Microscopy, 2014, 345–355
doi: 10.1093/jmicro/dfu020
Advance Access Publication Date: 2 June 2014

ArticleDownloaded from http://jmicro.oxfordjournals.org/ at Osaka Daigaku Ningen on October 1, 2014

Analysis of nonlinear intensity attenuationin bright-ﬁeld TEM images for correct 3D
reconstruction of the density in micron-sizedmaterials
Jun Yamasaki1,2,*, Michihiro Mutoh3, Shigemasa Ohta4, Syuichi Yuasa4,
Shigeo Arai2, Katsuhiro Sasaki5, and Nobuo Tanaka2
1

Research Center for Ultra-High Voltage Electron Microscopy, Osaka University, 7-1 Mihogaoka, Ibaraki,
Osaka 567-0047, Japan, 2EcoTopia Science Institute, Nagoya University, Furo-cho, Chikusa-ku, Nagoya
464-8603, Japan, 3Department of Crystalline Materials Science, Nagoya University, Furo-cho, Chikusaku, Nagoya 464-8603, Japan, 4JEOL Ltd., 3-1-2 Musashino, Akishima, Tokyo 196-8558, Japan, and5
Department of Quantum Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603,
Japan*To whom correspondence should be addressed. E-mail: yamasaki@uhvem.osaka-u.ac.jp
Received 25 February 2014; Accepted 7 May 2014

Abstract
To obtain the correct tomographic reconstruction of micron-sized materials, the nonlinear
intensity attenuation of bright-ﬁeld transmission electron microscopy (BF-TEM) images
was analyzed as a function of the sample thickness using a high-voltage electron microscope. The intensity attenuation was precisely measured relative to the projection thickness of carbon microcoils (CMCs) at acceleration voltages of 400–1000 kV using objective
apertures (OAs) with radii of 2.1–28 nm−1. The results show that the nonlinearity isstrongly dependent on the OA size and the acceleration voltage. The inﬂuence of nonlinearity on tomographic reconstructions was also examined using a specially developed
360°-tilt sample holder. Sliced images of the reconstructed volumes indicated that anincrease in the nonlinearity caused artiﬁcial ﬂuctuations in the internal density of materials and inaccurate shapes of the objects in more signiﬁcant cases. Conditions sufﬁcient
for reconstruction with the correct density have been estimated to be 0.67 of theminimum electron transmittance, and for reconstructions with correct shapes, 0.4. This
information enables foreseeing the quality of the reconstruction from a single BF-TEM
image prior to the tilt-series acquisition. As a result to demonstrate the appropriateness ofthese conditions, a CMC with a diameter of 3.7 µm was reconstructed successfully; i.e.
not only the shape but also the internal density were correctly reproduced using electron
tomography.
Key words: electron tomography, bright-ﬁeld TEM, nonlinear intensity attenuation, HVEM, 360°-tilt sample holder,carbon microcoil© The Author 2014. Published by Oxford University Press on behalf of The Japanese Society of Microscopy. All rights reserved.
For permissions, please e-mail: journals.permissions@oup.com

345

346

Introduction

correct 3D reconstruction of internal density have not yet
been established. Such ‘3D density reconstruction’ isrequired for the discrimination of materials and voids[15,20] and for the detection of gradual compositionalchanges in various materials [19], which are often induced
by intermixing in multilayer ﬁlms and core-shell particles.
In addition to eliminating the missing-wedge effects,
increasing the accuracy of the projection information for 3D
objects is also indispensable to the 3D density reconstructions. According to the principle of tomographic reconstruction, the signals in a projection image must be proportional
to the projection mass–thickness (density integrated along thebeam direction). Therefore, prior to the reconstruction procedure, the image intensity of the micrographs in a tilt
series must be converted to the ‘proportional’ signals.
However, this conversion is only possible when the image
intensity and sample thickness are correlated by a known
monotonic relationship. BF-TEM image intensity basicallychanges when a portion of the scattered electrons are stopped
by the objective aperture (OA) and/or other ﬁxed apertures in
the column of the microscope (scattering contrast) [23].
When the diffraction contrast is negligibly small, the scattering contrast becomes the mass–thickness contrast, which
reﬂects the composition and atom density of the material aswell as the thickness. Under a kinematical approximation, the
BF-TEM image intensity based on the mass–thickness contrast
exhibits exponential attenuation with increasing thickness. The
beam intensity I(t) over a sample area with a thickness t isshown as follows:IðtÞ ¼ I0 expðμtÞ;

ð1Þ

where I0 is the incident beam intensity and μ is the attenuation
coefﬁcient, which is dependent on the scattering cross sectionof the material and the OA size. Therefore, the electron transmittance T(t) given by I(t)/I0 exhibits linear attenuation on asemi-logarithmic scale:ln TðtÞ ¼ μt:

ð2ÞHowever, the BF-TEM image intensity in actual experiments
deviates from this linear attenuation due to multiple elastic/
inelastic scatterings with increasing sample thickness [23–26].
Importantly, if the linearity has collapsed in images used for atilt series, the inaccurate mass–thickness information will betransformed into an inaccurate density distribution during the
3D reconstruction [14,15,17–19,27–29].
Currently, a large percentage of electron tomography
results may be disturbed more or less by several factors
such as the missing-wedge effect, the nonlinear attenuation
effect and the diffraction contrast for crystalline samples.
In the present study, to focus on analyzing the nonlinear

Downloaded from http://jmicro.oxfordjournals.org/ at Osaka Daigaku Ningen on October 1, 2014

Currently, 3D analyses of nanometer- and sub-micron-sized
objects are extensively performed by tomography usingtransmission electron microscopes (TEM) in various imaging
modes suitable for each purpose. For example, since the ﬁrst
report in 2001 [1], electron tomography based on annulardark-ﬁeld scanning TEM (ADF-STEM) has been successfully
applied to observing the 3D structures of crystalline materials[2–4], owing to the image contrast insensitive to the diffraction conditions. Active utilization of the diffraction contrast
in bright-ﬁeld STEM and dark-ﬁeld TEM has enabled the3D analysis of embedded structures in bulk crystalline materials such as dislocations and precipitates [5–7]. On theother hand, various non-crystalline objects such as biological samples [8–10], polymers [11] and other inorganic
materials [12,13] have been widely analyzed by bright-ﬁeld
TEM (BF-TEM) tomography. Among the various imaging
modes in TEMs, BF-TEM is the most basic and simplest
mode. BF-TEM tomography has an advantage that theimages used for the reconstruction are free from local distortions due to sample drift unlike STEM. Another practicaladvantage is that adhesion of contaminants on samples via
parallel illumination, which is sometimes a problem forscanning focused probes, is not a serious issue in BF-TEM.
In electron tomography using medium-voltage TEMs,
the size and thickness of samples are limited to be <1 µm[1–5,11–19]. To extend the observable range to the micronscale, increasing the acceleration voltage is essential. Several
achievements in 3D observations of micron-sized materials
have been reported using BF-TEM tomography in a highvoltage electron microscope (HVEM) [8–10,20]. The 3D
surface morphology of such micron-sized materials can be,
however, observed also using scanning electron microscopy(SEM). Therefore, the important capability required intransmission electron tomography is not only the qualitative
determination of 3D shapes but also the quantitative analysis of the 3D data, particularly the data for the interior of
micron-sized objects.
One of the important issues on the quantitative 3D analyses is the missing-wedge effect, which is induced by thelimited tilt angles because of the sample holder design and/or the sample conﬁguration itself. The most reliable andstraightforward solution is to acquire a tilt series over theentire angular range. This approach has been achieved todate by some researchers using pillar samples attached to
special sample holders [16,21,22]. Because the reconstructed volumes in the results are free from the elongation
artifacts, quantitative information on the 3D shapes, such as
superﬁcial areas and volume fractions of embedded grains
in bulk, has been obtained [16]. Unlike such quantitative‘3D shape reconstruction’, procedures for achieving theMicroscopy, 2014, Vol. 63, No. 5

Microscopy, 2014, Vol. 63, No. 5347

attenuation effect in BF-TEM images, a special experimental
setup was prepared. The samples were amorphous carbon
microcoils (CMCs) [30–35] with well-deﬁned shapes andcompositional homogeneity, which is beneﬁcial for thickness estimation. Measurements of the nonlinearity over awide range of acceleration voltages were conducted using an
HVEM. Moreover, a 360°-tilt tomography sample holder
designed speciﬁcally for the HVEM was utilized to eliminate
the missing-wedge effect.

Experimental
HVEM and newly developed sample holder
for 360°-tilt tomography
Downloaded from http://jmicro.oxfordjournals.org/ at Osaka Daigaku Ningen on October 1, 2014

The HVEM at Nagoya University (JEOL: JEM-1000KRS)[10] was used for analyzing the intensity attenuation withincreasing sample thickness and for acquiring the tomographic tilt series. BF-TEM images of CMCs were takenusing a 2K × 2K CCD camera (Gatan: ORIUS™) at acceleration voltages of 400, 600, 800 and 1000 kV. For comparison, zero-loss images with an energy window of 10 eV werealso recorded using a postcolumn imaging ﬁlter (Gatan: GIF
Quantum™). For 3D reconstruction of a several micrometersized object, the images in the tilt series must have ﬁelds ofview larger than the object, e.g. ∼10 µm. In the HVEM, thelowest magniﬁcation possible while keeping the objective lens
power on (namely, not in LowMag mode, but in Mag modefor JEOL microscopes) is 400, which enables the recording ofa 25 × 25 µm2 region using the CCD camera. In the current
study, a magniﬁcation of 1000, which is suitable for observing a CMC with a diameter of 3.7 µm, was selected.
The OA size is a deciding factor for the BF-TEM image
contrast and therefore also for the intensity attenuation. The
HVEM has four OA sizes, which are denoted as OA1, OA2,OA3 and OA4 in descending order of the radii. In Fig. 1a,their collection semi-angles α = 24.3, 13.2, 6.80 and
1.86 mrad are compared with the halo-ring pattern taken at1000 kV from a CMC. Although the halo rings are formedby the diffracted beams from the amorphous structure, inthis paper the term ‘BF-TEM’ also refers to images takenwith OA1–OA3. The HVEM was designed in such a waythat scaling at the back-focal plane of the objective lens did
not change much when the acceleration voltage waschanged. In fact, the spatial frequency sin α/λ (λ: wave
length) for the collection angles at 400 kV were measured tobe nearly the same as those at 1000 kV (only ∼10% larger
than at 1000 kV). In this paper, the notation for theimaging conditions is in the format 1000 kV–OA3, whichindicates the use of 1000 kV and OA3.
Figure 1b shows the tip section of the sample holder specially developed for the HVEM. The holder has an internal

Fig. 1. (a) Sizes of the OAs compared with a halo-ring pattern from a CMC
taken at 1000 kV. The collection semi-angles α = 24.3, 13.2, 6.80 and
1.86 mrad (27.9, 15.2, 7.82 and 2.13 nm−1, respectively) were calibrated
based on the diffraction pattern of a silicon crystal. (b) Newly developed
sample holder for HVEM tomography. The shape of the internal rod isindicated by the white broken lines. The rod is rotatable over 360° withrespect to the external cylinder. (c) Scanning ion microscopy (SIM) imageof a CMC ﬁxed at the needle attached to the head of the internal rod.

rod that is rotatable over 360° with respect to the external
cylinder. For tomographic observations, a sample is initiallyattached to the tip of a tungsten needle in a focused ion
beam instrument (in the present study, Hitachi: FB-2100

Microscopy, 2014, Vol. 63, No. 5

348

Carbon microcoils
To exclude the missing-wedge effect and to determine therelationship between the sample thickness and image intensity, the sample must have a well-deﬁned, rod-like shape. In
addition, to focus on multiple scattering effects, uniform
amorphous materials are desirable due to the minimized diffraction contrast in their BF-TEM images. Thus, to meet allof the above requirements, CMCs are considered to be oneof the best choices. Figure 2a and b shows micrographs of
the CMCs (Microphase Co., Ltd.) used in the present study.
These helical, amorphous carbon ﬁbers, most of which aredouble helical, are grown via the pyrolysis of acetylene
using catalytic metal particles such as nickel [30]. Becauseof their characteristic helical morphology, a typical diameteron the micron scale and conductive properties [33], CMCs
are expected to have applications such as radio wave absorbents, tactile sensors [32,34].
Although the 3D structure of CMCs has been roughly
examined using scanning confocal electron microscopy [35],
precise 3D observations via electron tomography have notyet been achieved, probably due to the inability of the electrons in medium-voltage TEMs to penetrate micron-sized
thick samples. Figure 2b shows a BF-TEM image of the
CMC attached to the 360°-tilt sample holder in the HVEM.
If the cross section of the carbon ﬁber has a circular shape,
the surface of one ﬁber of the double helix is drawn using aseries of points (x, y, z) that satisfy the following equations:x ¼ ðR þ r cos φÞ cos θ
y ¼ ðR þ r cos φÞ sin θz¼p

θþ r sin φ
2π

ð0  θ < 2π;

0  φ < 2πÞ;

ð3ÞFig. 2. Carbon microcoils. (a) SEM image (taken by Microphase Co., Ltd.)and (b) BF-TEM image using the HVEM. (c) Parameters for the helicalstructure and (d) a view of the constructed model.

where R and ϕ are the radius and azimuth angle of the helicalstructure, and r and θ are the radius and azimuth angle of thecircular cross section of the ﬁber (see Fig. 2c). The parameterp is the pitch of the helix, which should be 4r for an ideallytight double helix. The second ﬁber of the helical pair isreproduced in the same manner using a translation of p/2 inthe z direction. Figure 2d shows an example of the model
constructed using Eq. (3).

Results
Comparison of the experimental images
and results for the structural model
Figure 3a shows a BF-TEM image of a CMC with a diameterof 3.7 µm taken at 1000 kV–OA1. Numerous particles
attached to the surface (indicated by the arrows) were usedlater as the ﬁducial markers in the 3D reconstruction. The projection thickness for the model in Fig. 2d is shown in Fig. 3b
and is bilaterally symmetric, unlike the image in Fig. 3a. Thisdifference is observed because the coil axis in (a) is not perpendicular to the beam direction. When the coil axis of the model
was tilted ∼5°, the projection (Fig. 3c) was in good agreementwith the image presented in Fig. 3a. As seen in Fig. 2a and b
as well as in the lower half of Fig. 3a, local ﬂuctuations of the

Downloaded from http://jmicro.oxfordjournals.org/ at Osaka Daigaku Ningen on October 1, 2014

was used) (Fig. 1c). Then the needle is inserted into thecentral hole of the mounting block and ﬁxed by a screw (seeFig. 1b). Finally, the mounting block is screwed into thehead of the internal rod. For each tilt series described below,
176 images were recorded from 0.0° to 179.4° with exposure times of 1–2 s. The last image of each series nearly coincides with the reverted ﬁrst image, which conﬁrms that no
signiﬁcant deformation of the sample occurred due to radiation
damage or sputtering effects during the tilt-series acquisitions.
The tilt-series images were processed for 3D reconstruction
using commercially available software (TEMography™, System In Frontier Inc.). As ﬁducial markers for image alignment,
small particles on the CMC, such as tungsten clusters adhered
during the ﬁxing process to the needle (shown later in Fig. 3a),were used. The reconstructions were performed using thestandard ﬁltered back-projection method.

Microscopy, 2014, Vol. 63, No. 5

349

Fig. 3. CMC with a diameter of 3.7 µm. (a) BF-TEM image taken at 1000 kV–OA1. Projections of the model
structure (b) without tilt and (c) tilted 5°. The black arrows in (a) indicate the clusters used as the ﬁducial
markers.

for the area near the line in Fig. 3a taken at 1000 kV–OA1.
The proﬁle of −ln T/μ measured along V0 − C0 in Fig. 4c isshown in Fig. 4a. The value of μ has been optimized to be
2.7 × 10−3 nm−1 such that the energy-ﬁltered proﬁle is inagreement with the unﬁltered proﬁle for the thinner area ofthe sample. However, for the areas thicker than 1 µm, theproﬁle is saturated and never corresponds to the thickness.
Therefore, with respect to the linearity, unﬁltered BF-TEM
images are preferable to ﬁltered ones, because the linearity of
the latter is largely limited to areas with a thickness of 1 µm
even when observed at 1000 kV.

Nonlinear intensity attenuation dependence
on the OA size and acceleration voltage
Using the data between P1 and P2 in Fig. 4a, the relationshipbetween ln T and the sample thickness was plotted in Fig. 5a(indicated by OA1). Because the thickness estimation in theareas P1  P01 and P02  P2 is difﬁcult as mentioned previously, only the thickness range between 1200 and 2700 nm isavailable for analysis of the relationship between thickness
and transmittance (hereafter, referred to as ‘speciﬁed thickness range’). Figure 5a shows also the least squares ﬁtted line
for the subset in the speciﬁed thickness range, which has aslope μ = 1.0 × 10−4 nm−1 and passes exactly through theorigin. This result indicates that the intensity attenuation at
1000 kV–OA1 has a good linearity up to ∼3 µm or more.
Figure 5a also shows the intensity attenuation obtainedwith OA2, OA3 and OA4 at the same position and in thesame manner as with OA1. Because a BF-TEM image isformed by electrons passing through the OA, the intensities
obtained with smaller OAs are smaller than that with thelarger OA over the entire thickness range. It can be seen in
Fig. 5a that the data with smaller OAs fall in curved rather
than straight lines. This behavior can be visualized moreclearly by considering the intercepts of the least squares
ﬁtted lines at zero thickness. For example, at a glance, thedata with OA2 in the speciﬁed thickness range fall in a

Downloaded from http://jmicro.oxfordjournals.org/ at Osaka Daigaku Ningen on October 1, 2014

helical pitch and loosening of the interlocked helical pairoccurred. For precise comparisons free from such uncertainties, the image intensity was analyzed along the line in Fig. 3a,in which the helical pair was tightly interlocked. The line runs
from a position V in the vacuum to the coil center C andpasses through the thickest part of the sample near P2.
The intensity proﬁle measured in Fig. 3a was normalizedusing the vacuum intensity I0 to show the electron transmittance T. As can be seen in Fig. 4a, when the attenuation
coefﬁcient μ is 1.0 × 10−4 nm−1, the proﬁle of −ln T/μ agrees
well with the thickness proﬁle in Fig. 3c. Although theagreement is best for a tilt angle of 5°, the thickness proﬁles
between P1 and P2 are insensitive to the slight estimation
error of the tilt angle. There are, however, discrepancies
between the measured proﬁle and the model in the areas
near P1 and P2. In Fig. 4a, the positions where the discrepancies start are represented as P01 and P02 , which correspondto the thickness of 1200 and 2700 nm, respectively. As mentioned also in the section Nonlinear intensity attenuation
dependence on the OA size and acceleration voltage, it isgenerally known that the nonlinear attenuation makes
BF-TEM image brighter than expected from the linear
attenuation [23–26]. Against the general tendency, theimage intensity in the areas P1  P01 and P02  P2 is darker
than expected from the model and thus cannot be explainedby the nonlinearity. As shown schematically in Fig. 4b, P1
and P2 correspond to the object edges for the transmitted
electrons. In Fig. 3a, the ﬁber cross section appears to have apolygonal shape rather than a circular shape, which possibly originates from the polyhedral shape of the catalytic
cluster [31]. The projection thicknesses near the edges are
considered to be sensitive to the cross-sectional shape andtend to deviate from the calculation based on the circularshape. Thus, it is considered that the darker image contrasts
in P1  P01 and P02  P2 do not originate from the nonlinearity, but indicate the actual thicknesses.
The electron transmittance was also measured under theenergy-ﬁltered condition. Figure 4c presents a zero-loss image

Microscopy, 2014, Vol. 63, No. 5

350

Fig. 5. Intensity attenuation in BF-TEM images with increasing thickness. (a) Dependence on the OA size at 1000 kV. (b)Dependence on the acceleration voltage with OA1. Least squares ﬁtted lines for the data in the speciﬁed thickness ranges(1200–2700 nm for 400 and 1000 kV and 1300–2600 nm for 600 and 800 kV) are also shown.

straight line. However, the negative value of the intercept
indicates that the intensity attenuation does not exactly
undergo exponential decay.

Figure 5b shows the attenuation at acceleration voltagesof 400, 600, 800 and 1000 kV with OA1. The smaller
intensities at the lower voltages reﬂect the differences in

Downloaded from http://jmicro.oxfordjournals.org/ at Osaka Daigaku Ningen on October 1, 2014

Fig. 4. Comparison of the projection thickness of the structural model and electron transmittance. (a) Proﬁles along
the lines in Figs 3a, c and 4c. The model projections were calculated for coil axis tilts of 4°, 5° and 6°. The μ values are
1.0 × 10−4 and 2.7 × 10−3 nm−1 for the BF-TEM and zero-loss images, respectively. (b) Schematic of the cross sectionof the CMC along the electron transmission. (c) Zero-loss image of the CMC. The shape of the ﬁber is indicated by thebroken lines.

Microscopy, 2014, Vol. 63, No. 5

3D reconstruction of a CMC sample
Tomographic tilt series were acquired for the CMC in
Fig. 3a at the three characteristic conditions shown in Fig. 5,
i.e. 1000 kV–OA1, 400 kV–OA1 and 1000 kV–OA4.
Sequential views of the tilt series indicated that the CMC
axis was inclined ∼25° from the rotation axis, which wasforeseen in Fig. 1c. The results for the 3D reconstructions
are shown in Fig. 6. Regardless of the differences in theimaging conditions, the 3D shape of the CMC (and also thedust attached to it) has been reconstructed well. This meansthat the substantial degrees of the nonlinearity observed inFig. 5 do not hinder overviews of 3D objects.
For detailed comparisons, the cross sections of the 3Dvolumes were then examined. Figure 7a shows a sliced
image of the structural model when cutoff as shown in thelower portion of the ﬁgure. The normal direction of the slice
is parallel to the rotation axis and thus inclined 25° from thecoil axis. Figure 7b–d shows corresponding images sliced
from the reconstructed volumes in Fig. 6a–c, respectively.
The proﬁle in Fig. 7b conﬁrms that the material and vacuum
have been reconstructed with the respective uniform

intensity (apart from the enhanced edges, which are discussed later). In contrast, the internal density of the CMC in
Fig. 7c is not uniform, but has a gradient from the center.
Moreover, there is a slight increase in the vacuum level inthe interior of the coil. These features are more signiﬁcant in
Fig. 7d, in which also the shape of the CMC has been incorrectly reproduced. In particular, the interior region is so
blurred that the inner walls are difﬁcult to identify. In
Fig. 7c and d, the artiﬁcial streaks are remarkable in thevacuum regions. However, the direct effect of the nonlinearity has not yet been clariﬁed. As can be seen in Fig. 5, notonly the linearity but also the total intensity decrease in theorder Fig. 7b > c > d. Such dark images tend to decrease theprecision of the alignment using ﬁducial markers. Apartfrom such a practical problem, these results do demonstratethat 3D density reconstructions of a 3.7 µm-sized material
are achievable using 1000 keV electron beams.

Discussion
Nonlinear attenuation caused by multiple
scatterings
In a material, a portion of the incident electrons is scattered
to the angular region larger than the OA radius. During
passage through the material, these electrons can be scattered again back to the angular region inside the OA witha certain probability. Consequently, at the exit surface ofthe sample, the number of electrons inside the OA becomes
greater than expected based on kinematical single scatterings. Thus, these multiple scatterings make the attenuation of the BF-TEM intensity more gradual than the ideal
linear attenuation, as seen in the curves with OA3 and
OA4 in Fig. 5a. Based on the above consideration, thedeciding factor should be the angular distribution of thescattered electrons. Roughly speaking, a greater number ofelectrons outside of the OA lead to more the scattered-backelectrons.

Fig. 6. Volume-rendered views of the reconstructed volumes for the same CMC observed at (a) 1000 kV–OA1,(b) 400 kV–OA1 and (c) 1000 kV–OA4.

Downloaded from http://jmicro.oxfordjournals.org/ at Osaka Daigaku Ningen on October 1, 2014

penetration capability. Because the data at 600 and 800 kV
were obtained using another CMC, the speciﬁed thickness
range is slightly different from that for the results at 400 and1000 kV. In these ranges, the data at any voltages seem toexhibit linear attenuation. However, similar to the results at
1000 kV–OA2, OA3 and OA4 in Fig. 5a, the data at
400 kV–OA1 show a considerably negative intercept value
at zero thickness. Such a pseudo-linear attenuation feature
has been observed also in previous reports [17,24,25],
although the cause is as yet unclear. However, such nonlinear attenuation certainly introduces error in the conversion from intensity to thickness, and thus inhibits correct
3D reconstruction. In summary, smaller OAs and lower
acceleration voltages increase the degree of the nonlinearityof the intensity attenuation.

351

352

Microscopy, 2014, Vol. 63, No. 5

This tendency can be seen in Fig. 5a, in which the nonlinear
characteristics increase when a smaller OA is used. However,this result conﬂicts with a previous report using 2 MeV electron
beams, in which no change of the linearity due to the OA size
was observed [24]. In that report, the linearity was estimatedfrom only several measurement data sparsely sampled in a thickness range near zero thickness. In such a thinner range, because
the nonlinear curves deviate only slightly from a straight line,
precise estimation of the nonlinearity could be difﬁcult. However, in the present study, the nonlinearity has been estimatedfrom a lot of measurement data densely sampled in the thicker
ranges (Fig. 5), where the nonlinear attenuation becomes much
more signiﬁcant. Thus, the contradiction between the previous
and the present reports may be due to the level of precisionof the measurements. Regarding Fig. 5b, the data at 1000 kVshow much larger transmittance and better linearity than that at
400 kV. This result is expected, because smaller scattering crosssections for higher energy electrons result in smaller numbers ofelectrons scattered to the outside of the OA. Thus, both the OA
and acceleration voltage must be selected carefully to control thenonlinearity and quality of 3D reconstructions.
The above discussion of multiple scatterings should bevalid without distinction between elastic and inelastic processes. Based on the elastic and inelastic scattering cross sections of atoms, the linear attenuation coefﬁcient μBF for
BF-TEM image intensity is calculated to be [23]ρNA Z1=3 λ2
E 2 b1
μBF ¼
ðZ  1Þ þ 2 ln b ;
1þ
E0
b
πA
 1=3 2
Z λ
b¼1þ
ð4Þ
2παaH

for a material which has a density ρ and consists of a lightelement with atomic number Z and atomic weight A (6 and
12 for CMCs, respectively). NA is Avogadro constant, aH
Bohr radius, E0 and E rest energy and kinetic energy of theincident electrons, respectively. Assigning the measured
value 1.0 × 10−4 nm−1 to μBF, ρ is estimated to be 1.9 g/cm3.
As shown in Fig. 4c, a large fraction of electrons is lost in
the zero-loss image if the sample is thicker than 1 µm. Thismeans that the corresponding unﬁltered BF-TEM intensity ismostly formed by inelastically scattered electrons. Note thatthe contribution of plasmon scattering is not considered in
Eq. (4). Nevertheless, the estimated ρ agrees with the density
generally known for amorphous carbons (1.8–2.1 g/cm3).
This good agreement indicates that, in the thickness rangeof the linear attenuation, deﬂection angles by plasmon scatterings are much smaller than the OA radius (24.3 mrad)
and therefore do not contribute to the intensity attenuationin the unﬁltered BF-TEM image so much. On the other
hand, if the thickness increases up to the nonlinear attenuation range, the angular spread by multiple plasmon scatterings could become too large to be negligible compared with
the OA radius. This behavior may be the reason why
experimental nonlinear attenuations are not reproducedwith a high degree of accuracy by the mathematical model
based on a statistic treatment of multiple elastic scatterings[24,25]. Clariﬁcation of the angular distribution of themultiple inelastic scatterings is required to elucidate thefunctional form of the nonlinear attenuation more accurately. Further studies will lead to corrections for the nonlinear curvatures, as has been achieved for incoherent
BF-STEM and HAADF-STEM tomography [15,19].

Downloaded from http://jmicro.oxfordjournals.org/ at Osaka Daigaku Ningen on October 1, 2014

Fig. 7. Comparison of sliced images. (a) Structural model and reconstructed volumes at (b) 1000 kV–OA1, (c) 400 kV–OA1 and (d) 1000 kV–OA4. The
bright and dark regions correspond to the material and vacuum, respectively. The normal direction of the slice is parallel to the rotation axis asshown in the lower part of (a). Line proﬁles between A and B are also shown in (b–d).

Microscopy, 2014, Vol. 63, No. 5

Inﬂuence of nonlinear attenuation on 3D
reconstructions

the energy window and the OA. Instead of Eq. (1), theattenuation of zero-loss intensity Izl is therefore shown as
Izl ðtÞ ¼ I0 expðμBF tÞ expðμinel tÞ¼ I0 exp½ðμBF þ μinel Þt;

ð5Þ

where μinel is the attenuation coefﬁcient due to the energy
ﬁltering. Using μBF = 1.0 × 10−4 nm−1 and μBF + μinel =2.7 × 10−3 nm−1 measured in Fig. 4a, the total-inelastic
mean free path is estimated to λinel = 1/μinel =(2.7 × 10−3 − 1.0 × 10−4)−1 ≃ 380 nm. As shown in Fig. 4a,
the maximum thickness for the linearity in the zero-loss
image is 1 μm, which is ∼260% of 1/μinel. On the other
hand, according to the linearity criterion as μBF × tmax = −ln
T(tmax) = 0.4–0.67, the maximum thickness tmax for the
linearity in unﬁltered BF-TEM images is only 40–67% of
1/μBF. The difference should be collateral evidence thatthe nonlinear intensity attenuation in unﬁltered BF-TEM
images is induced by the scattered-back electrons fromoutside of the OA; in contrast, energy-loss electrons haveno chance to come back into the zero-loss window to form aﬁltered image.

Inﬂuence of Fresnel fringes on 3D reconstructions
The remaining problem in the reconstructions at 1000 kV–
OA1 is the unnaturally bright rim of the ﬁber cross section in
Fig. 7b. It has been reported that edge-enhanced images intilt-series cause edge-enhanced 3D reconstructions [18]. In
our case, it is suspected that Fresnel fringes in the BF-TEM
images have induced the edge enhancement in the reconstructed volumes. As for the image in Fig. 3a, all images inthe tilt series were taken at the in-focus condition for theouter edges (P1 in Fig. 3a) of the CMCs. Therefore, the top
and bottom positions shown in Fig. 4b were at the out-offocus conditions of ∼1.8 μm. The Fresnel fringe contrast
around the ﬁber edges in the top and bottom positions should
be superimposed on the mass–thickness contrast, although
it is difﬁcult to see in Fig. 3a due to the overlap of the ﬁbers.
Because the Fresnel fringe contrast does not correspond to
the projection thickness of an object, it must cause errors in
the quantitative 3D reconstructions, particularly for samples
with intricate surface morphologies such as mesoporous
materials. Fresnel fringes are explained based on wave optics
as a blurring effect of the edges induced by lens defocusing.
In geometric optics, a corresponding simpliﬁed discussion of
the blurring db is presented in terms of the depth of ﬁeld DF =db/α [36]. According to the formula, decreasing the OA
radius α is effective for increasing the depth of ﬁeld toward
1.8 µm. In other words, the Fresnel fringe contrast for the outof focus of 1.8 µm can be reduced by using a smaller OA.
However, this solution enhances the nonlinear attenuation,

Downloaded from http://jmicro.oxfordjournals.org/ at Osaka Daigaku Ningen on October 1, 2014

In Fig. 5, the values of ln T at the maximum thickness tmax
are approximately −0.9 at 400 kV–OA1 and −2.4 at
1000 kV–OA4. The corresponding transmittance T(tmax) at
the maximum thickness is 0.4 and 0.09, respectively. The
threshold value for accurate 3D shape reconstructions
should exist somewhere between them. It is interesting that
this range is nearly coincident with the threshold transmittance of 0.1 using energy-ﬁltered images [14,17], despite thedifference of the contrast generation mechanism. The peculiar
coincidence may imply a type of universality for BF-TEM
image contrasts regardless of energy ﬁltering.
The common feature in Fig. 7c and d is the gradient of
the internal density from the center. This phenomenon is
known as the ‘cupping effect’ in the ﬁeld of X-ray tomography, and results from the nonlinearity induced by different absorption coefﬁcients for each X-ray energy [27–29].
In electron tomography, dips in the density and edge
enhancements have been reported to appear in the reconstructed volumes of convex dense objects, such as particles
and rods, when ADF-STEM images in the tilt series include
the nonlinearity or saturated intensity [18,19]. Our sample
has a more complicated 3D conﬁguration, i.e. a tubular
structure with inner concave surfaces. Figure 7c and dshows that edge enhancement does not occur for such innerconcave surfaces. Nevertheless, the reconstructed density inthe internal space is artiﬁcially higher, as if some lowdensity materials are present. This result indicates that one
should be careful, particularly when observing 3D structures with cavities or those that contain low-density materials in the interior, such as tubular, mesoporous andcore-shell structures.
Judging from the intercept values in Fig. 5b, the dataobtained at 600, 800 and 1000 kV exhibit the linear
attenuation, and thus they should bring the correct 3D
density reconstructions as shown in Fig. 7b. In Fig. 5b,ln T(tmax) is approximately −0.4 at 600 kV–OA1, whichcorresponds to T(tmax) = 0.67. Assuming the minimum transmittance as a good criterion for correct 3D density
reconstruction, the threshold T(tmax) should exist between
0.4 and 0.67. In other words, whenever two-thirds of theincident electrons remain inside the OA, the 3D density
reconstructions are achievable. This information should be
beneﬁcial for practical tomography experiments, because
the attainability of the 3D density reconstructions can beforeseen from a single BF-TEM image prior to tilt-seriesacquisition.
Based on the above criterion, the attenuation coefﬁcientsmeasured in Fig. 4a are discussed again. The number of electrons contributing to a zero-loss image is reduced by both of

353

354

as clariﬁed in the present study. The trade-off relationshipthus limits the maximum thickness for more complete density
reconstructions without edge enhancement. Clariﬁcation of
the optimal conditions and the maximum mass thickness
should therefore be the next target of research.

Conclusions

Acknowledgements
The authors are grateful to Mr M. Ohsaki in JEOL Ltd. for discussions about designing the sample holder and System In Frontier Inc.
for discussions on precise 3D reconstruction procedures. We also
thank Mr Y. Yamamoto and Dr C. Morita of HVEM laboratory in
Nagoya University for their assistance with the experiments. One of
the authors (N.T.) thanks Dr. S. Motojima for useful discussions.

References
1. Koguchi M, Kakibayashi H, Tsuneta R, Yamaoka M, Niino T,
Tanaka N, Kase K, Iwaki M (2001) Three-dimensional STEM
for observing nanostructures. J. Electron Microsc. 50: 235–241.

2. Midgley P A, Weyland M, Thomas J M, Johnson B F G (2001)
Z-Contrast tomography: a technique in three-dimensional nanostructural analysis based on Rutherford scattering. Chem.
Commun. 10: 907–908.
3. Yamasaki J, Tanaka N, Baba N, Kakibayashi H, Terasaki O(2004) Three-dimensional analysis of platinum supercrystals bytransmission electron microscopy and high-angle annular darkﬁeld scanning transmission electron microscopy observations.
Philos. Mag. 84: 2819–2828.
4. Sueda S, Yoshida K, Tanaka N (2010) Quantiﬁcation of metallic
nanoparticle morphology on TiO2 using HAADF-STEM tomography. Ultramicroscopy 110: 1120–1127.
5. Kimura K, Hata S, Matsumura S, Horiuchi T (2005) Dark-ﬁeld
transmission electron microscopy for a tilt series of ordering alloys:
toward electron tomography. J. Electron Microsc. 54: 373–377.
6. Barnard J S, Sharp J, Tong J R, Midgley P A (2006) Threedimensional analysis of dislocation networks in GaN using weakbeam dark-ﬁeld electron tomography. Philos. Mag. 86: 4901–4922.
7. Hata S, Miyazaki H, Miyazaki S, Mitsuhara M, Tanaka M,
Kaneko K, Higashida K, Ikeda K, Nakashima H, Matsumura S,
Barnard J S, Sharp J H, Midgley P A (2011) High-angle triple-axis
specimen holder for three-dimensional diffraction contrast imaging intransmission electron microscopy. Ultramicroscopy 111: 1168–1175.
8. Frank J, McEwen B F, Radermacher M, Turner J N, Rieder C L(1987) Three-dimensional tomographic reconstruction in highvoltage electron microscopy. J. Electron Microsc. Tech. 6: 193–205.
9. Barnard D P, Turner J N, Frank J, McEwen B F (1992) A 360°
single-axis tilt stage for the high-voltage electron microscope. J.
Microsc. 167: 39–48.
10. Tanaka N, Usukura J, Kusunoki M, Saito Y, Sasaki K, Tanji T,
Muto S, Arai S (2013) Development of an environmental high-voltage
electron microscope for reaction science. Microscopy 62: 205–215.
11. Jinnai H, Tsuchiya T, Motoki S, Kaneko T, Higuchi T,
Takahara A (2013) Transmission electron microtomography insoft materials. Microscopy 62: 243–258.
12. Koster A J, Ziese U, Verkleij A J, Janssen A H, de Jong K P(2000) Three-Dimensional transmission electron microscopy: anovel imaging and characterization technique with nanometerscale resolution for materials science. J. Phys. Chem. B 104:
9368–9370.
13. Klingstedt M, Miyasaka K, Kimura K, Gu D, Wan Y, Zhao D,
Terasaki O (2011) Advanced electron microscopy characterization for pore structure of mesoporous materials; a study of
FDU-16 and FDU-18. J. Mater. Chem. 21: 13664–13671.
14. Kaneko T, Nishioka H, Nishi T, Jinnai H (2005) Reduction ofanisotropic image resolution in transmission electron microtomography by use of quadrangular prism-shaped section. J. Electron
Microsc. 54: 437–444.
15. Ercius P, Weyland M, Muller D A, Gignac L M (2006) Threedimensional imaging of nanovoids in copper interconnects using
incoherent bright ﬁeld tomography. Appl. Phys. Lett. 88: 243116.
16. Kawase N, Kato M, Nishioka H, Jinnai H (2007) Transmission
electron microtomography without the ‘missing wedge’ for quantitative structural analysis. Ultramicroscopy 107: 8–15.
17. Kato M, Kawase N, Kaneko T, Toh S, Matsumura S, Jinnai H(2008) Maximum diameter of the rod-shaped specimen for transmission electron microtomography without the ‘missing wedge’.
Ultramicroscopy 108: 221–229.

Downloaded from http://jmicro.oxfordjournals.org/ at Osaka Daigaku Ningen on October 1, 2014

The nonlinear attenuation of the mass–thickness contrast in
BF-TEM images with increasing thickness and its inﬂuence
on 3D reconstructions were analyzed. It was clariﬁed thatthe nonlinearity was sensitively affected by the OA radius
and the acceleration voltage. In fact, using the largest OA
and 1000 kV electron beams, a CMC with a diameter of
3.7 µm was reconstructed so successfully that not only theshape but also the internal density was correctly reproduced. We also succeeded in rough estimations of the conditions that the minimum electron transmittance of 0.4 and0.67 are sufﬁcient for the 3D shape reconstructions and the3D density reconstructions, respectively. Based on theseresults, the quality of the reconstruction can be foreseenfrom a single BF-TEM image prior to the tilt-series acquisition. More precise determination of the threshold value will
be achieved by conducting additional tomography experiments under imaging conditions between 1000 kV–OA1
and 400 kV–OA1 with respect to the nonlinearity. It may
also be important to verify the criterion for non-crystalline
materials other than amorphous carbon and for samples
with multiple components.
Among some problems that must be solved to obtain
quantitative tomography of micron-sized materials, thepresent research focused on the nonlinear intensity attenuation in BF-TEM images. However, researchers must also
pay attention in actual tomography experiments to image
blurring related to chromatic aberration and multiple scatterings [25,26,37,38]. The next issue should be to establish
guidelines for selection of the proper OA size that balances
the inﬂuences of the nonlinear attenuation, Fresnel fringes
and the lateral precision.

Microscopy, 2014, Vol. 63, No. 5

Microscopy, 2014, Vol. 63, No. 5

29. Krumm M, Kasperl S, Franz M (2008) Reducing non-linear artifacts of multi-material objects in industrial 3D computed tomography. NDT&E Int. 41: 242–251.
30. Motojima S, Kawaguchi M, Nozaki K, Iwanaga H (1990)
Growth of regularly coiled carbon ﬁlaments by Ni catalyzed pyrolysis of acetylene, and their morphology and extension characteristics. Appl. Phys. Lett. 56: 321–323.
31. Kawaguchi M, Nozaki K, Motojima S, Iwanaga H (1992) A
growth mechanism of regularly coiled carbon ﬁbers throughacetylene pyrolysis. J. Cryst. Growth 118: 309–313.
32. Du J-H, Sun C, Bai S, Su G, Ying Z, Cheng H-M (2002) Microwave electromagnetic characteristics of a microcoiled carbon
ﬁbers/parafﬁn wax composite in Ku band. J. Mater. Res. 17:
1232–1236.
33. Yamamoto K, Hirayama T, Kusunoki M, Yang S, Motojima S(2006) Electron holographic observation of micro-magnetic ﬁelds
current-generated from single carbon coil. Ultramicroscopy 106:314–319.
34. Yoshimura K, Nakano K, Miyake T, Hishikawa Y, Kuzuya C,
Katsuno T, Motojima S (2007) Effect of compressive and tensile
strains on the electrical resistivity of carbon microcoil/silicone-rubber composites. Carbon 45: 1997–2003.
35. Hashimoto A, Shimojo M, Mitsuishi K, Takeguchi M (2009)
Three-dimensional imaging of carbon nanostructures by scanning confocal electron microscopy. J. Appl. Phys. 106: 086101.
36. Williams D B, Carter C B (1996) Transmission electron
microscopy, Chap. 6 (Plenum Press, New York).
37. Motoki S, Kaneko T, Aoyama Y, Nishioka H, Okura Y,
Kondo Y, Jinnai H (2010) Dependence of beam broadening on
detection angle in scanning transmission electron microtomography. J. Electron Microsc. 59: S45–S53.
38. Wang F, Zhang H-B, Cao M, Nishi R, Takaoka A (2011) Image
blurring of thick specimens due to MeV transmission electron
scattering: a Monte Carlo study. J. Electron Microsc. 60:
315–320.

Downloaded from http://jmicro.oxfordjournals.org/ at Osaka Daigaku Ningen on October 1, 201418. Saghi Z, Xu X, Möbus G (2008) Electron tomography of regularly shaped nanostructures under non-linear image acquisition.
J. Microsc. 232: 186–195.
19. Van den Broek W, Rosenauer A, Goris B, Martinez G T, Bals S,
Van Aert S, Van Dyck D (2012) Correction of non-linear thickness
effects in HAADF STEM electron tomography. Ultramicroscopy
116: 8–12.
20. Zhang H B, Zhang X L, Wang Y, Takaoka A (2007) Tomography experiment of an integrated circuit specimen using 3 MeV
electrons in the transmission electron microscope. Rev. Sci.
Instrum. 78: 013701.
21. Zhang H, Takaoka A, Miyauchi K (1998) A 360°-tilt specimen
holder for electron tomography in an ultrahigh-voltage electron
microscope. Rev. Sci. Instrum. 69: 4008–4009.
22. Kamino T, Yaguchi T, Konno M, Ohnishi T, Ishitani T (2004) A
method for multidirectional TEM observation of a speciﬁc site at
atomic resolution. J. Electron Microsc. 53: 583–588.
23. Reimer L, Kohl H (2008) Transmission electron microscopy, 5thed., Chap. 6 (Springer, New York).
24. Zhang H B, Wang F, Takaoka A (2007) Measurement of electrontransmission through tilted thick specimens with an ultrahigh
voltage electron microscope. J. Electron Microsc. 56: 51–55.
25. Wang F, Zhang H B, Cao M, Nishi R, Takaoka A (2010) Multiple scattering effects of MeV electrons in very thick amorphous
specimens. Ultramicroscopy 110: 259–268.
26. Oshima Y, Nishi R, Asayama K, Arakawa K, Yoshida K,
Sakata T, Taguchi E, Yasuda H (2013) Lorentzian-like image
blur of gold nanoparticles on thick amorphous silicon ﬁlms inultra-high-voltage transmission electron microscopy. Microscopy
62: 521–531.
27. Brooks R A, Di Chiro G (1976) Beam hardening in X-ray reconstructive tomography. Phys. Med. Biol. 21: 390–398.
28. Hammersberg P, Mangard M (1998) Correction for beam hardening artefacts in computerised tomography. J. X-Ray Sci.
Technol. 8: 75–93.

355