theelectron wave field at the specimen plane can be describedcompletely.

ll rights reserved.

du.au (C. Dwyer).

In the present paper, we demonstrate the importance of theshape of the source distribution for the quantitative interpretationof atomic resolution STEM images. We then apply the methoddescribed previously to demonstrate the experimental measure-ment of source shape in an aberration-corrected FEG-TEM.

2. Components of a probe-forming lens system

It is worthwhile reviewing the key components that generatethe electron probe incident on the specimen. The incidentelectron wave field at the specimen plane is generated by imagingan effective electron source onto the specimen plane via animaging lens(es), typically the pre-field of the objective lens. Theprimary factors determining the incident electron wave field atthe specimen plane are the spatial distribution of the effectiveelectron source, the energy spread of the effective electron source,the size and shape of the illumination aperture, and theaberrations in the lens(es) that images the source. Theseparameters have distinct effects on the incident wave field whichare summarized below and illustrated schematically in Fig. 1.

In an idealized case (Fig. 1 a), the effective source is a pointsource, whose intensity distribution is described by a Dirac deltafunction S0ðxÞ ¼ dðxÞ (bold symbols denote two-dimensionalvectors perpendicular to the optic axis z). In the ideal case thereare no lens aberrations and no aperture is used. The source andspecimen planes are in-focus and the relative magnification is M

(in the schematic in Fig. 1 a the magnification is simply M=�z2/z1,where jMjo1 indicates demagnification and the minus sign

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z1

z2

Effective Source

Objective Prefield

Specimen

Fig. 1. Schematic representation of the effects of (a) an ideal probe-forming system, (b) an aperture, (c) a finite source size, and (d) a finite source size and an aperture.

C. Dwyer et al. / Ultramicroscopy 110 (2010) 952–957 953

indicates image inversion). The point source emits a sphericalwave with perfect spatial (transverse) coherence. The aberration-free lens then forms a perfect image SðxÞ ¼M�2S0ðM

�1xÞ of thissource at the specimen plane.

Now consider lens aberrations. Within the isoplanatic approx-imation, aberrations in the lens system that image the effectivesource onto the specimen plane introduce a phase deviationwhich depends on the incident wave vector, and not on the sourcepoint. These phase deviations act to broaden the image of theeffective source. When the dominant influence on the size of aprobe is the lens aberrations, it is said to be aberration-limited.Typically, an illumination aperture is used to limit the effect ofaberrations by restricting the set of incident wave vectors whichcontribute to the image (Fig. 1 b). By preventing the transmission,and hence interference, of some components of a collapsingspherical wave, the aperture acts to broaden the image of thepoint source into an aberrated Airy disc (see Fig. 1 b). When thedominant influence on the size of a probe is the aperture, it is saidto be diffraction-limited.

Now consider a system without lens aberrations but with aneffective source S0ðxÞ that has a finite spatial distribution (Fig. 1 c).In practice, the size and shape of the effective source isdetermined by the collective effect of several factors: the physicalsize of the emitter, the demagnification of the emitter, themechanical and electronic instabilities in the system, includingthe high tension, the emitter, the gun and accelerator system andthe lenses. We can consider that each point of this spatiallyincoherent effective source gives rise to a spatially coherentspherical wave, and that waves originating at different sourcepoints are mutually incoherent. After propagating away from thesource over a distance which is large compared to both the spatialextent of the source and the electron wavelength, the total wavefield acquires some degree of spatial coherence as governed by thevan Cittert–Zernike theorem [4–6]. However, in the case of aSTEM, it is simpler to consider that each spherical wave emittedby the source is independently imaged onto the specimen planeby the lens system. Waves generated at off-axial source points willbe imaged at points related by the magnification M. The image ofthe effective source on the specimen plane is then an incoherentsum over the demagnified images of the individual source points,where the weighting of each image is SðxÞ ¼M�2S0ðM

�1xÞ. Whenthe dominant influence on the size of a probe is the size of theeffective source, it is said to be source-limited.

When the effects of lens aberrations and a finite sourcedistribution are combined (Fig. 1 d), the wave field in thespecimen plane is an incoherent sum of truncated sphericalwaves, where the weighting of each image is SðxÞ. Such a wavefield is described by the density matrix (or mutual coherence

function [6])

rðx;x0Þ ¼Z

d2x00

cðx�x00

ÞSðx00

Þc�ðx0�x00

Þ; ð1Þ

where c is a (normalized) diffraction-limited wave. The intensityof such a wave field is given by

rðx;xÞ ¼Z

d2x00

Sðx00

Þjcðx�x00

Þj2; ð2Þ

which is simply the convolution of the diffraction-limitedintensity and the effective source distribution in the specimenplane.

Over the period of an experiment, an effective source pointmay emit waves with different wavelengths, within the range ofthe energy stability of the system, typically better than 0.8 eV for aSchottky FEG. This is equivalent in its effect to having a rangeof focal settings of the imaging lens [7] which generates acorresponding range of slightly out-of-focus images of the sourcepoint. As will be described later, the effect of a finite focal spreadcancels when the incident wave direction satisfies a Bragg anglewithin the crystal [8,9]. This enables experimental configurationsto be selected, which measure partial spatial coherence, indepen-dently of the effects of partial temporal coherence [1]. In practice,the effective wavelength spread (and hence the temporalcoherence) is determined primarily by the high tension stability,the temperature of the emitter, the extraction voltage, and thestability of the lens(es).

3. Spatial coherence in STEM

While the signal used to generate a STEM image is produced bya detector in the diffraction plane, the effect of partial spatialcoherence on the STEM image is most easily understood withreference to the plane of the specimen. From Fig. 1 d, a finiteeffective source gives rise to a set of mutually incoherent Airydiscs which are displaced and weighted according to the sourceintensity distribution. Hence, the intensity at a given point in aSTEM image contains contributions from the set of displaced andincoherent Airy discs. Mathematically, the intensity in a STEMimage is given by a convolution of the image Icoh that would begenerated by spatially coherent electron probe and the effectivesource distribution in the specimen plane:

IðxÞ ¼Z

d2x0 Sðx0ÞIcohðxþx0Þ: ð3Þ

This result holds for all types of STEM images (where the imageintensity corresponds to the value of some signal as a function ofprobe position).

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0.07 0.42 0.09 0.24

5 A

0.01 0.12 0.01 0.04

Fig. 2. Frozen phonon multislice simulations of BF-STEM (a)–(e) and ADF-STEM (f)–(j) images of [0 0 1] b�Si3N4 exhibiting the cumulative effects of a perfect imaging

system [(a) and (f)], residual lens aberrations [(b) and (g)], focal spread [(c) and (h)], Gaussian source with 0.8 A FWHM [(d) and (i)], and modified-Lorentzian source with

0.5 A FWHM [(e) and (j)]. The intensity scales indicate the fractional probe intensity.

C. Dwyer et al. / Ultramicroscopy 110 (2010) 952–957954

To illustrate the relative importance of the factors contributingto the incident wave field, Fig. 2 shows a sequence of frozenphonon multislice simulations of atomic resolution BF-STEM(Figs. 2 a–e) and ADF-STEM (Figs. 2 f–j) images of [0 0 1] b�Si3N4.These images exhibit the cumulative effects of residual lensaberrations, finite focal spread, and finite source size. Thesimulations [32] assume a modern CS-corrected Schottky FEG-TEM operating at 300 kV, a probe semi-angle of 25 mrad, a BFcollection angle of 10 mrad, ADF collection angles of 40–190 mrad,and a specimen thickness of 220 A.

Figs. 2 a and f correspond to an incident wave field generatedby a perfect point source (perfect spatial and temporal coherence)and an aberration-free lens system. Figs. 2 b and g furtherincorporate the effect of higher-order lens aberrations (as perFig. 1 b), with aberrations set to values typical of the residualaberrations after correction for CS using a commercial hexapolecorrector [10]. It is apparent that the introduction of these higher-order aberrations have a discernible but rather minor effect on theimage intensity. Figs. 2 c and h further incorporate the effect ofpartial temporal coherence corresponding to a focal deviationparameter of 25 A, consistent with an energy spread of 0.8 eVtypically encountered from a Schottky FEG. Quantitatively, theaddition of partial temporal coherence reduces the heights of theimage maxima but the resolution of the image is largelyunchanged. It should be noted that the effects of residualaberrations and focal spread are reduced if a smaller probeconvergence angle is used.

The simulations in Figs. 2 a–c (and similarly Figs. 2 f–h) haveassumed a point source and hence perfect spatial coherence. Incontrast, the calculations in Figs. 2 d and e (and similarly Figs. 2 iand j) further incorporate, respectively, a finite source of Gaussiandistribution with 0.8 A full-width half-maximum (FWHM) and afinite source of modified-Lorentzian distribution with 0.5 AFWHM. (Since the Lorentzian distribution is non-integrable intwo dimensions, it was multiplied by a Gaussian with 10 AFWHM. This ensures integrability while leaving the value of theLorentzian largely unchanged near the origin.) These particularsource distributions were chosen to give the same nominalresolution in the ADF image (more precisely, they were chosen to

give equal widths for the major peaks in the ADF imagecorresponding to atomic columns of Si). It can be seen that thesedistributions give rise to significantly different image intensities.For example, the Lorentzian distribution, with its broader tails,results in significantly lower image contrast. It is also readily seenthat the incorporation of a finite source has far greater impact onthe image intensity than residual aberrations or focal spread.These simulations emphasize the dependence of the quantitativeimage intensity on the shape and size of the effective sourcedistribution. In Section 5, we demonstrate a method for measur-ing the effective source distribution of a sub-A probe.

4. Spatial coherence in coherent CBED

The method of source measurement that will be demonstratedin Section 5 relies on an understanding of the way in which acoherent convergent-beam electron diffraction (CBED) pattern ofa crystal is affected by a finite source distribution. The effect canbe understood from Fig. 1 d along with the fact that the position ofinterference fringes in the overlapping discs of a coherent CBEDpattern are dependent on the nominal probe position. The latterpoint is easily grasped from the mathematical expression for theintensity in the region of overlap between the central beam and adiffracted beam [11–15]:

IðkÞ ¼ jc0ðkÞj2þjcgðk�gÞj2þ2Rec0ðkÞcg

�ðk�gÞe2piðwðkÞ�wðk�gÞþg�x0Þ;

ð4Þ

where cgðk0Þ is the amplitude of the diffracted beam g for anincident plane wave with transverse wave vector �k0, 2pwðk0Þ isthe corresponding phase shift due to lens aberrations, x0 is theprobe position, and Re implies the real part of a complex number.Heuristically, the displacement of the probe corresponds to theintroduction of a phase ramp across the probe-forming aperture,which results in a shift of the interference fringes.

When the effective source has a finite distribution, each pointof the effective source is imaged independently as a (possiblydefocused) Airy disc in the specimen plane (Fig. 1 d). If the axialsource point is imaged to the nominal probe position, then off-axial

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C. Dwyer et al. / Ultramicroscopy 110 (2010) 952–957 955

source points will be imaged to points slightly displaced from thisnominal position. Hence, the coherent CBED pattern is anincoherent sum of the corresponding patterns, each one arisingfrom an Airy disc which is slightly displaced from the nominalprobe position and weighted by the source distribution in thespecimen plane [1,16]. From this point of view it is understood thatif the source distribution in the specimen plane is smaller than thelattice spacings in the field of view, then the effect will be todampen the contrast of the interference fringes. On the other hand,if the source size is comparable or greater than the lattice spacing,then the fringes will not be visible. Quantitatively, the intensity in acoherent CBED pattern of a crystal in the presence of partial spatialcoherence is given by [1,16]

IðkÞ ¼Xg;h

cgðk�gÞ ~Sðg�hÞch�ðk�hÞ; ð5Þ

where cgðk0Þ now includes phase shifts due to lens aberrations andprobe displacement, and ~SðkÞ is the Fourier transform of thenormalized effective source in the specimen plane. Since theinterference fringes arise from the terms where gah, Eq. (5) saysthat the interference fringes are damped according to the Fouriertransform of the effective source distribution in the specimenplane. Hence, as will be demonstrated in Section 5, the size andshape of the effective source distribution can be determined bymeasuring the contrast of interferences fringes for different latticespacings [1,17].

If the effects of partial temporal coherence are included, thenwe must perform a weighted integral of Eq. (5) with respect to thefocus deviation, where the weighting function is the normalizedfocal distribution F. Carrying out the integration, we obtain for theintensity in the Ronchigram

IðkÞ ¼Xg;h

cgðk�gÞ ~Sðg�hÞch�ðk�hÞ ~F ð@C1

wðk�gÞ�@C1wðk�hÞÞ; ð6Þ

where ~F denotes the Fourier transform of the focal distribution,and @C1

denotes a partial derivative with respect to defocus. Forthe purposes of the present work, we simply note that ~F ð0Þ ¼ 1,and that the argument of ~F in Eq. (6) vanishes for pointsequidistant from g and h. Hence, if we consider interferencebetween the central beam and a diffracted beam, partial temporalcoherence has no effect for points where the Bragg condition issatisfied. However, ~F is generally a decaying function, so that theeffects of partial temporal coherence increase with departurefrom the Bragg condition.

5. Measurement of source distribution using Ronchigrams

As described in Section 4, the effect of partial spatial coherenceon a coherent CBED pattern (hereafter referred to as a Ronchi-gram) of a crystal is to dampen the interference fringes observedin the overlapping diffraction discs. Hence, by quantifying thiseffect by means of electron scattering simulations, we can accessthe Fourier components of the effective source distribution [1].

In principle, any well-known crystal structure can be used togenerate a Ronchigram for the source measurement. However, wehave chosen to use diamond because (i) its small latticeparameter (a=3.57 A) allows easier access to spatial frequencieshigher than 1 A�1, and (ii) it produces a relatively small amount ofthermal diffuse scattering (TDS) (the kinematical mean-free-pathfor room-temperature TDS of 300 keV electrons is 2:7mm). Point(i) permits characterization of the effective source in the case of asub-A probe, while point (ii) minimizes the effect of inelasticscattering that cannot be excluded by means of an energy filter.

Ronchigrams of [1 1 2]- and [0 1 2]-oriented diamond wererecorded using the Titan3 80–300 FEG-(S)TEM (FEI Co.) equipped

with CS probe and imaging aberration correctors (CEOS GmbH)installed at Monash University. The patterns were acquired usinga beam energy of 300 keV, an extraction voltage of 4.0 kV, gun lens6, and spot size 9 (although it is noted that the patterns wererecorded on consecutive days). The Ronchigram intensities wererecorded using an Ultrascan 1000 charge-coupled device (CCD)camera (Gatan Inc.) and an exposure time of 1 s. The modulationtransfer function (MTF) of the CCD camera was measured usingthe technique described in Ref. [18]. The effect of the CCD MTFwas subsequently removed from the recorded Ronchigrams usingFourier-ratio deconvolution.

Quantification of the experimental Ronchigrams was per-formed using Bloch state-based dynamical scattering simulationsincorporating absorption due to TDS [19]. The Bloch stateeigenvalues and eigenvectors were used as input for a suite ofspecifically written computer programs written in the IDLprogramming language. The latter programs were used to refinethe several parameters involved in the fitting process. Theseparameters included a set of geometric parameters associatedwith the scaling, translation and rotation of the experimentaldata, and a set of physical parameters such as specimen thickness,relative probe position, lens aberrations and, most importantly,the Fourier components of the effective source distribution. Afterobtaining suitable initial estimates, these parameters wererefined using a so-called Levenberg–Marquardt least-squaresfitting procedure [20].

A comparison of the experimental and simulated Ronchigramsof diamond [1 1 2] and [0 1 2] are shown in Figs. 3 a and b,respectively. The intensities are displayed on a linear scale, withblack and white corresponding to the lowest and highestintensities, respectively. For the [1 1 2] pattern, black correspondsto zero intensity, while for the [0 1 2] pattern, a finite minimum hasbeen used to highlight the detail in the central disc. The specimenthicknesses for [1 1 2] and [0 1 2] were determined to be 200720and 350720 A, respectively. The fitting procedure described abovewas performed for selected regions of the patterns. In the case of[1 1 2], these were regions of single overlap of a diffracted disc andthe central disc, while for [0 1 2], the selected region was thecentral disc. In both cases, regions very close to the edge of a discwere excluded in order to avoid difficulties associated with a non-circular aperture in the experiment. Owing to the relatively smallconvergence angle in the [1 1 2] pattern, a reasonable fit to thisRonchigram was obtained by neglecting all lens aberrations otherthan defocus. On the other hand, the bending of interferencefringes due to higher-order residual lens aberrations can be readilyobserved in the [0 1 2] pattern, and it was necessary to include upto third-order aberrations in the fitting procedure. While theaberration coefficients so determined produce a reasonable fit forthe majority of the [0 1 2] pattern, it can be seen that the positionsof two sets of {2 4 2} interference fringes are in poor agreementwith the simulation. These sets of fringes are observed to havelower contrast than the other sets of {2 4 2} fringes, implying adegree of anisotropy in the effective source distribution. Due to thepoor agreement of their positions with those in the simulation, thecontrast of the {2 4 2} fringes in question was incorrectlydetermined by the fitting algorithm to be approximately zero(but note that a finite value of contrast has been used to makethese fringes visible in the simulation in Fig. 3 b). Nonetheless, thereasonable fit obtained for the other two sets of {2 4 2} interferencefringes allowed the extraction of the Fourier component of theeffective source at the corresponding spatial frequency.

The Fourier components of the effective source distributionobtained by the fitting procedure described above are listed inFig. 4 a and plotted in Fig. 4 b. The plot in Fig. 4 b also featuresa Gaussian function which is the least-squares fit to the measuredvalues (since the f2 4 6g�type fringes are not observed in the

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5 mrad 10 mrad

Fig. 3. Interspersed pie-shaped segments of experimental and simulated Ronchigrams of (a) diamond [1 1 2] and (b) diamond [0 1 2]. Both patterns show regions around

the central beam. In pattern (a), the horizontal and vertical fringes correspond to {2 2 0} and {1 1 1} lattice planes, respectively. In pattern (b), the horizontal fringes

correspond to {4 0 0} lattice planes and all other fringes correspond to {2 4 2} lattice planes. The dark circles visible in pattern (a) are the result of a computer-generated

mask which excludes the disc edges. The patchy contrast in the experimental segments of pattern (b) are the result of amorphous surface layers.

|g| (Å−1)

S (|

g|)

{ hkl } |g| (Å−1) S

{111} 0.49 0.61 ± 0.01{220} 0.79 0.40 ± 0.02{004} 1.12 0.17 ± 0.01{242} 1.37 0.13 ± 0.01

Fig. 4. (a) Fourier components of the effective source distribution measured using Ronchigrams of diamond. (b) Comparison of the measured components and the best-fit

Gaussian function (solid line).

C. Dwyer et al. / Ultramicroscopy 110 (2010) 952–957956

experimental pattern, the corresponding Fourier componentswere set to zero for the purpose of obtaining the best-fitGaussian function). While a Gaussian function is commonlyadopted to describe the effective source distribution, it is seenthat the measured Fourier components show a significantdeparture from such a functional form. A departure fromGaussian form has also been reported by James et al. [21], inthat case for a different model of electron microscope and adifferent type of field-emission gun.

6. Discussion

The method of measuring the effective source distributiondemonstrated in the previous section has several advantages: (i) itis valid for dynamical scattering, (ii) the degree of spatialcoherence is measured directly and independently for eachFourier component of the effective source distribution, enabling,in principle, the detailed shape of the source distribution to bemapped in all directions, without any assumption regardingshape or symmetry, (iii) the measurement is made largelyindependent of the degree of temporal coherence, and (iv) themeasurement can be made in the presence of amorphous surfacelayers. When combined with measurements of the focal spreadand lens aberrations, the incident electron wave field on the

specimen can be accurately characterized. While there existsimpler methods to measure the gross characteristics of thesource distribution, using brightness measurements, for example,the major advantage of the method demonstrated here is that itdoes not involve an assumption about the functional form of thesource distribution. It is emphasized, however, that this methodincorporates into the measurement of the effective sourcedistribution the effect of instabilities during the exposure time,such as specimen and/or beam drift, which reduce the contrast ofinterference fringes in the Ronchigram.

The previous section outlined a method for measuring certainFourier components of an effective source distribution. In order toaccount for this effective source in STEM image simulations usingEq. (3) (or, alternatively but probably less reliably, remove itseffect from experimental images by deconvolution) interpolationbetween the measured Fourier components is required. Inter-polation of the measured Fourier components is also requiredbefore obtaining the real space distribution of the effective sourceby an inverse Fourier transformation. In the present work, thisstep was deemed inappropriate on account of shortcomings of theexperimental data set, namely, that the data was acquired overtwo consecutive days and the number of measured Fouriercomponents is rather small.

In Section 3 it was demonstrated that the degree of spatialcoherence is important for the quantitative interpretation of

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C. Dwyer et al. / Ultramicroscopy 110 (2010) 952–957 957

experiments involving focused probes. Moreover, the effects ofpartial spatial coherence are quite independent of the lens aberra-tions. The situation is very different for high-resolution TEM (HRTEM)imaging where near-parallel illumination is used. In this case, theleading order effects of finite source size inextricably incorporate thelens aberrations [22–24]. In contrast, the effect of source size onSTEM images (which we are concerned with in the present work) isrelated via the principle of reciprocity to the effect of the camera MTFon high-resolution TEM images [25]. The significance of the lattereffect has been known for some time in the HRTEM community (see,for example, [26–31]).

7. Conclusions

In this paper, we have reviewed the manner in which partialspatial coherence affects STEM images and coherent CBEDpatterns, and emphasized the importance of measuring theeffective source distribution for the quantitative interpretationof atomic-resolution STEM images and other small probe experi-ments. We have demonstrated a Ronchigram-based scheme forthe measurement of the source distribution in a 300 kV FEG-TEMfitted with a probe CS corrector.

Acknowledgments

The authors wish to express their gratitude to C.J. Rossouwfor providing access to his Bloch-state computer code, and toP. Nakashima and M. Weyland for enabling measurement of theCCD PSF. C.D. and J.E. gratefully acknowledge financial supportfrom FEI Company and the Australian Research Council (ARCLinkage Project LP0990059).

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