Hollow-cone dark-field transmission electron microscopy for grain ...

Hollow-cone dark-field transmission electron microscopy for grain size 

and dislocation-density quantification of nanocrystalline materials 

B. Yao 1,2 , H. Heinrich 2, 3 , K. Barmak 4 , K. R. Coffey 1,2 , Kyu Cho 5 , and Y.H. Sohn 1,2 

1 

Advanced Materials Processing and Analysis Center, University of Central Florida, FL 32816, USA 

2 

Department of Mechanical, Materials and Aerospace Engineering, University of Central Florida, Orlando, FL 32816, 

USA 

3 

Department of Physics, University of Central Florida, Orlando, FL 32816, USA 

4 

Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA 

5 

U.S. Army Research Laboratory, Weapons and Materials Research Directorate, Aberdeen Proving Ground, MD 21005, 

USA. 

Methodologies based on hollow-cone dark-field (HCDF) transmission electron microscopy (TEM) are described to study 

grain size and dislocation density in nanocrystalline materials. In order to accurately quantify the nanocrystalline grain 

size, high quality TEM images are required where each grain has adequate contrast to its neighbor grains. Conventional 

bright-field and dark-field TEM images often cannot provide this contrast for all the grains in the field of view. In this 

study, HCDF-TEM imaging, with a slightly convergent beam, as an improved technique was found to be suitable to form 

high contrast micrographs for nanocrystalline grain size quantification. A variant geometry of HCDF can also be used to 

examine the dislocation structure. Compared to the conventional two-beam approach to form weak-beam dark field 

(WBDF) TEM images for dislocation characterization, the method described in this study is readily applied to provide 

high-quality WBDF-TEM images, and it is suitable to study both samples with coarse grains and those with ultrafine 

nanocrystalline grains. Examples from both thin film and bulk nanostructured samples are provided. 

Keywords hollow-cone dark field; nanocrystalline; grain size; dislocation density 

1. Introduction 

Microscopy: Science, Technology, Applications and Education 

______________________________________________ 

A. Méndez-Vilas and J. Díaz (Eds.) 

Nanocrystalline materials with a grain size below ~100 nm exhibit a variety of unique mechanical, chemical, and 

physical properties that allow new applications and technologies [1-3]. For example, it has been reported that the yield 

stress and microhardness of nanocrystalline materials can be 2-10 times higher than the corresponding coarse-grained 

polycrystalline materials with the same chemical composition [4]. Electronic properties are also affected, as noted in 

recent work wherein the scattering of conduction electrons at the grain boundaries of nanometric Cu films has been 

shown to provide the most significant contribution to the resistivity increase in the films as thickness is reduced [5]. 

The interesting properties of nanocrystalline materials are due to their unique microstructure. The grain size and 

density of dislocations are two of the most important microstructural parameters. The grain size is directly correlated to 

the volume fraction of atoms at interfaces, which are responsible for many of the special properties of nanocrystalline 

materials [6]. The small grain size of nanometric materials may also induce other nano-scale effects [7]. The 

dislocation density strongly influences the strength and ductility of materials [8, 9]. It is known that most coarse-grain 

metals or alloys can be hardened by increasing the dislocation density. Simultaneously, an increase in dislocation 

density (e.g., through work hardening) typically results in a decrease of the ductility of materials, as it reduces the 

capacity for further dislocation movement and accumulation [8]. The small size of nano-grains, however, may limit the 

conventional operation (e.g., nucleation and growth) of dislocation sources, which results in a low ductility of 

nanocrystalline materials. Recently, it has been reported that partial dislocations can be generated at grain boundaries 

[9], which provides an approach to obtain nanocrystalline materials with both a high strength and plasticity. 

Although many methods are available for grain size and dislocation density characterization, transmission electron 

microscopy (TEM) is obviously one of the most powerful techniques. X-ray diffraction methods, based upon 

diffraction line width, may provide an “average grain size” and dislocation density. However, such measurements are 

sensitive to other factors that influence diffraction line width [10]. TEM-based grain size measurement techniques also 

allow the distribution of the grain sizes, and the grain shapes to be determined. Similarly, the dislocation density and 

distribution, the interaction of dislocations with precipitates or grain boundaries can be achieved from TEM 

micrographs, given that these features are imaged appropriately. 

In this paper, methodologies based on hollow-cone dark field (HCDF) TEM imaging for grain size and dislocation 

density characterization are described and demonstrated. The HCDF-TEM technique has been employed widely for 

precipitates or secondary phase identification [11, 12]; orientation mapping of nano- and micro- grains [13-15]; and 

phase quantification of materials with multiple phases [16]. In this paper, an introduction to a general HCDF-TEM 

technique will be given first (section 2). The balance of the paper will describe the use of this technique to examine the 

grain size (section 3) and dislocation density (section 4) of nanocrystalline materials. A comparison of the HCDF-TEM 

technique with other TEM techniques will also be presented. 

©FORMATEX 2010 1319


A. ______________________________________________ 

Méndez-Vilas and J. Díaz (Eds.) 

2. Introduction to the HCDF-TEM technique 

The HCDF-TEM imaging described in this work was conducted with a FEI/Tecnai F30 TEM/STEM microscope with a 

field emission gun operating at 300 keV. In the HCDF-TEM mode, the incident electron beam is tilted at a fixed angle 

with respect to the optical axis of the objective lens, and a rotation of the tilted beam forms a hollow-cone illumination. 

A ray diagram of the HCDF-TEM is shown in Fig. 1(a). On the viewing screen of the microscope, the diffraction 

pattern is deflected to allow a portion of a selected diffraction ring (e.g., 111 or 200) to coincide with a centered 

objective aperture, as is normally done in conventional centered dark-field imaging. HCDF-TEM images are formed as 

the position of the diffraction pattern over the objective aperture is circumferentially scanned as the electron beam is 

rotated at the fixed angle to the optical axis. The geometry of the rotation of the diffraction pattern over the aperture is 

illustrated in Fig. 1(b). The beam rotation can be conducted automatically via the microscope control software, or 

manually through a mechanical probe. As will be described in detail below, a continuous rotation of electron beam is 

necessary in order to form high-quality HCDF-TEM images for grain size quantification, where the grain-to-grain 

contrast of all grains in the field of view is concerned. For dislocation density characterization, however, the rotation is 

performed manually, and the HCDF-TEM images are not formed using the whole diffraction ring(s), but only from the 

centered diffraction vector g. 

(a) 

e beam 

Optical axis 

Rotation trace 

Specimen 

View screen 

Fig. 1 (a). The ray diagram of HCDF-TEM technique, and (b) The diffraction plane configuration for HCDF TEM imaging [17]. 

As reviewed previously, the HCDF-TEM techniques can be employed for a variety of applciations. The specific setup 

and operation procudures may vary significantly in different cases. This paper will focus on the application of 

HCDF-TEM to grain size and dislocation density characterization. Detailed descriptions of the techniques will be given 

in following sections. 

3. Grain size quantification of nanocrystalline materilas 

(b) 

TEM micrographs ideal for grains size quantification should have most (and ideally all) grains in the field of view 

having an adequately high contrast between adjacent grains such that the grain boundary network is clearly identifiable. 

The idea of using the HCDF technique to obtain high quality micrographs is to maximize the diffraction contrast for all 

imaged grains. Guided by this principle, the set-up of the HCDF imaging is modified as below. First of all, a 

continuous rotation of a slightly-convergent beam is used to form the image. A high rotation speed of electron beam 

(10 rotations per second) and a relative long CCD integration time (about 1 second) are typically used to maximize the 

number of intensity averaging scans over the annular region. A slightly-convergent beam for has been found to aid 

practical operation. Generally, HCDF imaging is normally done using a parallel (non-divergent) electron beam. 

However, the HCDF image formed using a parallel beam has a very low intensity and, thus, a long CCD integration 

time is required to form the images. The low intensity and the long integration time make it difficult to acquire high 

quality micrographs, because of sample drift and because of the inability to judge whether the image is in sharp focus. 

In order to increase the intensity of dark-field images, the beam is converged to about 40 mm in diameter on the 

viewing screen, which results in different semi-convergence angles at different magnifications. The beam convergence 

de-focuses the diffraction pattern at the diffraction plane (or the plane of the objective aperture) and introduces some 

uncertainty regarding which diffracted electrons are contributing to the HCDF images. Nevertheless, the imaging 

electrons are limited by the size of the objective aperture. The aperture size can then be optimized to provide the best 

image for grain identification. As in the normal practice for HCDF imaging, the electron beam tilt angle is adjusted (in 

our case, typically 0.3˚~0.4˚) so that the direct beam is just out of the objective aperture and does not contribute to the 

micrographs. It should be noted that even with this slightly convergent illumination, high quality HCDF micrographs 

formed with several diffraction rings still require a much longer CCD integration time (approximately a factor of ten 

increase) compared to that used for BF images. 

1320 ©FORMATEX 2010

Three micrographs, a HCDF, a BF, and a conventional DF of the same field of view in a 40 nm-thick Cu film 

annealed at 600˚C for 30 minutes are shown in Figs. 2 (a), (b), and (c), respectively. A description of the preparation of 

the Cu films and TEM samples can be found elsewhere [20]. The BF-TEM image has a low grain-to-grain contrast for 

most grains because of the uniformly high intensity of the transmitted beam. Although a few grains satisfying the 

Bragg diffraction condition can be identified, the weak diffraction contrast for most grains prevents identification of all 

but a few of the grain boundaries. Conventional DF-TEM images provide high contrast for the grains whose diffracted 

electrons pass through the objective aperture, but provide lower grain-to-grain contrast than BF imaging for the majority 

of the grains present. The HCDF-TEM technique described above uses one or more diffraction rings of polycrystalline 

samples (depending on the size of the objective aperture) to form the image. Consequently, most grains in the field of 

view can exhibit contrast relative to adjacent grains, as shown in Fig. 2(a), and this allows the boundaries between 

grains to be identified. In some cases, the grain boundaries themselves also show some contrast due to a different 

density, the absence of a long-range ordering of atomic arrangement, or the presence of impurities at the interface. The 

appearance of grain boundary contrast is also very helpful to the tracing work. 

(a) (b) (c) 


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Fig. 2. (a) A HCDF , (b) a BF, and (c) a conventional DF micrograph of the same field of view in a 40 nm-thick Cu film annealed at 

600˚C for 30 minutes. 

As reported elsewhere [20], the image quality of the HCDF-TEM micrographs can be influenced by many factors, 

and the key points are summarized here. The HCDF imaging puts stringent requirements on the microscope alignment 

and focusing in order to reveal the small (less than 50 nm) grains and grain boundaries clearly. The small extent of 

under-focus (e.g., ~100 nm) may not degrade the quality of BF-TEM images significantly. However, the same extent of 

under-focus may induce a significant image blurring for the HCDF-TEM imaging technique described here. This 

influence is typically more evident when smaller grains are examined. The image blurring is understood in the TEM as 

the defocus causing the image to shift in the beam tilt condition. A rotating tilted beam makes this effect even more 

evident. In order to acquire high quality HCDF images, it is generally required to conduct all alignments for highresolution 

TEM as a first step, followed by the specific alignments for the HCDF technique (i.e., the dynamical conical 

DF Pivot Point X, Y), and finally a very careful focusing. The size of the objective aperture, along with the beam tilt 

angle, controls the angular range of diffracted electrons that form the images and strongly influences the quality of 

micrographs. Generally, it is recommended to select three or four lower-index diffraction rings (e.g., from 111 to 222 

reflections of fcc phase) to form the image. The HCDF TEM micrograph using a small aperture can be very sharp, but 

some regions of the image show poor grain-to-grain contrast. On the other hand, using a large aperture provides strong 

grain-to-grain contrast for almost all of the grains, but the image is blurred and cannot be focused clearly. For a 

microscope operating at 300 keV, a beam tilt angle of 0.3˚~0.4˚ and an aperture in a diameter about 60 µm are 

appropriate. Another important factor influencing the image quality is from the material being examined, and the 

artefacts induced during the TEM sample preparation. Factors favouring a higher image quality include a high atomic 

number (thus high scattering ability) of the material being imaged, a columnar grain structure, large-angle grain 

boundaries, and the absence of texture, strain, and defects. Although various materials may yield different qualities of 

HCDF images, it has been demonstrated that HCDF technique consistently provide images with much higher quality 

than those by BF or DF techniques. The last factor to consider, and perhaps the most important one, is in regard to the 

quality of TEM samples. Like most other TEM techniques, the ideal TEM sample for HCDF imaging should be as thin 

as possible and free of artefacts. Unfortunately, most TEM sample preparation techniques, such as the electro-polishing 

[18], ion-milling [18], and focused ion beam milling [19], may induce some defects. Recently, a back-etch technique 

was able to provide TEM samples of thin films with a very clear surface [20, 21]. An example HCDF TEM image from 

a sample prepared using this back-etch technique is shown in Fig. 3(a), which can be compared to an image of a sample 

prepared by ion-milling (shown in Fig. 3 b) from the same film. 



A. ______________________________________________ 


(a) (b) 

Fig. 3. The HCDF TEM images from TEM samples prepared with (a) the back-etch technique, and (b) conventional low angle ionmilling 

method. The image from the ion-milled sample is typically less sharp and clear owing to ion-induced artefacts, which cause 

non-uniform contrast within each grain. The tiny dots as circled in Fig. (b) are not observed in Fig. (a). 

The quantification of grain size from the TEM micrographs is very critical. For some materials, the quantitative 

microstructural analysis of TEM micrographs to extract grain size information can be conducted through automated 

programs that identify and trace the grain boundaries [22]. For many other materials, however, the very complex 

contrast present as a result of defects, strain and thickness contours, or artefacts introduced during sample preparation, 

may preclude automated quantification and require instead manual tracing of the boundaries [5]. An example will be 

used here to describe the method of the manual tracing of grain boundaries and to quantify the grain size. First of all, a 

series of images tilted at different angles (e.g., -1.5°, 0°, 1.5°) of the same field of view are acquired, as shown in Fig. 

4(a), (b), and (c) respectively. These images were then enlarged and printed out on high-quality papers. By 

overlapping a transparency film on the printed images, the grain boundaries can be traced manually on the transparency, 

and the traced network of grain boundaries in this example was shown in Fig. 4(d). The traced images were then 

scanned to form digital images, which can be further processed and quantified conveniently by programs (e.g., Image J, 

Q Materials). Fig. 4(e) shows the processed image of Fig. 4(d). 

(a) (b) (c) 

100 nm 

100 nm 

(d) (e) 

Fig. 4. HCDF TEM images of the same field of view, but with slightly different sample tilt angles: (a) -1.5º; (b) 0º; and (c) 1.5º. 

Figures (d) and (e) show the traced grain boundary network and the processed digital image for grain size quantification. 

1322 ©FORMATEX 2010 

100 nm

4. Dislocation density quantification of nanocrystalline materilas 


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A variety of TEM techniques have been used to study dislocation structures [23], such as BF, weak-beam dark-field 

(WBDF), and high-resolution TEM (HR-TEM). The HR-TEM image has a limited field of view, and furthermore, it 

can only reveal dislocations on the imaged lattice planes. The BF-TEM imaging can show the strain contrast induced 

by the dislocations. Unfortunately, the dislocations exhibited by the BF images typically have an inherent low contrast 

(due to the intense background), and a low spatial resolution [23]. Consequently, it is not considered as a preferred 

technique for dislocation examination, especially for samples with a high dislocation density. 

The WBDF-TEM imaging has been considered to be one of the most suitable techniques to study dislocations and many 

other defects. The basic idea of the WBDF technique is simple: using a large value of excitation error (s) for a selected 

diffraction vector (g) gives a small effective excitation distance (ξeff) and hence a narrow image of most defects, since 

the width of a dislocation is related to ξeff /3 [24]. The conventional approach to form WBDF-TEM images is based on 

a two-beam condition, and the procedures to obtain this condition is: 

1) An adjustment of the specimen orientation to obtain a two-beam diffraction condition where only the direct beam 

and a single diffraction relfection (diffraction vector, g) are excited. Thus the diffraction pattern consists primarily of 

two bright spots. 

2) A tilt of the electron beam to bring the primary excited reflection, g, onto the optical axis and excit the higher 

order 3g reflection. This is called g (3 g) condition since the 3 g vector is now on the Ewald sphere and g vector is 

repositioned away from Ewald sphere by a displacement referred to as the excitation error. 

3) Finally, the objective aperture is inserted to select the centered reflection, g, to obtain the WBDF-TEM image. 

Fig. 5 shows an illustration of the geometry for the g (3 g) condition. The intensity of the g spot in the diffraction 

pattern will be quite weak, hence the terminology of weak-beam dark field imaging. 

Fig. 5. An illustration of the geometry of the twobeam 

condition g(3g) for WBDF-TEM images to 

reveal the dislocation structure. 

In practical operation, however, it is very challenging to utilize this WBDF technique to study dislocations in 

ultrafine nanocrystalline materials (e.g., grain size less than 100 nm in diameter). The technique requires that a 

diffraction pattern from a single grain be obtained and that the grain be aligned to meet the two-beam condition. This is 

not possible for many nanostructured materials of interest. For example, the nanocrystalline Al phase in a trimodal Al 

metal matrix composite has a grain size typically ranged from 30 to 100 nm [25]. At this grain size, it is practically 

impossible (limited by the size of selected-area aperture of the microscope) to obtain a selected area diffraction pattern 

(SADP) from an individual grain due to the overlap of grains and inclusion of surrounding grains in the selcted area. 

Convergent-beam electron diffraction (CBED) provides a possible solution for some materials, however, the input of 

high energy from the convergent beam into a small grain may easily change its defect structure [23]. Even when the 

diffraction pattern from an individual nanoscale grain can be obtained, the alignment of the nano-grain required to 

obtain a two-beam condition is much more challenging. A small mechanical drift during the sample tilt can easily 

change the field of view of the sample, especially when a convergent electron beam and high magnifications are used. 

Unfortunately, such a drift cannot be monitored since the microscopy is conducted in the diffraction mode to search for 

the two-beam condition. 

The HCDF-TEM technique provides a more effective approach to examine the dislocation structures of nano- and 

micro- grains. Based on the above description and Fig. 5, two necessities are required to obtain WBDF-TEM images 

showing the dislocation structure. First, the selected reflection g is on the optical axis. Second, the g vector has a 

relatively large positive excitation error. It can be understood from the geometry of the HCDF-TEM technique (as 

shown in Fig. 1b) that the first requirement, the diffraction vector g on the optical axis, is inherently satisfied, when the 

electron beam tilt angle is adjusted so that the centered objective aperture coincides with the selected diffraction ring g. 

Once such a beam tilt angle is obtained, changes in sample orientation to find grains with their g vectors having an 

appropriate excitation error do not change this condition. 

The procedure to obtain grain orientations with an appropriate positive excitation error, s, for a selected g vectors is 

straightforward. The basic principle is that the magnitude of the excitation error can be judged from the intensity of 

illuminated grains. For example, if the g vector is very close to the Ewald sphere (s close to 0), the corresponding grain 



A. ______________________________________________ 


is highly illuminated in the dark field image. On the other hand, if the vector is far away from the sphere (with a very 

large s), the grain is dark in the micrograph. In theory, the grains with a moderately large s should have an intermediate 

intensity, but practically, it is difficult to identify these grains simply by their intensities. Our approach is to proceed as 

follows: 

1) A dark field image is obtained by using a reflection (g vector) of interest. A brightly illuminated grain in the 

dark field image is chosen for dislocation imagine. The highly illuminated grain can be obtained by manually rotating 

the diffraction pattern and/or by tilting the samples. 

2)The sample is then tilted about both axis to maximize the brightness of the chosen grain in the dark field image. 

This establishes that the g vector for the grain of interst is on the Ewald sphere. 

3)Whenever highly illuminated grains are obtained (i.e., s close to 0), a small tilt of the specimen can increase the 

excitation error, either in a positive or negative direction, where a rotation of electron beam may be necessary when the 

diffraction spot is out of the objective aperture. The image can be recorded when the grain intensity is reduced, but the 

intensity may not be reduced to the extent that the defect structure is not observable. High-quality micrographs of the 

dislocation structure can only be obtained if the excitation error is in a positive direction. The sign of the excitation 

error can be judged by comparing the corresponding image qualities when samples are tilted in two different directions. 

The image with the positive excitation error will provide dislocation contrast and can be used for measurements of 

dislocation density. 

Fig. 6 illustrates the geometry between the electron beam, g vector, and the axis of sample tilt. With an arbitrary 

angle between the sample tilt axis and the g vector (i.e., the direction normal to the diffraction plane), a tilt of the 

sample may induce two effects. One is to change the relative angle between the diffraction plane (or g vector) and the 

optical axis on the vertical plane in Fig. 6, which effectively alter the sign and magnitude of excitation error s. The 

change of excitation error generally induces a smooth change of the illumination intensity of grains. The other effect is 

to change the orientation of g vector on the horizontal (or X-Y) plane, which causes the shift of diffraction spot around 

the corresponding diffraction ring. In HCDF-TEM technique, it is possible that the originally centered diffraction spot 

will move out of the objective aperture when the sample is tilted, which causes an abrupt change of illumination 

intensity, and a sharp contrast in the field of view can be observed. In this case, a slight adjustment of the rotation of 

the electron beam can easily bring the diffraction spot beck to the objective aperture under the hollow-cone TEM mode. 

It should be noted that all the procedures mentioned above are conducted under the image mode and judged by the 

illumination intensity in HCDF images, therefore, the small mechanical drift in the conventional two-beam approach is 

not a practical problem in the this technique. 

Fig. 6. The geometry relations between g-vector 

(lying in or very close to X-Y plane when 

excitation error is close to zero), optical axis of 

microscope, and alpha tilt axis. 

Compared to the conventional two-beam approach for WBDF-TEM images, the HCDF method presented in this 

paper is more readily employed, especially when ultrafine nanocrystalline grains are examined. Figs. 7(a) and (b) show 

the WBDF images of a small nanocrystallie Al material and a coarse grain Al, respectively, acquired using the HCDF 

technique described above. The sample used is a trimodal Al metal matrix composite, and the preparation of 

composites and TEM samples can be found elsewhere [20]. 

The dislocation density can be quantified in two different approaches from TEM micrographs. The first one is based 

on the dislocation length (in three dimensions, or 3D) per unit volume, which is also named as volume dislocation 

density. As the TEM micrographs are two-dimensional (2D) projected image, a factor should be included to transform 

the 2D length to 3D. Further, the thickness of the sample or the grains should be measured in order to determine the 

grain volume, which typically is not readily achieved. The other approach, also named as areal dislocation density, can 

be employed more conveniently, which is defined as the number of dislocations per unit area. The area of grains or 

interested regions can be measured conveniently. The number of dislocations is counted as half of the total interception 

point of dislocations with the two surface planes of the TEM samples. The point of the interception of dislocations with 

the perimeter of grains is not included in the sum. In both approaches, the invisibility criterion (i.e., dislocations 

invisible when g.b=0) is also a concern. For example, when g vector of Al {111} is used to form the micrographs, and 

b is typically in directions, only half of the dislocations can be revealed. 


(a) (b) 

100 nm 

Fig. 7. The WBDF micrographs showing the dislocation structure in (a) nanocrystalline Al phase and (b) coarse grain Al phase. The 

dislocations in nanocrystalline Al phase are formed through the cryomilling, while those in coarse grain Al phase are produced by the 

thermo-mechanical deformation process. 

5. Conclusions 

Methodologies based on HCDF-TEM are described to study grain size and dislocation density in nanocrystalline 

materials. It has been demonstrated that HCDF-TEM imaging, with a slightly convergent beam, as an improved 

technique was found to be suitable to form high contrast micrographs for nanocrystalline grain size quantification. 

Factors which may influence the quality of HCD-TEM micrographs are also reviewed briefly. A variant geometry of 

HCDF can also be used to examine dislocation structure. Compared to the conventional two-beam approach to form 

WBDF-TEM images for dislocation characterization, the method described in this study is more readily applied to 

provide high-quality WBDF-TEM images, and it is suitable to study both samples with coarse grains and those with 

ultrafine nanocrystalline grains. Examples from both thin film and bulk nanostructured samples are presented. 

Acknowledgements The grain size quantification work was supported by the Semiconductor Research Corporation, Task 

1292.008. Partial support from the MRSEC program of the NSF under DMR-0520425 is also acknowledged. The dislocation study 

was sponsored by U.S. Army Research Laboratory (ARL) and was accomplished under Cooperative Agreement W911NF-07-2- 

0073. The views and conclusions made in this document are those of the authors and should not be interpreted as representing the 

official policies, either expressed or implied, of ARL or the US Government. The US Government is authorized to reproduce and 

distribute reprints for Government purposes notwithstanding any copyright notation hereon. The use of the Materials 

Characterization Facility of the Advanced Materials Processing and Analysis Center at University of Central Florida is gratefully 

acknowledged. 

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Hollow-cone dark-field transmission electron microscopy for grain ...

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