Springer Handbook of Microscopy pp 2-2 | Cite as

# Electron Nanodiffraction

## Abstract

This chapter introduces the practice and theory of electron nanodiffraction. After a brief introduction, the chapter provides a comprehensive description of electron diffraction techniques and their use for nanodiffraction. This is followed by discussions on electron probe properties, electron energy filtering and electron diffraction data analysis. Throughout the chapter, we emphasize different electron nanoprobes that can be formed inside an electron microscope, from a focused beam to parallel illumination, and how these probes can be used to extract structural information from different materials. For this purpose, we outline the electron diffraction theories based on both kinematic approximation and dynamic diffraction, which serve as the basis for the interpretation of electron nanodiffraction patterns. The principles and applications of scanning electron nanodiffraction and coherent diffraction imaging are covered in detail with applications for orientation mapping, imaging strain, 3-D nanostructure determination, and study of defects.

## electron diffraction

nanostructure analysis convergent beam electron diffraction (CBED) diffractive imaging four-dimensional scanning transmission electron microscopy (4-D STEM) strain mapping atomic resolution tomographyHere, electron nanodiffraction refers to a set of electron diffraction techniques that enable structural analysis at nanoscale. Specifically, the techniques we discuss here extend the study of electron diffraction to nanostructures, *small* crystals, defects, and materials' microstructure. We show how such studies can be carried out quantitatively at spatial resolutions ranging from sub-Å to nm and in two-dimensional () projections to three-dimensional () reconstruction.

Many of the instrumental requirements for electron nanodiffraction are similar to those for analytical electron microscopy. In fact, modern analytical transmission electron microscopes ( s) provide special convergent beam electron diffraction ( ) or nanobeam modes, or both, together with the modes for low and high magnification imaging, diffraction, and energy-dispersive x-ray ( ). This development resulted partly from the fact that the requirements for EDX (large tilt, small probe, low contamination) exactly match those for nanodiffraction. Analytical TEMs designed for scanning transmission electron microscopy ( ) also feature a bright field emission gun ( ), an improved vacuum system, and the instrument stability required for electron nanodiffraction. Most importantly, there is a clear scientific merit here since microanalysis complements electron diffraction in analytical transmission electron microscopy (TEM) by providing both chemical and crystallographic information.

The predecessor of electron nanodiffraction is electron microdiffraction using CBED. The first CBED pattern was recorded by *Möllenstedt* as early as 1939 using a two-magnetic lenses setup [18.1]. *MacGillavry* first attempted structure factor measurement by using the two-beam theory of Blackman for dynamic electron diffraction [18.2]. The multibeam theory of electron dynamic diffraction has its origin in the Bloch wave method originally formulated by Hans Bethe in his PhD thesis. Electron nanodiffraction started with the development of field emission gun (FEG) in the 1970s, which led to the development of dedicated STEM. Electron nanodiffraction took advantage of the small and highly coherent electron beam in these instruments (see *Cowley*'s reviews [18.3, 18.4]). However, in these early STEMs, diffraction patterns were often recorded using TV cameras and on video tapes; the quality of the recorded diffraction patterns was poor, and the handling of video data was difficult. The development of array detectors, such as CCD cameras or imaging plates, enabled parallel recording of diffraction patterns and quantification of diffraction intensities over a large dynamic range that became widely available only in the 1990s [18.5, 18.6]. At about the same time, field emission instruments that combine STEM with TEM were developed. Electron energy filters, such as the in-column \(\Upomega\)-energy-filter , also became available, which allowed the inelastic background intensities from plasmon or higher electron energy losses, to be removed from recorded diffraction patterns with an energy resolution of a few eV [18.7]. These developments in electron diffraction hardware were accompanied by the development of efficient and accurate algorithms to simulate electron diffraction patterns [18.8] and modeling structures on a first-principle basis. By the late 1990s, many technical difficulties encountered in performing electron nanodiffraction using dedicated STEM [18.3] had been solved. Other more recent developments are time-resolved electron diffraction at the time resolution approaching femtoseconds [18.10, 18.11, 18.9], coherent nanoarea electron diffraction for the study of individual nanostructures [18.12] and electron diffraction using fast pixelated detectors or segmented detectors [18.13, 18.14]. Further developments of these techniques will significantly improve our ability to interrogate structures at high spatial and time resolution, which hitherto was not available before.

- 1.
Diffraction pattern indexing and mapping

- 2.
Convergent beam electron diffraction (CBED)

- 3.
Coherent electron nanodiffraction.

Some background on transmission electron microscopy is needed and introductory materials and theoretical background that are not covered here can be found in several books [18.15, 18.16, 18.17, 18.18, 18.8].

## 18.1 Electron Diffraction Techniques

### 18.1.1 Selected Area Electron Diffraction (SAED)

Selected area electron diffraction ( ) is performed by illuminating the sample with a large defocused electron beam. The diffraction pattern is recorded from a selected sample area by placing an aperture at the image plane of the objective lens, as shown in Fig. 18.1. This plane is conjugate to the sample. Only electron beams passing through this aperture contribute to the diffraction pattern seen by the next intermediate lens. The electron beams come from the sample area defined by the virtual image of the selected area aperture at the specimen level. For an ideal lens, with the aperture centered on the optical axis, a small area at the center of the observed area is selected. This area is much smaller than the size of the aperture because of the objective lens magnification. A TEM equipped with an imaging (not probe) aberration corrector comes close to providing an ideal objective lens. Without the corrector, rays belonging to diffracted beams are at an angle to the optical axis, and they are displaced away from the center because of the spherical aberration of the objective lens (\(C_{\mathrm{s}}\)). The displacement is proportional to \(C_{\mathrm{s}}\alpha^{3}\), where \(\alpha\) is twice the Bragg angle (Fig. 18.1). The smallest area that can be selected in SAED is thus limited by objective lens aberrations. This limitation is largely removed when using an electron microscope equipped with a TEM aberration corrector placed behind the objective lens.

SAED is a popular diffraction technique in TEM. The technique can be applied to study both crystalline and noncrystalline materials. A large area illumination is useful for recording diffraction patterns from polycrystalline samples or for averaging over a large volume (for example, a large number of nanoparticles). SAED can also be used for low-dose electron diffraction, which is required for studying radiation sensitive materials, such as organic molecules. For small area analysis, the nanoarea electron diffraction technique described next is more appropriate. Alternatively, an aberration corrected TEM coupled with a small aperture can be used for electron nanodiffraction. For example, *Morishita* et al demonstrated that coherent diffraction can be achieved from areas as small as \(\approx{}{\mathrm{10}}\,{\mathrm{nm}}\) using this technique [18.19], also see Fig. 18.2a,b.

### 18.1.2 Nanoarea Electron Diffraction ( ) and Nanobeam Diffraction ( )

NAED uses a nanometer-sized parallel beam, with the condenser/objective setup shown in Fig. 18.3a-d [18.12], together with the use of a small condenser aperture. An auxiliary condenser lens, called condenser minilens or CM, placed immediately above the condenser-objective lens is also employed [18.20]. The CM lens takes the crossover formed by the last condenser lens (CL ) and images it onto the front focal plane of the objective prefield lens, which then forms a parallel beam on the specimen. Adjustment to the parallel beam can be made by changing the CL and CM lens excitations; the CL lens moves the beam crossover closer or further away from the CM lens, thus changing the beam divergence angle seen by the CM lens. For a condenser aperture of \({\mathrm{10}}\,{\mathrm{\upmu{}m}}\) in diameter, the probe diameter is \(\approx{}{\mathrm{50}}\,{\mathrm{nm}}\) with an overall magnification factor of \(1/200\) in the JEOL 2010 or 2100 electron microscopes (JEOL, USA). The beam size is much smaller than what can be achieved using a selected area aperture. Diffraction patterns recorded in this mode are similar to the SAED patterns. For crystals, the diffraction pattern consists of sharp diffraction spots. The major difference is that the diffraction volume is defined directly by the electron probe in NAED, since most electrons illuminating the sample are recorded in the diffraction pattern. NAED in an FEG microscope also provides higher beam intensity than SAED (the probe current intensity using a \({\mathrm{10}}\,{\mathrm{\upmu{}m}}\) condenser II aperture in a JEOL 2010F is \(\approx{}{\mathrm{10^{5}}}\,{\mathrm{e/(s{\,}nm^{2})}}\)) [18.12]. The small beam size allows the selection of an individual nanostructure and reduction of the background in the electron diffraction pattern from the surrounding materials. Figure 18.4a,b shows an example. The diffraction pattern was recorded from a single-wall carbon nanotube encapsulated with C60 molecules using a \({\mathrm{25}}\,{\mathrm{nm}}\) diameter electron probe (Fig. 18.4a,ba). The diffraction lines marked by the arrows are from C60 molecules. Together, \(\approx 25\) C60 molecules were selected and contributed to diffraction.

A focused probe can be formed by weakening the CL lens and placing the crossover at the front focal plane of the CM lens. This results in a focused probe on the specimen, which is placed at the focal plane of the objective prefield lens. When using a small condenser aperture with a small convergence angle, the probe size becomes diffraction limited in an FEG TEM. The diffraction patterns recorded in this case consist of small disks (Fig. 18.13a-eb). This nanodiffraction technique was pioneered by *Cowley* [18.4], it is now called NBD.

### 18.1.3 Convergent-Beam Electron Diffraction ( )

CBED is recorded using a focused electron probe at the specimen. Compared to the diffraction techniques that we have discussed so far, CBED differs in terms of the beam convergence angle (\(\theta_{\mathrm{c}}\)) and the electron probe size. The convergence angle is several times larger than what is used in NBD, but it is still significantly smaller than the convergence angle used in an aberration corrected STEM. The convergence angle is largely determined by the size of the condenser aperture (CA ). The CA is considered to be conjugate to the diffraction pattern in CBED. Using an additional mini-lens placed above or in the objective prefield, it is also possible to vary the convergence angle by changing the strength of the mini-lens for CBED. In addition, in CBED performed using a thermionic electron source, the incident plane-wave components of the illumination are considered to be incoherently related.

The relatively large convergence angle used for CBED gives rise to transmitted and diffracted disks (Fig. 18.5); the size of the disk determines the range of excitation errors for each reflection (more in the Sect. 18.4.2 on the geometry of CBED). Thus, the convergence angle is a very important parameter in CBED. Its choice depends on the application. Along a zone axis, the ideal CBED disk size is twice the Bragg angle of the lowest order zero-order Laue zone ( ) reflection, in order to fill the diffraction space as nearly as possible with scattered rays. In an off-zone axis orientation, a large CBED disk can be used to extend the number of (high order Laue zone) lines recorded in the transmitted disk. As the desired convergence angle changes from one crystal to another or one application to another, a TEM designed for CBED provides a range of excitations of the CM lens so it can be used to vary the convergence angle as shown in Fig. 18.3a-d. The size of the CBED disk for a fixed CM lens excitation is determined by the condenser aperture size and the focal lengths of the probe-forming lenses. Experimentally, by having several condenser apertures from a few \(\mathrm{\upmu{}m}\) to several tens of \(\mathrm{\upmu{}m}\), it is possible to cover a range of convergence angles for many materials science applications.

If the CA is coherently illuminated using a field emission electron source, the electron probe diameter is less than the lattice spacing, when the coherent CBED disks are allowed to overlap, it then becomes possible to form a scanning transmission (STEM) lattice image. By observing this STEM lattice image, it thus becomes possible (in thin crystals) to stop the probe on the region at which a CBED pattern is required. Alternatively, electron diffraction can be recorded at each probe position for STEM (four-dimensional STEM, see Sect. 18.1.5). By these methods, it is quite possible to obtain CBED patterns from different regions within a single unit cell, and that these show different site symmetries, or alternatively, by averaging over one or several unit cells, to obtain their average symmetry. In order to obtain sufficient intensity from a probe of subnanometer dimensions, an instrument fitted with a field emission gun is needed for this type of work. For the analysis of large crystals, the benefit of a FEG is the improved plane-wave coherence at the specimen level. This also makes it sensitive to the contributions from defects in a real crystal. However, because of the small focused probe, the pattern has reduced contributions from thickness variations and bending under the probe.

For very thin crystals, the resulting patterns may be interpreted as electron holograms. Coherent CBED patterns formed with a very large illumination aperture have a special name, ronchigrams. The interpretation of ronchigrams is discussed in Chap. 2, on STEM, since these provide the simplest and most accurate method of aligning the instrument and measuring the optical constants of the probe-forming lens.

*blank disk*) method has been investigated [18.22], and it was found to have the following advantages:

- 1.
It allows use of the smallest electron beam diameter for solving true nanocrystal structures.

- 2.
Since the beam energy is spread out throughout the disks, the (000) disk intensity may be measured without saturating the detector, so that

*absolute*intensity measurements can be made, comparing the intensity of the zero-order beam with the Bragg intensities. - 3.
One has a test, which is independent of the (unknown) crystal structure, for the presence of unwanted multiple scattering, if the structure is known to be noncentrosymmetric. In that case, these CBED patterns will only be centrosymmetric (in accordance with Friedel's law) if the scattering is kinematic. (Friedel's law is violated in the presence of multiple scattering.)

*McKeown*and

*Spence*[18.22]. Here, a three-dimensional map of the crystal potential was obtained, including the positions of the oxygen atoms. To solve the phase problem , the remarkable

*charge-flipping*algorithm was used [18.22]. (The charge-flipping algorithm is described in Sect. 18.4.4,

*Phase Retrieval Algorithms*.)

### 18.1.4 Large-Angle Methods

Various instrumental techniques have been developed to obtain an angular view of a diffracted order that is greater than the Bragg angle. The earlier methods for doing this were reviewed by [18.23]. Such an angular expansion is required for space-group determination of crystals with a large unit cell, in which overlap of low orders may occur at such a small illumination angle \(\theta_{\mathrm{c}}\) that little or no rocking-curve structure can be seen within the orders. It has also been discovered that many narrow high-order reflections may be observed simultaneously using large-angle techniques.

Closely related to the large-angle methods are ronchigrams and shadow images, which are described in Chap. 2, however they differ according to the angular range over which the illumination is coherent. In this section, we deal only with *incoherent* conditions and the application of techniques used to prevent the overlap of orders.

The *Tanaka* or (large-angle convergent-beam electron diffraction) method [18.24] allows parallel detection of the entire wide-angle pattern and requires no instrumental modifications. The pattern is again, however, obtained from a rather large area of sample. A description of the method is given in [18.25]. Figure 18.7 shows the principle of the method, while Fig. 18.8 shows a pattern from (111) silicon taken at \({\mathrm{120}}\,{\mathrm{kV}}\) by this method. In Fig. 18.7, the CBED probe was focused on the object plane of the objective lens, while the sample was moved up by a distance \(\mathrm{d}S\), forming an image of the electron source in the plane of the selected area aperture. A source image is formed in every diffracted order, as shown. The aperture can then be used to isolate one source image and so prevent other diffracted beams from contributing to the image. Because the source images are small at the crossover, the illumination cone can be opened up to a semiangle that is larger than the Bragg angle. The tradeoff is the large area of sample illuminated by the out-of-focus probe at the sample. In addition, different regions of the sample contribute to different parts of the diffraction pattern. Patterns may be obtained with the probe focused either above or below the sample; the best results seem to be obtained with it below the sample (for TEM instruments).

An important finding is that the use of the smallest selected area aperture together with the largest permissible defocus minimizes the contribution of inelastic scattering to the pattern. This effect was studied in detail in [18.26]. A similar technique is used to image other diffracted orders. Here, the order of interest is brought onto the optical axis using the dark-field tilt controls.

For defects, the LACBED or CBED technique can characterize individual dislocations, stacking faults and interfaces ([18.30, 18.31, 18.32], also Fig. 18.9). For applications to surfaces and interfaces, and structure without three-dimensional periodicity, parallel-beam illumination with a very small beam convergence is required.

The LACBED techniques that we have described so far are designed to avoid the overlap of diffraction disks by recording the intensities of a single reflection. Many applications require the intensities of multiple reflections, which can be obtained using the large-angle rocking-beam electron diffraction ( ) technique described by *Koch* [18.33] (Fig. 18.10a,b). The principle of LARBED is similar to the double-rocking beam technique developed earlier [18.34, 18.35]. It works by rocking the incident beam over a certain angular range, while ensuring that the same selected area of the sample contributes to the diffraction pattern and the same diffracted beam stay on the detector. For every incident-beam direction in this angular scan the intensity of the transmitted beam or a diffracted beam is displayed on a video monitor. The differences are: 1) LARBED uses a partial descan to produce a small diffraction circle from the precession of the incident beam, and 2) the diffraction rings are recorded on a charged-coupled device (CCD) camera instead of a point detector as in LACBED.

### 18.1.5 Scanning Electron Nanodiffraction, Scanning CBED, and 4D-STEM

Using the deflection coils, scanning electron nanodiffraction ( ) or scanning CBED ( ) patterns can be recorded from an area of the sample for every probe position, to provide spatially resolved structural information. This can be done either using a TEM or STEM. SCBED performed in a STEM with overlapping disks is also known as 4-D STEM. Diffraction patterns are recorded using a 2-D digital detector, for example, a CCD camera. Compared to position averaged CBED ( ) (Sect. 18.1.7), which records one diffraction pattern over many probe positions, SEND or SCBED collects the full 4-D data, in the form of two spatial coordinates, \((x,y)\) in the real space and \((k_{x},k_{y})\) in the reciprocal space. The only difference between SEND and SCBED is the beam convergence angle (which is larger for SCBED). For this reason, we will focus simply on SEND in the following discussion.

When a pair of magnetic coils (Fig. 18.11) are arranged perpendicular to each other, they apply uniform forces on the beam electrons along the horizontal (\(x\) and \(y\)) directions. Four pairs can be used to shift or tilt the beam along any direction in the \(x\)–\(y\) plane. Two pairs make a set of deflection coils covering the \(x\) and \(y\) directions. Double deflection coils are placed below the CL lens and above the CM lens. They are used to provide beam shift, bright-field beam tilt, and dark-field beam tilt. When driven by an external scan generator, they are used to scan the probe in a raster over the specimen and to form STEM images by coupling the scan together with a detector. In electron diffraction mode, they can be configured in a number of ways for beam rocking, conical scan, as used in precession [18.36], and scanning electron nanodiffraction [18.37].

Two deflectors working in opposite senses are used to shift or tilt the beam; the individual deflector excitations are different for these two operations. The beam shift is used for SEND, and the beam tilt is used in double-rocking LACBED or precession electron diffraction ( ). Figure 18.12 compares the beam shift with the beam tilt using the double-deflection coils in a TEM with a condenser-objective lens. For simplicity, the CM and the objective prefield lenses are shown as a single lens above the specimen. Consider a ray along the optical axis. To shift this ray at the specimen, it must be first deflected away from, and then toward, the optical axis by the first and second deflectors successively. Finally, the beam must intersect the optical axis at the front focal plane of the lens above the specimen, which then brings it to the specimen running parallel to the optical axis. To shift the beam, we actually tilt the beam. For other rays in the beam, because of the small convergence angle, the same tilt is achieved, so they all converge to the same point on the specimen. The amount of beam shift is proportional to the tilt angle. To tilt the beam, it is first deflected away from the axis, and then back towards the optical axis in such a way that all rays in the beam converge to the same point on the front focal plane as undeflected rays, but now shifted laterally.

Scanning electron diffraction can be carried out by first selecting an area of interest, dividing this area into a number of pixels, placing the electron probe at each of these pixels, and recording the diffraction patterns at each pixel [18.37]. Data acquisition is automated using either dedicated hardware to synchronize the scan and diffraction pattern (NanoMegas SPRL, Brussels, Belgium) or by using computer control of the TEM and the electron camera. An implementation of SEND using the second approach was reported by [18.38], which involves the automation of TEM deflection coils and diffraction pattern acquisition using a custom script written in the DigitalMicrograph® (DM, Gatan Inc., Pleasanton, CA) script language. The electron microscope is controlled using the script by communicating with the host processor built into the TEM. This technique does not require additional hardware other than the computer and the electron detector that are already installed on the TEM. The main drawback is that the speed of acquisition is limited by the camera readout speed or the speed of beam deflection inside the TEM, whichever is slower.

In the method reported in [18.38], the electron beam scanning is performed in the TEM mode and carried out using the deflection coils to shift the beam under computer control. Two types of computer access to the TEM are used for the scanning process; the first retrieves the values of the illumination deflection coils and stores the values as real numbers \(x\) and \(y\), and the second shifts the electron beam by the amount \(x\) and \(y\). The \(x\) and \(y\) values, however, only refer to the setting of the deflection coils, which need to be calibrated into distances in nanometers. For this purpose, two scanning vectors are established along the vertical and horizontal directions. The calibration is carried out under a standard magnification in TEM mode. The reference value of \((x_{1},y_{1})\) is first obtained from the initial beam position. The electron beam is then horizontally shifted to position 2, and \((x_{2},y_{2})\) are obtained. Using the calibrated magnification, the distance (\(d\)) between 1 and 2 can be set to a fixed value. Then, the horizontal and vertical scanning vectors are calculated. Once calibrated, the electron beam can be shifted to a specific position by a combination of the two scanning vectors.

Once the 4-D dataset is collected using SEND, bright and dark-field STEM images can be obtained simultaneously from SEND in the simplest form of analysis by integrating the diffraction intensities of the direct beam and diffracted beams, respectively. This way, SEND works like STEM. A major distinction is that with the diffraction patterns recorded and stored, other information can be extracted offline to form images, beyond the simple integrated intensities. For example, diffraction patterns can be indexed and analyzed for orientation and phase mapping (Sect. 18.4.1, *Orientation Mapping*). This analysis can be done at nm resolution, which is unique to transmission electron diffraction. This last option is simply not available using the fixed, STEM, detectors. The tradeoff here, of course, is that one will be dealing with a far more complex, and larger, data set.

- 1.
An annular area between the direct beam and the first ring (marked as 1)

- 2.
The second ring (marked as 2)

- 3.
The remaining area of the third ring, akin to the use of an annular dark field ( ) detector in STEM.

There are also major benefits in reducing radiation damage by using low-dose SEND to study radiation sensitive materials, including organic molecules. The work described in [18.39] showed that electron images recorded using illumination spots of \(100{-}200\,{\mathrm{nm}}\) from thin paraffin crystals and purple membrane improve the image contrast by a factor of \(3{-}5\) compared to electron images taken with a large illumination spot of \({\mathrm{3}}\,{\mathrm{\upmu{}m}}\). The improvement in image contrast was attributed to the reduced beam damage induced by specimen movement. In SEND, the beam damage is limited to only area of the specimen illuminated by the electron beam, and thus each diffraction pattern is recorded under nearly identical specimen conditions. The new *direct electron* detectors take advantage of this effect also, by summing many very brief exposures for which the effects of beam-induced motion are corrected during data merging.

### 18.1.6 Precession Electron Diffraction

Precession electron diffraction ( ) is a technique pioneered by *Vincent* and *Midgley* [18.36]. The principle of the method is illustrated in Fig. 18.14, in which, for simplicity, we omitted the CM and condenser-objective lenses above and below the specimen. In PED, the incident electron beam is made to rotate around the microscope's optical axis, maintaining a constant angle – the *precession angle*, by using beam deflectors [18.40]. To compensate for motion of the diffracted beams as the incident beam rotates, the outgoing beams are deflected back using the deflectors below the specimen. The technique is similar to the double-rocking technique we discussed for the recording of LACBED patterns [18.35], in which case the beam is made to scan over a rectangular area instead of precession around a circle. By recording electron diffraction patterns with the incident electron beam in precession, PED is able to provide the electron diffraction intensity integrated in angles across the Bragg condition for many reflections, provided that the recording time is much longer than the time it takes for one precession. Compared to CBED, which records the diffraction intensity for every incident beam direction, PED records one intensity integrated over the precession angle in a way similar to the rotational method in x-ray diffraction. It may be shown that this angular integration reduces the effects of multiple scattering, as first discussed by *Blackman* [18.42] and tested experimentally by *Horstmann* and *Meyer* [18.43]. Figure 18.15a-f shows an example. The sample is coesite, which is a high-pressure polymorph of silica with a monoclinic symmetry, space group \(C2/c\). The diffraction patterns, one without and one with precession, were taken on two sides of a twinned crystal. The patterns are only distinguishable using precession. Especially, the kinematically forbidden 001 reflection (\(00h\) with \(h\) odd) is not visible in the precessed pattern (arrowed in Fig. 18.15a-fc). The intensity of these reflections is quite strong in the conventional pattern; their absence in the precession pattern indicates a more kinematic-like behavior for the diffraction intensity [18.41]. This can be understood if we imagine that there is one extinction distance (in two-beam theory) associated with every point (every excitation error) within a CBED disk. By integrating over many such points, the precession signal averages over many extinction distances, and so smoothes out the oscillations with thickness due to the Pendellösung effect [18.44]. Because of this unique feature, PED has found major applications in electron crystallography for solving crystal structures.

PED is implemented by driving the \(x\) and \(y\) deflection coils before and after the specimen synchronously using the oscillating sine wave obtained from a signal generator, which is phase shifted and amplitude adjusted for the \(x\) and \(y\) scan drivers. The same waveforms are used to drive the coils below the specimen. This is schematically illustrated in Fig. 18.14. The result after careful adjustments is that, at the lower part of the beam deflector coils, the incident beam scans sequentially around a circle, which is then brought back to the specimen ideally to a fixed point so the rotating incident beam form a cone of a constant angle. Thus, a focused beam should stay focused in PED and sharp diffraction spots should stay similarly sharp.

### 18.1.7 Selected Area Diffraction in STEM

The drawback of performing SAED in a conventional TEM, where the objective lens spherical aberration limits the selected area to about \({\mathrm{100}}\,{\mathrm{nm}}\) or more, can be largely avoided by performing electron nanodiffraction in a STEM. There are several ways to perform selected area electron diffraction in a STEM. Sharp diffraction spots can be obtained by using the objective prefield lens to form a small parallel probe on the specimen. The diameter of the region of the specimen with near-parallel illumination depends on the diameter of the condenser aperture. Using a small aperture (\({\mathrm{10}}\,{\mathrm{\upmu{}m}}\) or less), the illumination may be as small as a few tens of nanometers, and diffraction pattern spots are then as sharp as those obtained by a parallel beam in a TEM. For applications where sharp diffraction spots are not so critical, such as phase identification or orientation mapping, a focused probe can be used with correspondingly higher spatial resolution.

- 1.
By positioning the electron probe at specific specimen positions, selected based on the STEM image.

- 2.
By applying a small, fast scan of the beam during the recording of the pattern (\({\mathrm{0.1}}\,{\mathrm{s}}\) exposure time or longer). Then, the area giving rise to the diffraction pattern can be increased significantly beyond the diameter of the electron probe [18.4].

- 3.
By recording scanning electron nanodiffraction patterns, which will be the subject of the next section.

Unlike SAED performed in TEM, the beam convergence angle is separately controlled from the selected area for electron diffraction in STEM. Because of this, some unique applications can be made. One is to acquire diffraction patterns over a small rectangular area defined by the STEM scan coils, which has special applications in atomic resolution STEM. The condenser aperture is coherently illuminated, so that large overlapping CBED disks interfere. The interference pattern changes sensitively as the electron probe moves from one atomic column to another, contributing to the image contrast observed in bright-field STEM [18.46]. Interpretation of coherent CBED (or *coherent nanodiffraction*) patterns, however, is complicated because we need to know the exact probe position as well as the phase of electron waves, including the phase from lens aberrations and electron multiple scattering. In this sense, the interpretation of these patterns is exactly as complicated as the interpretation of (high resolution electron microscopy) images. These patterns do, however, reveal the local point symmetry of the crystal as reckoned about the center of the beam, and this effect has been used to locate the STEM probe on particular atoms for collection of (electron energy loss spectroscopy) spectra [18.47]. This method has been used to determine the atomic structure, and to classify, the anti-phase domains that occur in alloys of CuAu [18.48], and has been reviewed by *Cowley* and *Spence* [18.49].

By averaging over a region of specimen, the PACBED removes all the interference between overlapping CBED disks [18.45, 18.50]. As the example in Fig. 18.16a-c shows, the patterns show a remarkable resemblance to CBED patterns recorded with an incoherent probe. In an aberration corrected STEM, the electron probe can be smaller than \({\mathrm{1}}\,{\mathrm{\AA{}}}\). The smallest specimen region that can be scanned in order to fully remove the coherence effect is a unit cell. The actual volume probed in a PACBED experiment depends on electron probe propagation. Since the electrons are no longer confined to a single atomic column as in a channeling situation, the actual volume is larger than the region scanned by the electron probe. Nonetheless, PACBED has the highest spatial resolution among all diffraction techniques for probing structure on the scale of the unit cell.

A major application of PACBED is the determination of crystal thickness for quantitative analysis of STEM image contrast. This technique when combined with quantitative techniques described in later chapters could be used to study local symmetry, polarization, and crystal stoichiometry.

## 18.2 Electron Probes

### 18.2.1 Probe Formation

*incoherent*illumination. Under

*incoherent*conditions (i. e., the electron lateral coherence length is much smaller than the diameter of the condenser aperture) the total probe diameter \(d_{0}\) of a focused probe is given approximately at Gaussian focus by adding in quadrature the various contributions to \(d_{0}\). Thus

The smallest probe under the incoherent condition is obtained by minimizing all the quantities in (18.2). The \(d_{\mathrm{s}}\) can be made smaller than \(d_{\mathrm{d}}\) and \(d_{\text{sa}}\) by combining a small physical source with large demagnification. Then the probe formation becomes *diffraction limited* and the illumination necessarily coherent. Then detailed computations are required for the probe shape for particular values of the lens aberrations, the defocus \(\Updelta f\), \(\lambda\), and \(\theta_{\mathrm{c}}\).

A focused electron probe on the sample is formed by placing the electron beam crossover far away from the front focal plane of the objective lens. This gives a demagnified, sharp, electron source image on the sample with magnification \(M\ll 1\). The size of the electron probe, in this case, is largely determined by the objective lens resolution function \(T(x,y)\). In reciprocal space, the demagnified electron source has a broad, spherical wave-like spectrum of wave vectors.

*most compact*probe

*parallel-beam*diffraction. Meanwhile, the sample remains at the back focal plane of the objective prefield lens, far away from the electron source image. Thus, the electron beam seen at the specimen level is a defocused image of the source. This large underfocus must be included as a part of the lens aberration function in (18.5) [18.12]. To demonstrate this, we assume a Gaussian distribution for the magnified electron source after the objective prefield lens (with a magnification of \(M\))

Figure 18.18 shows an example of a small focused probe formed inside a probe \(C_{\mathrm{s}}\)-corrected FEI Titan microscope operated at \({\mathrm{300}}\,{\mathrm{kV}}\) using a condenser aperture of \({\mathrm{50}}\,{\mathrm{\upmu{}m}}\) in diameter. The microscope was operated in the so-called \(\upmu{}\)-probe TEM scan mode, in which the probe corrector was operated like an additional condenser lens. The probe recorded has a FWHM of \({\mathrm{2.7}}\,{\mathrm{nm}}\). Diffraction patterns recorded using this probe consist of small diffraction disks with a convergence semi-angle of \({\mathrm{0.37}}\,{\mathrm{mrad}}\), according to (18.15).

### 18.2.2 Probe Current

### 18.2.3 Probe Coherence and Coherent Current

- 1.
If \(2L\ll 2R_{\mathrm{a}}\), the illumination aperture can be considered to be incoherently filled and so treated as an ideally incoherent effective source. This is the situation for conventional TEM systems using a tungsten or \(\mathrm{LaB_{6}}\) source under most operating conditions. Then CA can be treated as an ideally incoherent source, within which each point acts as a statistically independent emitter of electrons. (A useful exercise is to calculate \(L\) for a \(\mathrm{LaB_{6}}\) source operating at the smallest probe size.) The probe formed further downstream by this incoherent source, filling the illumination aperture, will then be partially coherent.

- 2.
If \(2L> 2R_{\mathrm{a}}\), the illumination aperture is coherently filled, and the radiation can be considered to originate from a point source. Then the entire optical system beyond CA is filled with perfectly coherent radiation and the probe may be treated as perfectly coherent. This is often a good approximation for field emission gun (FEG) instruments. With a focused probe, the sample is then illuminated by an aberrated, converging spherical wave.

Note that in forming a focused probe, the source is imaged onto the sample by the probe-forming lens, while the illumination aperture CA (called the *objective aperture* on STEM instruments) is imaged onto the detector in STEM by the objective lens. CA subtends a semiangle \(\theta_{\mathrm{c}}\), at the sample, while the geometrical electron source image (of diameter \(d_{\mathrm{s}}\)) subtends a semiconvergence angle \(\alpha\) at CA. Thus, a measurement of the coherence can be made by examining interference in overlapping CBED disks obtained with a crystal (Chaps. 2 and 13). In this chapter, we shall refer loosely to *coherent CBED* as the case \(2L\gg 2\textit{R}_{\mathrm{a}}\), and *incoherent CBED* as the case \(2L\ll 2\textit{R}_{\mathrm{a}}\), noting that these labels refer to the coherence conditions in the illumination aperture, and that in the second case the probe itself is partially coherent.

*reduced brightness*\(\beta_{\mathrm{s}}\uplambda^{2}\) of the electron source. The electron source brightness increases linearly with accelerating voltage \(\Phi\), while the electron wavelength is inversely proportional to the square root of the accelerating voltage. Thus, it can be concluded that both the reduced brightness and the coherent current are a constant property of the emitter, independent of the electron wavelength or accelerating voltage.

## 18.3 Energy Filtering

Removal of the inelastic background is an option when using an electron microscope equipped with an electron energy filter. The purpose of the filter is to remove from diffraction patterns all those electrons which, on traversing the sample, lose more than a few electron volts in energy and contribute to the background intensity. The important inelastic processes are phonon or plasmon scattering and single-electron excitation. (For a review of energy-loss processes in electron microscopy, see [18.55, 18.57].) Phonon scattering involves relatively large inelastic scattering angles, but very small energy losses (perhaps \({\mathrm{30}}\,{\mathrm{meV}}\)). These are not excluded by elastic filtering. Plasmon losses involve larger energies (about \({\mathrm{15}}\,{\mathrm{eV}}\)) and small scattering angles. Plasmon excitations or higher energy losses can be filtered out by dispersing the electrons according their energies using the magnetic or electrostatic fields inside an electron energy filter and using a slit equivalent to a few eV in width around the elastic (zero-loss) electron beam. The advantages of zero-loss energy filtering are outlined in more detail below in the discussion of imaging filters, however a glance at Fig. 18.22 will indicate the improvement in the quality of data to be expected. This improvement affects all the techniques discussed here, especially for quantitative CBED. *Duval* et al [18.58] was the first to demonstrate that, by placing the diffraction pattern at the object plane of the imaging filter, most of the background intensities disappeared, especially at small scattering angles. The other major application of energy-filtering is electron spectroscopic imaging ( ) used for composition mapping [18.7].

- 1.
The dramatic reduction in background significantly improves the contrast of fine high-order Laue zone ( ) lines and other features and thus enables the measurement of strain and determination of symmetry at greater accuracy.

- 2.
It enables the study of diffuse scattering whether it is thermal or comes from defects or from modulations of the crystal structure. In relatively thick samples, the diffuse scattering is buried by the inelastic background if elastic energy filtering is not used.

- 3.
Filtering allows much thicker crystals to be examined without incurring the penalty of radiation damage, which would result if higher accelerating voltages were used.

- 4.
The use of greater thickness (without background) for the study of defects. This is new information, which was not previously extractable due to the presence of the background.

There are several types of in-column energy filters that are named according to the shape of the electron path, such as the \(\Upomega\)-, \(\upalpha\)-, or \(\upgamma\)-energy filters. The other is the postcolumn Gatan imaging filter (GIF ). The in-column filter is placed between the intermediate and the projector lenses of the TEM and can be used in combination with all forms of electron detectors. The GIF is integrated with an electron camera and placed below the camera chamber. Its use for electron diffraction typically requires switching the TEM to a special low camera-length setting.

The optics of an imaging energy filter is illustrated in Fig. 18.21. While the details can differ, all energy filters have the optical elements of entrance image plane, achromatic image plane (where a focused image is formed without the separation of *color* or energy) and an energy dispersion plane, where the energy selection slit is placed. The intermediate and projector lenses here refer to the lenses immediately above and below the energy filter, which are part of the intermediate and projection lens systems in a TEM for the in-column energy filters. In GIF, they are replaced by focusing coils and additional multipoles. The function of the intermediate lens is to transfer the image or diffraction pattern to the image entrance plane, while the projection lens looks at the achromatic image plane or the energy dispersion plane in the imaging and diffraction mode or spectroscopic mode, respectively. A diffraction pattern is formed at the energy dispersion plane when we have an image at the achromatic image plane, or, in reverse, an image is formed at the energy dispersion plane when we have a diffraction pattern at the achromatic image plane.

Figure 18.22 shows the striking improvement that results in the quality of CBED patterns from the use of a \(\Upomega\)-energy filter, even when film recording is used. These patterns were recorded on the Zeiss Omega model 912 TEM-STEM. The exposure time used in Fig. 18.22 was \({\mathrm{1}}\,{\mathrm{s}}\) for the unfiltered recording and \({\mathrm{3}}\,{\mathrm{s}}\) for the filtered pattern. These times were arranged to produce an approximately equal optical density on the film, so that a valid comparison could be made. Before the imaging filters became widely available, energy filtering was performed by scanning the electron image or diffraction pattern over an EELS system with a point or array detector, by measuring the zero-loss peak intensity. The image recording time for a \({\mathrm{10^{3}}}\times{\mathrm{10^{3}}}\) pixel image using the Omega system is at least \(\mathrm{1000}\) times less than that required by the scanned readout system, for the same dose. (This assumes a parallel detector for the EELS capable of one-dimensional ( ) imaging.) The advantages of the Omega filter increase rapidly with the number of pixels. Since the experimental observation in Fig. 18.22 is that the background between the Bragg reflections is almost entirely removed by elastic filtering, we must conclude that this background is due to multiple, coupled phonon, and plasmon scattering. The phonon scattering events provide the large angular change, and the associated plasmon losses then allow these electrons to be removed by elastic filtering. This interpretation is consistent with the relatively large thickness used (\(t={\mathrm{270}}\,{\mathrm{nm}}\)).

For CBED, an optimum sample thickness exists with energy-filtered data. For very thin crystals, there is little inelastic scattering and so no requirement for filtering, however the CBED disks show no useful contrast variation. At very large thickness, all scattering is inelastic, and no elastic signal can be recorded. In the simplest model [18.60], the thickness dependence of, for example, the plasmon-loss electrons is given by the product of the multiply-scattered Bragg-beam intensity with an appropriate term of the Poisson distribution.

## 18.4 Diffraction Analysis

- 1.
Broad halos for amorphous materials or liquids

- 2.
Multiple concentric rings for powder samples

- 3.
Sharp diffraction spots (spot diffraction pattern)

- 4.
Disks for CBED.

- 1.
The diffraction peak position can be used to measure the \(d\)-spacing of individual reflections. The combination of diffraction peak positions and their indexing can be used to determine the crystal lattice, its repeating unit cell, cell parameters, and cell orientation.

- 2.
Diffraction pattern symmetry recorded in CBED and the dynamic extinction in the form of Gjonnes–Moodie lines can be used to determine the crystal symmetry or the lack of symmetry.

- 3.
Diffraction pattern indexing, when it is done together with SEND, can be used to determine nanograins, their crystallographic orientation, and distribution.

- 4.
The change in diffraction peak intensity and the diffuse scattering around the Bragg peaks can be used to identify structural defects.

- 5.
For very thin samples, the Fourier transform of the diffraction pattern gives the projected interatomic distances. For nanocrystalline or amorphous structures, the Fourier transform of the radial diffraction intensity gives the pair distances and their distribution. The diffraction patterns can also be used to obtain information about medium-range ordering.

- 6.
Electron diffraction intensity can be used to determine the atomic positions. In cases where multiple scattering effects in the measured diffraction intensities are strong, multiple scattering must be included in order to determine the atomic structure.

- 7.
Accurate structure factor measurement from diffraction intensities can be used to determine the atomic thermal vibrations (the Debye–Waller factors), and crystal potential, or charge density.

### 18.4.1 Crystal Diffraction Pattern Indexing and Orientation Mapping

#### The Geometry and Kinematical Intensity of Spot Diffraction Patterns

Here, we first provide a general description of spot diffraction pattern geometry and intensity and the relationship between the crystallographic and experimental coordinates.

*Pendellösung*or thickness fringe oscillations. The kinematical result is equal to the two-beam theory in the limit of \(|\omega|=|S_{g}|\xi_{g}\gg 1\) or \(|S_{g}|\gg 1/\xi_{g}\) or at very small thickness with \(|S_{g}|\approx 0\) and

*Automated Indexing of Electron Diffraction Patterns*). For this purpose, only an estimate of the diffraction peak intensity is needed. Often, an ad hoc formula is used to assign the spot peak intensity. The formula below is an example

- 1.
Proximity to the Ewald sphere with \(|S_{g}|<S_{\max}\)

- 2.
Length of \(g\) with \(g<g_{\max}\)

- 3.
The dimensionless parameter \(\omega=S_{g}\xi_{g}\) with \(\omega<\omega_{\max}\).

#### Kikuchi Lines and HOLZ Lines

Kikuchi lines are produced by Bragg diffraction of electron diffuse scattering produced by inelastic scattering and appear as background intensities in the diffraction pattern. The average electron energy loss is small, and the wavelength of the inelastic background is approximately the same as the incident electron beam. High-order Laue zone (HOLZ) lines are sharp lines observed in the CBED disks. They are produced by Bragg diffraction of lattice planes of high-order reflections. The rapid increase in the excitation error of a high-order reflection away from the Bragg condition results in a rapid decrease in the diffraction intensity. The maximum diffraction intensity occurs at the Bragg condition under the kinematic approximation, which appears as a straight line within the CBED disk. Since Kikuchi lines also mark the Bragg conditions, Kikuchi lines superimpose on HOLZ lines inside the disk and continue outside the disk. A major difference between the two is the contrast. Kikuchi lines are mostly observed for low and medium-order reflections, while HOLZ lines can be observed for high-order reflections.

#### Automated Indexing of Electron Diffraction Patterns

With the development of scanning-based electron diffraction techniques, a large number of diffraction patterns ( s) can be acquired over a short time. For example, \(\mathrm{1000}\) diffraction patterns can be acquired within minutes using the SEND technique described in Sect. 18.1.5, *Scanning Electron Nanodiffraction and Scanning CBED*. Indexing of such large diffraction data sets requires a fast and robust approach to DP indexing.

Various schemes of automatic indexing based on the comparison of experimental and calculated diffraction patterns have been proposed and developed for x-ray and electron diffraction [18.62, 18.63, 18.64, 18.65, 18.66]. For electron diffraction, the most successful approach so far is based on the template matching algorithm to compare the acquired DPs to precalculated ones [18.67]. In template matching, the crystal orientation (and phase) is determined from the best fit, which is identified among a large number of precalculated diffraction pattern templates. The template-based approach has also demonstrated some success of indexing superimposed diffraction patterns that were recorded from grain, or twin, boundaries [18.67].

In what follows, we will describe the normalized cross-correlation ( ) algorithm based on the comparison of an experiment DP with a set of simulated DPs and by searching for the best match [18.66, 18.68]. The NCC algorithm was initially developed by *Rauch* et al [18.66, 18.69] and improved by *Wu* and *Zaefferer* [18.68] and *Meng* and *Zuo* [18.70].

*Meng*and

*Zuo*[18.70] are:

- 1.
Determine the beam center position and camera length (magnification) of the experimental DP.

- 2.
Calculate the circular projection of diffraction intensities in the experimental DP. Intensities of all pixels located at one polar angle are integrated to generate a 1-D circular profile.

- 3.
Calculate the radial projection of diffraction intensities in the experimental DP. Intensities of all pixels on one radius are integrated to generate a 1-D radial profile.

- 4.
Calculate the circular and radial projections for all simulated DPs based on the experimental camera length.

- 5.
Compare the circular and radial profiles of the experimental DP with those of the simulated ones using the 1-D NCC. Record the pattern rotation angle for each simulated DP.

- 6.
Compare the experimental DP with the simulated DPs using the radial and circular profiles. Select an amount of simulated DPs that are most similar to the experimental ones.

- 7.
Compare the experimental DP with the selected simulated DPs using the direct 2-D NCC [18.71].

Circular and radial profiles calculated in steps (4) and (5) can be used achieve a faster indexing result. Since a 1-D profile contains a much smaller amount of pixels than the whole DP, the calculation of the correlation coefficient of circular and radial profiles is significantly faster than comparing the whole DPs. Thus, a comparison using the circular and radial profiles can be used to select a small number of simulated patterns. A final NCC comparison is executed with the selected patterns. Figure 18.24 shows the indexing result of an experimental DP.

The DP center plays a crucial role in NCC-based algorithms. The most accurate way to determine the DP center is to use the 2-D lattice observed in an experimental pattern selected for calibration. The 2-D lattice determination involves fitting two lattice vectors and the origin of the lattice to match the calibration pattern. The fitting provides an accurate determination of the DP center position. If no well-defined lattice is observed, the disk center of the direct beam can be simply taken as the DP center. The DP magnification (camera length) is utilized to scale the simulated patterns to match the experimental patterns. The DP magnification can be calibrated using a standard sample such as Si.

The significant challenges for achieving reliable diffraction pattern indexing results are the quality of experimentally acquired DPs, which is poor when the crystals are thick, with strong diffuse inelastic scattering background, and overlapping DPs from different grains.

#### Orientation Mapping

Fine-grained and nanostructured materials are very much at the core of materials research. Significant properties arise in nanostructures from surface/*interface effects*: A significant fraction (or the majority) of atoms are located on or near the surfaces or interfaces in a nanostructure. Since the chemical bonds of surface or interface atoms can differ significantly from interior (or bulk) atoms, interfacial atoms give rise to distinct chemical, mechanic, thermodynamic, electronic, magnetic, and optical properties. The surface/interface effects are expected to increase as the ratio of surface to bulk atoms increases.

However, nanostructure characterization is challenging because of the small sizes and the difficulty of characterizing small interfaces. Electron backscatter diffraction ( ) is commonly used to characterize the microstructure of granular materials, such as metals and alloys, down to the submicron scale [18.72]. However, the spatial resolution of the EBSD technique is limited by the relatively large electron probe size in a SEM and the interaction volume, as well as the need to tilt the sample to a high angle, which is typically \(70^{\circ}\) from horizontal, for the collection of the EBSD patterns. The achievable spatial resolution is above \(\approx 20{-}30\,{\mathrm{nm}}\) using a field emission SEM and the accuracy of orientation determination is in a range of \(0.5^{\circ}\). To improve upon the resolution of EBSD, transmission EBSD ( ) was developed [18.73]. An electron transparent sample is used in t-EBSD and the sample is mounted in such way that the diffraction pattern projected from the lower surface of the sample can be recorded. Studies have shown that a spatial resolution of \(\approx{\mathrm{10}}\,{\mathrm{nm}}\) can be achieved by t-EBSD, but the resolution is sample thickness and acceleration voltage dependent, and the success of t-EBSD also depends strongly on these factors.

For very fine grains, electron nanodiffraction techniques, such as SEND performed in a TEM, offers a spatial resolution of \(\approx 1{-}5\,{\mathrm{nm}}\). The major limitation with the use of electron nanodiffraction for orientation mapping has been diffraction pattern indexing, which was not as fast as the EBSD technique. This limitation is largely improved now with the development of automated indexing methods (see the previous section for the discussion). CBED, which gives much better contrast for the recorded Kikuchi lines as well as HOLZ lines, provides the best accuracy for orientation determination, and it can be combined with electron nanodiffraction, in principle to improve the accuracy of orientation determination.

In what follows, we will illustrate the principle of TEM-based orientation mapping using the Au thin film of Fig. 18.25a-c as an example. The film has a thickness of \({\mathrm{50}}\,{\mathrm{nm}}\) deposited on a silicon substrate. A plane view Au thin film sample was prepared using tripod polishing followed by ion milling, and a region of interest was selected for electron diffraction study (Fig. 18.25a-cb). The SEND dataset was acquired on a \(31\times 31\) mesh using an electron probe of \({\mathrm{2.8}}\,{\mathrm{nm}}\) in FWHM, at an exposure time of \({\mathrm{0.1}}\,{\mathrm{s}}\) for each diffraction pattern and a step size of \({\mathrm{8}}\,{\mathrm{nm}}\). The corresponding scanned area of the sample is \(240\times 240\,\mathrm{nm^{2}}\). The diffraction patterns were recorded on a Gatan UltraScan^{®} 1000XP (Model 894) CCD camera with Peltier-cooled CCD and \(2048\times 2048\) pixels. The recorded diffraction patterns were binned eight times.

In order to generate an orientation map from the Au film sample, a correlation image is first generated from the SEND patterns. The aim of the correlation analysis method is to identify similar diffraction patterns, average these similar patterns, and obtain a correlation map where the distribution of similar diffraction patterns is identified. This correlation image is further processed to form the orientation map. The similarity of DPs is quantified by the value of the NCC coefficient calculated for two individual diffraction patterns. The value of an NCC coefficient between two DPs will range from \(-1\) to 1, with NCC \(=0\) indicating complete dissimilarity, and NCC \(=1\) indicating complete similarity. We use (18.41) to compute the NCC value between two diffraction patterns, say 1 and 2, with 1 for A and 2 for B. Next, the DPs are grouped using the values of NCC between the pairs of diffraction patterns. A group is defined as one with all DPs belonging to the group having values of the NCC values amongst themselves to be equal to or greater than a fixed, predefined, threshold value. This fixed threshold of NCC is called *correlation threshold* (CT ). The formation and number of groups will be directly affected by a change in this CT value, which is schematically shown in Fig. 18.26 using the analysis of seven DPs as an example. When CT is set at \(\mathrm{0.3}\), all seven DPs fall into one single group, as the NCC values amongst these seven DPs are equal to or higher than \(\mathrm{0.3}\). Also, if a CT value of 0.8 is selected, each DP is considered as a different group, i. e., grouping will not occur for these DPs, as the NCC values are \(<0.8\).

A correlation image is obtained from the diffraction grouping analysis, each group is assigned a unique index number and color (Fig. 18.27a,b), and diffraction patterns belonging to the same group are summed together and averaged to obtain a diffraction template for that group. Thus, the correlation image displays areas belonging to different diffraction templates.

For the example of Fig. 18.25a-c, the correlation analysis yielded 54 diffraction templates, which are analyzed for orientation determination using the automatic indexing technique described in Sect. 18.4.1, *Automated Indexing of Electron Diffraction Patterns*. Figure 18.25a-cc shows the results of the diffraction pattern indexing result. The orientation of the grain is color coded, with red for [001], green for [011], and blue for [111].

The results of orientation mapping can be validated using the constraints of virtual dark-field imaging. A virtual dark-field image is obtained from the SEND data by integrating diffraction intensities within a (region of interest) of the diffraction patterns; each DP gives the intensity of a pixel in the virtual dark-field or bright-field image, where the DP is recorded.

Figure 18.28a-cc shows the result of overlay of one of the reconstructed DF images onto the orientation map. The diffraction spot selected for virtual DF image construction as indicated in Fig. 18.28a-ca yields the virtual DF image of Fig. 18.28a-cb. The grain corresponding to the selected spot is associated with bright intensities as compared to other regions of the nanostructure. It is important to note that this is the same grain that corresponds to the template 7 in Fig. 18.27a,b. This grain is blue-colored in the orientation map, which is close to [111]. This procedure can be performed for other grains identified in the orientation map.

The orientation map of Fig. 18.25a-c reveals that most of the grains are closer to the [111] zone axis. This is expected for the Au system, as Au has an fcc crystal structure and, consequently, it would crystallize in a way such that the normal direction would be predominantly [111]. However, interestingly, we also see grains very close to [001], which is colored in red. Thus, the orientation information of the fine nanostructure of the Au thin film sample is successfully determined by the method described above.

Applications of TEM-based orientation mapping have been demonstrated for the characterization of copper interconnect lines fabricated by the damascene process in microelectronic devices [18.74], highly-deformed metals [18.75], nanocrystalline Cu [18.76], and crack growth in nanocrystalline TiN thin films [18.77]. Figure 18.29a-c shows intergranular fracture in nanocrystalline TiN observed in a FIB fabricated beam. In order to better distinguish the grain boundary and grain shape, SEND was performed before the bending test. The scanning region was \({\mathrm{200}}\,{\mathrm{nm}}\times{\mathrm{150}}\,{\mathrm{nm}}\) with an electron probe size \(\approx{\mathrm{2}}\,{\mathrm{nm}}\) and step size \({\mathrm{8}}\,{\mathrm{nm}}\). The orientation of each grain was determined by automatic electron diffraction pattern indexing. The orientation map and the corresponding diffraction pattern for each grain are shown in Fig. 18.29a-ca,b. The majority of diffraction patterns are from single crystal, indicating that very few grain-overlaps exist along the electron beam direction. From the correlation map color-coded by the crystal orientation of each grain, most of the grains are separated by high angle grain boundaries. The crack propagation path is shown in Fig. 18.29a-cc with a post-mortem bright-field TEM image and a schematic figure illustrating the configuration of grains and crack. The crack front proceeds along the grain boundary between G5 and G6, and then diverges into two crack propagation paths (Fig. 18.29a-cc). The crack on the right keeps propagating along the grain boundary of G3 until it meets G2, where the crack is arrested due to large resistance exerted by the grain boundary perpendicular to the crack propagation direction. The crack on the left continues to propagate along the grain boundary of G2, G4, and G5 and keeps being deflected by the grain boundaries.

#### Strain Mapping Using SEND

Using the TEM deflection (or STEM scan) coils, SEND patterns can be recorded from an area of the specimen to provide spatially resolved strain information for strain mapping. The diffraction peak positions are determined from the recorded diffraction patterns and used to map the local strain in real space. Additionally, from the recorded diffraction patterns, bright and dark-field STEM images can be obtained from SEND by integrating the diffraction intensities of the direct beam or the diffracted beams, respectively, which can be used to register the strain map to these images. SEND can be performed in either STEM or TEM mode. When NBD is performed in STEM, the STEM ADF detector can be used to visualize the ROI by collecting the scattered electrons to large angles during scanning. This way, diffraction can be correlated with the probe location during data collection.

The basis of strain mapping is that a series of diffraction patterns are acquired and compared to a reference diffraction pattern from an unstrained region. The strain is then calculated as the relative lattice mismatch by determination of the exact position of the diffraction spots positions. Using the FEI-Titan microscope operated in the \(\upmu{}\)-Probe STEM mode with a \({\mathrm{2.7}}\,{\mathrm{nm}}\) probe size and \({\mathrm{0.5}}\,{\mathrm{mrad}}\) convergence angle, a precision of \(6\times{\mathrm{10^{-4}}}\) was reportedly achieved [18.54]. The accuracy depends on the sample preparation and related strain relaxation, and is estimated to be about \(\mathrm{10^{-3}}\).

*Rouviere* et al reported on an implementation of scanning procession electron diffraction on a FEI-TITAN TEM [18.78]. The scan coils of the STEM unit were used to precess the incident beam and to perform a descan of the diffracted beams at a precession speed of \({\mathrm{0.1}}\,{\mathrm{s}}\). Diffraction patterns were recorded on a \({\mathrm{2}}\,{\mathrm{k}}\times{\mathrm{2}}\,{\mathrm{k}}\) Gatan ultrascan CCD camera with acquisition times of \(\approx{\mathrm{1}}\,{\mathrm{s}}\). The incident beam used for strain measurements had a convergence angle of \({\mathrm{1.8}}\,{\mathrm{mrad}}\) and a size of \({\mathrm{2.4}}\,{\mathrm{nm}}\). Thus, the diffraction patterns consisted of small disks. By using PED, the intensities within a given diffraction disk are made more uniform than the CBED-like patterns recorded without precession. Diffracted beam positions were measured by detecting the edges of the diffraction disks instead of their peak intensity. Because of the improvements in the intensity distribution within the diffraction disk, a better measurement accuracy could be obtained with the help of precession. Precession also helps by spreading the intensity across to high-index diffraction spots and making them more amenable for detection. Additionally, precession improves the robustness of measurements by reducing the crystal misorientation effects by averaging over the precessed incident beam directions at the cost of slightly larger beam diameter, and increased crystal volume from the tilted incident beam and its precession.

Figure 18.30b shows the measured strain profiles from a Si/SiGe multilayer grown by reduced pressure chemical vapor deposition ( ) using NBD with and without precession. The sample contains four SiGe layers, each \({\mathrm{11}}\,{\mathrm{nm}}\) thick and of different Ge compositions: \(\mathrm{20}\), \(\mathrm{31}\), \(\mathrm{38}\), and \({\mathrm{45}}\%\). The composition was determined using secondary ion mass spectroscopy ( ). The SiGe layers are biaxially strained by the Si substrate, and the lattice parameter is larger than the reference substrate only in the direction perpendicular to the layers. Figure 18.30 also plots the strain profile obtained by finite element simulations of the structure to take into account the strain relaxation in the thin TEM lamella. The profile was plotted by averaging the strain along the \(\langle 011\rangle\) beam direction and convoluting the obtained profiles with the measured electron beam size, i. e., \({\mathrm{2.5}}\,{\mathrm{nm}}\). This reduced the strain in the layers slightly; this effect is greater in the layer with the higher Ge concentration, where the strain is reduced from 2.76 down to \({\mathrm{2.6}}\%\). As can be seen in Fig. 18.30, in the SiGe layers, the difference between the measurements of NBD with and without precession is small. At the center of the SiGe layers, the difference between the calculated and the experimental strain obtained with precession is about \({\mathrm{0.1}}\%\) for the three layers that have the lowest Ge concentration. Large differences are observed inside the silicon; the profile obtained with precession is slightly negative inside the silicon, which fits very well with the simulation result, while the NBD measurement without precession gives far larger negative strain than the simulation result indicates, especially near the Si-SiGe layer interface. Away from the SiGe layers, the strain profile obtained with precession is very smooth, with a root mean square of fluctuations of \({\mathrm{1.5\times 10^{-4}}}\).

Application of SEND with precession is demonstrated in Fig. 18.31 for the analysis of a transistor with recessed \(\mathrm{Si_{0.65}Ge_{0.35}}\) source and drain. Figure 18.31a is a bright-field image of the device. Strain and rotation maps have been obtained by using (18.44). The DPs used in the analysis were obtained using a beam semiconvergence angle of \({\mathrm{1.8}}\,{\mathrm{mrad}}\), a precession angle of \(0.5^{\circ}\), and a beam diameter of \({\mathrm{2.5}}\,{\mathrm{nm}}\). The root mean square of the strain in the Si substrate, far from the layers, can be used to measure the strain measurement precision, i. e., the reproducibility of the technique. For NBD, the strain precision can be as low as \({\mathrm{6\times 10^{-4}}}\). The best precision obtained experimentally on NBD with precession was \({\mathrm{9\times 10^{-5}}}\) with a probe size of \({\mathrm{2.5}}\,{\mathrm{nm}}\). In the above examples, only small maps, \(20\times 20\) pixels, were obtained. With improved precession speed, larger memory, and a shorter camera acquisition time, larger maps can be realized. Overall, initial results demonstrate that NBD with precession is a very efficient technique to measure strain in nanostructures.

#### Determination of Three-Dimensional Nanostructures

The structure of nanocrystalline materials is determined by the constitutive phases, composition, 3-D grain morphology, orientation, and distribution, which can only be obtained from a 3-D structure determination. Previously, 3-D x-ray diffraction microscopy ( ) [18.79, 18.80] was developed for the study of polycrystalline materials at mesoscale. Recently, two new x-ray diffraction ( ) techniques, differential-aperture x-ray microscopy ( ) [18.81] and diffraction contrast tomography ( ) [18.82], achieved sub-\(\mathrm{\upmu{}m}\) spatial resolution in 3-D. Using a combination of a SEM and focused ion beam, 3-D electron backscattered diffraction ( ) [18.83] is the technique for obtaining 3-D orientation maps in bulk polycrystalline samples. However, the destructive nature of 3-D-EBSD makes it unfavorable for multitechnique or in-situ analysis. There are fewer attempts in 3-D microstructure determination using TEM. *Liu* et al reported a 3-D orientation mapping technique called *3-D-OMiTEM*, which was developed based on the conical scanning dark field imaging technique [18.84]. More recently, *Midgley's* group at Cambridge [18.85] demonstrated a determination of the three-dimensional precipitate morphology in an Ni-based superalloy using scanning precession electron diffraction [18.36] and a principal component-based separation algorithm to separate the matrix and precipitate diffraction patterns (Fig. 18.32).

In what follows, we show how SEND can be used for 3-D nanostructure determination . Specifically, by taking advantage of diffraction information, we show that the principles of electron tomography can be extended for nanostructure characterization beyond the capabilities of the traditional mass-thickness contrast approach.

Electron tomography works by recording a series of electron images at various sample rotations (or tilts) and reconstructing a 3-D image from the projected images. The contrast of recorded images is assumed to be proportional to the sample thickness and mass. The 3-D image construction is carried using algorithms developed through cryoelectron microscopy, based on well-established methods such as the Radon transform. The range of sample rotations plays a crucial role in the accurate reconstruction of targeted objects. The tilt angle of a specimen can be limited by the sample thickness, shadowing effects from the sample holder or the supporting grid [18.86].

For electron diffraction, a custom tomography holder that allows \(\pm 87^{\circ}\) rotation of the specimen was developed by *Meng* and *Zuo* [18.87]. Their design employs a needle-shaped specimen mounted on a regular JEOL single-tilt holder (Fig. 18.33a,b). The sample is placed on the top of the tungsten substrate using the FIB lift-out technique. The sample is annularly milled to the desired diameter (usually between \(\mathrm{100}\) and \({\mathrm{300}}\,{\mathrm{nm}}\)). The small diameter of the mounting tube (part C) allows a free sample rotation in a small polepiece gap. The needle-like sample is also parallel to the rotation axis of the holder, which provides a rotation with the minimum precession movement.

Three-dimensional diffraction data can be obtained by tilting the sample and performing SEND at each sample tilt. In the experiment reported by Meng and Zuo, an exposure time of \({\mathrm{0.1}}\,{\mathrm{s}}\) and a diffraction pattern size of \(256\times 256\) pixels were used in the diffraction pattern recording. A small camera length was also used to include as many diffraction spots as possible without too much degradation in the resolution of the diffraction pattern. The step size of sample tilting was selected based on a balance between the time cost of the data acquisition, the data size, and the accuracy of the reconstructed grain morphology. A smaller step size gives a more reliable 3-D morphology of a grain at the cost of an increase in time and data size. It has been suggested to use a tilting step size smaller than \(10^{\circ}\).

The first step of 3-D reconstruction is to examine the projected (2-D) morphology of the grains. The 2-D morphology of a grain is identified by constructing dark-field images from the recorded DPs through the following sorting process. At a specific sample rotation angle, the position and intensity of all diffraction spots are recorded using the template matching method [18.88]. Using this information, a dark-field image is then constructed for each diffraction spot. Two dark-field images will be similar to each other if their diffraction spots belong to the same DP of the single crystal grain. Thus, dark-field images with similar contrasts are grouped using normalized cross-correlation. A correlation threshold is used for grouping based on the trial and error method. The 2-D morphology is extracted from the averaged dark-field image, after applying an intensity threshold. Meanwhile, diffraction spots belonging to a single grain are grouped into a single crystal diffraction pattern. Unlike an experimental DP, this DP only contains a subset of measured diffraction spots. Figure 18.34 shows the sorting results of the TiN sample at \(-5^{\circ}\). This step is repeated for every sample rotation angle.

Next, the filtered DPs are indexed for the determination of crystal orientation. If the number of diffraction spots in an image-filtered DP is not sufficient for a reliable indexing, we index the averaged experimental DPs within the identified 2-D grain. In this case, the averaged experimental DP may contain diffraction spots from other overlapping grains. Therefore, multiple local maxima may appear in the indexing correlation factor map [18.89]. Electron DP indexing is done using automatic indexing (Sect. 18.4.1, *Automated Indexing of Electron Diffraction Patterns*). Most diffraction patterns can be indexed this way with the exception of a few DPs far away from zone axes. However, a small number of failed index attempts does not affect the overall analysis.

- 1.
The difference between the two beam directions is equal to the sample rotation step size

- 2.
The 2-D grain images overlap with each other. Figure 18.35 shows the 2-D images of one grain from \(\mathrm{-75}\) to \(-5^{\circ}\).

*Amanatides*and

*Woo*[18.91]. An element \(x_{j}\) in \(x\) represents the distribution of the object in the \(j\)th voxel; \(x\) is the unknown variable. An element \(p_{i}\) in \(p\) represents the measured projection under the \(i\)th ray; \(p\) is determined based on the acquired dark-field images. The value of \(p_{i}\) is set to 1 if the projection of the \(i\)th ray is within the outline of the 2-D grain morphology. Otherwise, it is set to 0. Various algorithms were developed for solving (18.45). Here, we use the algebraic iterative algorithm first proposed by

*Kaczmarz*[18.92]; \(x\) is additively modified in each cycle to approximate the ideal solution. We stop the iteration when \(x\) is stable.

The output of the ART is a 3-D map of voxel contribution to the target grain. By creating an isosurface of the map, the morphology of the grain can be plotted in the 3-D space. The isosurface value is adjusted so that the isosurface is continuous.

The orientation of a grain is determined from the indexing results of all available projections. In stereo projection, the indexing results are expected to form a line. The orientation of a grain is defined by the transformation matrix that transforms a vector in the crystal coordinate onto one in the holder coordinate. We define the transformation relation as \(\boldsymbol{h}=\mathbf{T}\boldsymbol{c}\); \(\boldsymbol{c}\) is a 3-by-1 vector, which represents a direction in the crystal coordinate; \(\boldsymbol{h}\) is a 3-by-1 vector, which represents the same direction in the holder coordinate; \(\mathbf{T}\) is a 3-by-3 matrix, which transforms the direction from the crystal coordinate into the holder coordinate.

Figure 18.36a-d shows a reconstructed 3-D image of TiN nanograins using 3-D-SEND. The experiment was performed on a TiN thin-film nanocrystalline sample. TiN is a material that is widely applied in the electronics industry, as well as in protective and decorative coating [18.94]. Compared with polycrystalline TiN, nanocrystalline TiN exhibits improved mechanical properties such as hardness, and wear and corrosion resistance [18.95, 18.96, 18.97]. This is achieved by controlling intrinsic properties such as grain size, morphology, and texture [18.98]. Nanocrystalline TiN thin film can be grown by chemical vapor deposition ( ) and physical vapor deposition ( ) [18.99]. Here, the TiN sample was grown on a p-type Si(100) substrate by an unbalanced magnetron sputtering ( ) system [18.96].

Seven major grains in the TiN sample were reconstructed, as well as their crystallographic orientations. Previous experimental studies found that the grain is elongated along the growth direction, and multiple grains are stacked along the elongation direction [18.94]. However, the morphology of the grains is not known. The 3-D-SEND reconstruction results show that grains are elongated and the grain boundaries are not ideally round or flat.

### 18.4.2 Convergent Beam Electron Diffraction of Complex Crystals

Compared to electron nanodiffraction, which uses a small condenser aperture to achieve a small convergence angle that yields a spot-like diffraction pattern, CBED records diffraction patterns using a larger convergence angle that gives rise to diffraction disks, instead of diffraction spots. The extra diffraction intensities recorded in CBED make it ideal for the study of crystals, especially complex ones.

#### The Geometry of CBED

At medium convergence angles, the interpretation of electron nanodiffraction patterns recorded from crystals uses the same theory for CBED. The starting point for understanding CBED is the Ewald sphere construction. Figure 18.37 shows one example. By the requirement of elastic scattering, all transmitted and diffracted beams are on the Ewald sphere. Let us take the incident beam \(P\), which satisfies the Bragg condition for \(g\). For an incident beam \(P^{\prime}\), to the left of \(P\), the diffracted beam also moves to the left. The difference between the incident wave and the diffracted wave is the vector \(\boldsymbol{g}\). The deviation of the diffracted beam away from the Bragg condition is defined by the so-called excitation error given in (18.25).

#### CBED Intensities

*Cowley*and

*Moodie*, and others [18.107, 18.108]. As a numerical method, multislice has the advantage that it can treat both crystals and nonperiodic structures, including amorphous structures. Because of this, the multislice method is particularly suitable for electron nanodiffraction simulation. The multislice method models the forward propagation of the electron waves through successive thin slices of potentials (Fig. 18.38). The basic equation is the relationship between the incident wave \(\phi_{n}(x,y)\) and the exit wave \(\phi_{n+1}(x,y)\) of the \(n\)th slice

The main limitation of the multislice method is the number of atoms that can be included realistically in a simulation. The limitation comes from the atomic potential sampling considerations, as illustrated in Fig. 18.38. The 3-D sample potential in a multislice calculation is represented in a 2-D numerical array for each slice along the beam direction. The representation of the atomic potentials requires a minimum number of sampling points. For example, a minimum of five points are required to represent the center, the size, and the gap of the atomic potential. For a \({\mathrm{1}}\,{\mathrm{\AA{}}}\) sized atom, the spacing between these points is \({\mathrm{0.2}}\,{\mathrm{\AA{}}}\), defining a minimum pixel size in the real space. In this case, \({\mathrm{1}}\,{\mathrm{k}}\times{\mathrm{1}}\,{\mathrm{k}}\) represents a sample area of \(20\times{\mathrm{20}}\,{\mathrm{nm^{2}}}\).

#### Crystal Symmetry and Symmetry Mapping

A major application of CBED is to determine the crystal symmetry. Since the electron probe size is very small, the symmetry determination can be carried out locally at high spatial resolution, for example inside nanodomains in ferroelectrics. The crystal symmetry is reflected in the diffraction patterns. For example, if the crystal has a rotation axis, two diffraction patterns related by rotation should be the same. The same is true for mirror symmetry. Additional symmetries are produced in electron diffraction because of 1) the principle of reciprocity and 2) the projection along the zone axis for ZOLZ [18.100]. The principle of reciprocity states that the intensity of the diffracted beam (B) with a source (A) is the same as the intensity detected at A with the source at B by the same scatter. The projection of crystal structure along the zone axis orientation used in observation produces a mirror symmetry at the middle of the sample, which may or may not exist in the crystal. The combination of reciprocity and projection with the crystal point groups produces 31 diffraction groups , whose relationships with the 32 point groups were tabulated by *Buxton* et al [18.109]. The correspondence is often not unique. The determination of crystal point groups comes down to elimination of multiple choices using the symmetry of diffraction patterns recorded along several major symmetric orientations and/or using information about the lattice determined from the diffraction pattern geometry. The diffraction pattern symmetries used in the determination are those of the whole pattern, the transmitted beam (bright), the diffracted beams (dark-field), and the symmetry between \(+g\) and \(-g\) beams. It should be emphasized that the Friedel symmetry (\(I_{g}=I_{-g}\)) is absent in electron diffraction because of dynamic scattering. The point groups can be uniquely determined by electron diffraction.

The screw and glide axes present in the crystals can be determined by observing dynamic extinction in kinematically forbidden reflections (zero structure factor due to the glide or screw axes). These reflections generally show some intensities due to electron multiple scattering. The dynamic extinction is observed when the incident beam is in the glide plane in the case of a glide; this was first reported by Gjonnes and Moodie using CBED (the extinction appears as dark lines, subsequently named G–M lines). The dynamic extinction of a screw axis is more complicated and is described in detail in [18.18].

The combination of point group determination and identification of translation symmetry allows the unique identification of space groups [18.110, 18.111]. Both CBED and LACBED techniques can be used for this purpose. Applications of symmetry determination by CBED include phase identification and as part of the determination of unknown structures. Methods for quantifying and autodetection of the CBED symmetry can be found in [18.112, 18.113].

Real crystals often have local symmetry dependent on sample position. An obvious case is the breakdown of symmetry to surface and interfacial stress and strain or the presence of defects. Another case is ferroelectric crystals, electric polarization removes the inversion symmetry, and its direction coincides with the principal symmetry axis in tetragonal and rhombohedral crystals. Thus, measurement of local symmetry can be used to determine the polarization direction. There are many other examples, where local symmetry can help with phase identification and microstructure determination.

In using CBED for local symmetry determination, it is helpful to quantify the amount of symmetry recorded in CBED patterns [18.112, 18.114, 18.115]. The basic idea is to measure, from the diffraction intensities, the similarity between points inside the CBED disks that are related by symmetry. The similarity can be measured by the standard normalized cross-correlation coefficient (\(\gamma\) as defined in (18.41)) [18.71]. In order to measure the symmetry, regions in CBED patterns must be selected and are aligned. Figure 18.39a-c shows an experimental CBED pattern from the Si[110] zone axis. We use this pattern to demonstrate the image processing procedures employed for the dark-field symmetry quantification. The discussion below is specific to mirror symmetry, but the principle also applies to rotational symmetry. First, two diffraction discs are selected on two sides of the mirror plane (marked by the yellow line) as shown in Fig. 18.39a-ca. For the discussion, the selected CBED discs are named template \(\mathrm{A}\) and template \(\mathrm{A}^{\prime}\) (Fig. 18.39a-cb,c), respectively. Each template is then rotated by an angle \(\theta\), so that the mirror is aligned. The template A is used as the reference motif so that the symmetry element is calculated by comparison with template \(\mathrm{A}^{\prime}\). For the mirror operation, the template \(\mathrm{A}^{\prime}\) is flipped to obtain a mirror image. The mirror-applied image will be referred to as \(\mathrm{A}^{\prime}_{m}\). For the rotational operation, the template \(\mathrm{A}^{\prime}\) is rotated by \(180^{\circ}\), \(120^{\circ}\), \(90^{\circ}\), and \(60^{\circ}\) for the two, three, four, and sixfold rotation, respectively. The rotated template \(\mathrm{A}^{\prime}\) will be referred to as \(\mathrm{A}^{\prime}_{n}\) (\(n=2,3,4,6\)). The circular mask shown in Fig. 18.39a-cb,c is used to remove areas affected by the CBED disk edge. Thus, the final templates are obtained by multiplying the mask image to the templates A and \(\mathrm{A}^{\prime}_{m}\).

The normalized cross-correlation coefficient (\(\gamma\)) is used to quantify the similarity between A and \(\mathrm{B}=\mathrm{A}^{\prime}_{m\text{ or }n}\). For a pattern with perfect symmetry, \(\gamma=1\). For the experimental pattern in Fig. 18.39a-c, which was recorded using a JEOL 2100 \(\mathrm{LaB_{6}}\) TEM at \({\mathrm{200}}\,{\mathrm{kV}}\), the \(\gamma\) values range from 0.981 to 0.991 for the mirror. A test of the robustness of the symmetry quantification procedure over 20 experimental Si[110] CBED patterns gave the \(\gamma\) values ranging from \(\mathrm{0.981}\) to \(\mathrm{0.991}\) for all quantification results.

The symmetry quantification method we have described can be combined with the scanning electron diffraction technique described in Sect. 18.1.5 for symmetry mapping. In scanning CBED, a series of CBED patterns are recorded and stored in a 4-D dataset. The 4-D dataset consists of \(m\) by \(n\) patterns; the \(m\) and \(n\) correspond to the number of sampling points along the two edges of the rectangular grid. Figure 18.40a-e shows an application of the symmetry mapping technique to the determination of the ferroelectric domain boundary in \((1-x)\mathrm{Pb(Zn_{1/3}Nb_{2/3})O_{3}}\)-\(x\mathrm{PbTiO_{3}}\) (\(x=0.08\)) (PZN-PT). The principle of domain identification is based on the above described CBED measurement of crystal symmetry. The ferroelectric polarization direction lies in the mirror plane, which can be determined by measuring the mirror symmetry in the recorded CBED patterns. Thus, ferroelectric domains can be identified by the change of CBED pattern symmetry (Fig. 18.40a-e). The change in CBED pattern symmetry is quantified using the normalized cross-correlation (\(\gamma\)) value of a pair of diffraction discs related by mirror symmetry. The SCBED experiments of Fig. 18.40a-e were carried out using a JEOL 2010F FEG TEM operated at \({\mathrm{200}}\,{\mathrm{kV}}\) with a convergent beam of \({\mathrm{2.6}}\,{\mathrm{nm}}\) in FWHM. Energy-filtering ( ), which improves the contrast of CBED patterns, was performed using a Gatan imaging filter (GIF). EF-SCBED was performed by scanning the focused electron probe over a selected area on a \(15\times 15\) grid, a step size of \({\mathrm{2}}\,{\mathrm{nm}}\), and through a postcolumn GIF energy window of \({\mathrm{10}}\,{\mathrm{eV}}\). The shift and tilt of diffraction patterns during beam scanning were minimized and calibrated using a silicon single crystal. The symmetry of PZN-8%PT was determined as monoclinic \(Pm\), which agrees with the x-ray diffraction result [18.116]. Nanodomains are observed using EF-SCBED as demonstrated by the symmetry variations across these domains in an EF-SCBED dataset from a \(30\times{\mathrm{30}}\,{\mathrm{nm^{2}}}\) sample area are shown in Fig. 18.40a-ea. The scan consists of 15 by 15 points, with a step size of \({\mathrm{2}}\,{\mathrm{nm}}\). The \(\gamma\) values are shown in grayscale. Two domains are identified (type-1 and type-2) with different mirror symmetry. The type-1 and type-2 domains are associated with two distinguishable CBED patterns that were observed along the \([100]_{\mathrm{C}}\) incident direction (Fig. 18.40a-eb,d). The highest \(\gamma\) values of type-1 and type-2 patterns are detected along two different directions (A and B) as shown in Fig. 18.40a-eb,d. The A and B directions are rotated by \(45^{\circ}\) along the \([100]_{\mathrm{C}}\) zone axis. The corresponding simulated patterns for type-1 and type-2 domains are along the monoclinic \(Pm\) zone axes \([100]_{Pm}\) and \([010]_{Pm}\), as shown in Fig. 18.40a-ec,e, respectively. In the \(Pm\) structure model, the polarization direction is \(\boldsymbol{P}_{\mathrm{S}}=[u,0,v]_{Pm}=[3,0,4]_{Pm}\), which lies in the mirror plane of \(Pm\) symmetry [18.116]. Along the \([100]_{Pm}\) incident direction, the mirror plane is superimposed on \((001)/(00\overline{1})\) reflections, which is parallel to the A direction in Fig. 18.40a-eb. This mirror is not observed along the \([010]_{Pm}\) incident direction. The projection of the polarization lies approximately on the \((101)/(\overline{1}0\overline{1})\) reflections, which is parallel to the B direction in Fig. 18.40a-ed. The highest mirror symmetry in this case is detected along direction B in the simulated pattern (Fig. 18.40a-ee) with \(\gamma={\mathrm{60}}\%\).

Another application of symmetry mapping is to determine the highest symmetry in a single crystal and obtain highly symmetrical experimental CBED patterns. The symmetry of recorded CBED patterns is very sensitive to the quality of sample surface and strain in prepared single crystal samples, including modifications to the surface by ion milling or focused ion beam irradiation. Because of this, the measured symmetry in experimental patterns often vary from area to area. For symmetry determination, it is critical to determine the highest symmetry. This can be achieved using SCBED and symmetry mapping. Figure 18.41a,b shows an example for Si[110] CBED patterns. The sample was prepared by precision ion milling. To quantify the mirror symmetry in the CBED patterns, the symmetry related pairs of diffraction discs from \((\mathrm{A},\mathrm{A}^{\prime})\) to \((\mathrm{G},\mathrm{G}^{\prime})\) are selected (orange circles) about the mirror plane (blue circles). The intensity weighted cross-correlation value is used to map the symmetry, the highest symmetry measured is at \(\gamma={\mathrm{98.1}}\%\) [18.117].

Figure 18.42a-c shows an application of highest symmetry determination using SCBED for barium titanate (\(\mathrm{BaTiO_{3}}\)). Ferroelectricity of \(\mathrm{BaTiO_{3}}\) was discovered in the 1940s. Since then, it is regarded as a model of perovskite-type ferroelectrics. \(\mathrm{BaTiO_{3}}\) undergoes successive phase transformations from the high-temperature cubic (C) paraelectric phase to three low-temperature ferroelectric phases with tetragonal (T), orthorhombic (O), and rhombohedral (R). In Fig. 18.42a-c, the highest symmetry was identified from a total of 625 CBED patterns over an area of \({\mathrm{25}}\,{\mathrm{nm}}\times{\mathrm{25}}\,{\mathrm{nm}}\) and the most symmetrical CBED patterns taken along the pseudocubic incidence \([100]_{\text{pc}}\) at different temperatures are shown. In the T (\(P4mm\)), O (\(Amm2\)), and R (\(R3m\)) phases, \(\boldsymbol{P}_{\mathrm{S}}\) lies in one of the \(\langle 100\rangle_{\mathrm{T}}\), \(\langle 110\rangle_{0}\), and \(\langle 111\rangle_{\mathrm{R}}\) directions, respectively. Thus, for CBED patterns taken along the \([100]_{\text{pc}}\) incidence, we should expect mirror planes normal to \([010]_{\mathrm{T}}\), \([001]_{\mathrm{T}}\), \([011]_{0}\), \([0\overline{1}1]_{0}\), \([011]_{\mathrm{R}}\), and \([0\overline{1}1]_{\mathrm{R}}\) directions. Figure 18.42a-ca shows the CBED pattern of T phase with mirror plane \(m\parallel[001]_{\mathrm{T}}\), and \(\upgamma={\mathrm{98.5}}\%\) is consistent with the space group \(P4mm\). Figure 18.42a-cb,c shows the CBED patterns taken approximately at \({\mathrm{263}}\,{\mathrm{K}}\) (O phase) and \({\mathrm{95}}\,{\mathrm{K}}\) (R phase) with mirror plane \(m\parallel[011]_{0}\) and \(\upgamma={\mathrm{95.8}}\%\), \(m\parallel[011]_{\mathrm{R}}\) and \(\upgamma={\mathrm{94.6}}\%\), respectively.

### 18.4.3 Measurement of Crystal Structure Factors

Electron structure factors are obtained from experimental CBED patterns by using the refinement method [18.118, 18.119, 18.120, 18.121]). The refinement method works like the Rietveld method in powder x-ray or neutron diffraction, where the structure factors are treated as structural parameters, which together with other parameters, are obtained by comparing experimental and theoretical intensities and optimizing for the best fit. Multiple scattering effects are taken into consideration by using dynamical theory to calculate diffraction intensities during the refinement. In this way, the failures of the kinematical approximation in electron diffraction are avoided. Further, electron interference due to coherent multiple scattering actually enhances the sensitivity of the diffracted intensities to the crystal potential and crystal thickness, and thus improves the electron diffraction measurement accuracy.

*Nakashima*and

*Muddle*that avoids this requirement [18.121]). Experimental issues involved in the diffraction pattern recording include the geometric distortions of the diffraction patterns, the detector resolution and noise, the optimum sample thickness, and diffraction geometry. These are discussed in details in the literature [18.122, 18.123, 18.124, 18.125]. The noise in the experimental data can be estimated using the measured detector quantum efficiency ( )

Three types of diffraction conditions have been used for electron structure factor measurements. One is the systematic row diffraction condition (Fig. 18.43), in which a set of parallel lattice planes is close to the Bragg diffraction condition. The CBED pattern appears one-dimensional with relatively uniform intensity normal to the systematic direction, because diffraction is dominated by reflections belonging to the same set of lattice planes (a *systematic row*). The largest effect of a small change in the structure factor of a reflection on its diffraction intensity is near the Bragg condition, which can be accomplished for two reflections in a systematic row CBED pattern. Other choices of diffraction conditions include the symmetric zone-axis orientation and slightly off-zone axis orientations [18.125, 18.127]. The advantage of these orientations is that a larger number of reflections can be refined and measured simultaneously. This is done at the cost of increased complexity in comparing two-dimensional patterns and computing time, because a large number of reflections contribute to diffraction in the zone axis orientation.

To calculate the theoretical intensities, an approximate model of the potential is needed. In the case of electron density measurements, the crystal structure (the atomic species and their coordinates, the cell constants) is first determined very accurately. This is usually done by x-ray or neutron diffraction. The unknowns for electron refinement are the low-order structure factors (which are the most sensitive to bonding effects, as we have seen), the absorption coefficients, and the experimental parameters related to diffraction geometry and specimen thickness. The structure factors calculated from a spherical atom or ionic model can be used as a starting point. Absorption coefficients are estimated using the Einstein model with known Debye–Waller factors either from direct measurement of x-ray or neutron diffraction [18.128] or theory.

- 1.
The zone axis center (in practice, the tangential wave vector \(\boldsymbol{K}_{\mathrm{t}}\) for a specific pixel)

- 2.
The length and angle of the \(x\)-axis in the zone axis coordinate used for simulations

- 3.
The specimen thickness, an intensity normalization coefficient, and the background intensity model.

- 1.
Structure factor amplitude and phase (in the case of an acentric crystal) of the selected reflection (

*hkl*at or near the Bragg condition) - 2.
Fourier amplitude (and phase in some cases) of the absorption potential of the selected reflection for the same

*hkl*as above.

Figure 18.44 shows the \(\chi^{2}\) map as a function of the structure factors of the (110) and (220) reflections, for a rutile (110) systematic refinement. It clearly shows that near the global minima there is no other local minimum. This property ensures that the refinement program can find the true global minima. It is interesting to note that, for the (110) reflection, the minimum point is almost independent of the (220) reflection.

Estimates of errors in refined parameters can also be obtained by repeating the measurement. For CBED, this can be done by using different regions of the pattern or patterns recorded at different diffraction conditions and sample thicknesses. A test of electron diffraction accuracy has been reported by [18.120, 18.127, 18.129]. They measured the low-order structure factors of silicon, which are known from x-ray Pendellösung measurements. The experimental data of [18.129] has since then been re-refined using 279 beams, which were selected using more stringent beam selection criteria. This lowered \(\chi^{2}\) to \(\mathrm{1.58}\). The electron structure factors obtained are \(U(\overline{1}11)={\mathrm{0.04738}}(5)\ \AA{}^{-2}\) and \(U(\overline{2}22)={\mathrm{0.00095}}(5)\ \AA{}^{-2}\). The x-ray Pendellösung measurements, converted to electron structure factors, give \(U(\overline{1}11)={\mathrm{0.04736}}(4)\ \AA{}^{-2}\) and \(U(\overline{2}22)={\mathrm{0.000943}}(5)\ \AA{}^{-2}\). In this case, the different electron diffraction measurements and the best x-ray data agree within the experimental measurement error.

A major application of the accurate measurement of crystal structure factors is the determination of the crystal charge density. Since the crystal charge density is a ground state property of the electronic structure of the crystal, quantitative CBED has the potential to probe local electronic structures. This is based on the conversion of the accurately measured electron structure factors to x-ray crystal structure factors (Fourier transform of charge density) and from them to map electron distributions in crystals. In particular, the significant improvement in the accuracy of experimental structure factors, achieved by the development of quantitative CBED, has resulted in a number of accurate studies of electron density of several inorganic crystals of different bonding types (for a review, see [18.130]). Critical to the success was to correct extinction in x-ray diffraction, from which the intensity of a few strong low-order reflections deviates significantly from the ideal imperfect crystal (mosaic blocks) model and limits the accuracy of experimental structure factors extracted from diffraction intensities using the kinematical approximation.

### 18.4.4 Inversion of Diffraction Patterns and Nanostructure Determination

Inversion of electron diffraction patterns provides a direct solution to the analysis of electron diffraction data. In the limit of coherent diffraction, the diffraction pattern records the intensity of the Fourier transform of the electron exit wave function. Inverse Fourier transform of the diffraction pattern thus requires the phase missing in the diffraction pattern. This is known as the phase problem in diffraction. Critical to the inversion of diffraction patterns is to find the phases of the diffracted waves.

In crystallography, the phase problem is solved based on a priori information about the crystal structure. The a priori information includes the sharply peaked atomic charge density and the periodicity of the crystal. By solving the inversion problem of crystal diffraction, crystallographers routinely image atoms in 3-D molecules as long as they can be crystallized. There is a long history of attempts at inverting the recorded electron diffraction patterns. One was to use interference between diffraction disks in electron ptychography [18.131]. The concept of ptychography was first proposed by *Hoppe* [18.132] and then further developed by *Rodenburg* [18.131]. In the original ptychography, electron diffraction patterns are recorded over an area of a crystal using a coherent probe with a diameter less than the size of the crystal unit cell, and the diffraction intensity at the middle of the overlapping disks is processed as a function of the probe position to form atomic resolution images [18.131]. For electron nanodiffraction, we must consider objects that are not perfect, infinite, crystals.

The ability to invert diffraction patterns to form images has attracted considerable interest recently in the x-ray diffraction community, where the lack of a high-resolution imaging lens has been a major obstacle toward x-ray imaging (Chap. 20). In electron diffraction, the additional phase introduced by the lens aberrations does not affect the diffraction intensity, and diffractive imaging by solving the phase problem provides atomic resolution imaging at diffraction-limited resolution. The inversion of electron diffraction patterns of nanometer-sized objects is helped by the fact that the small object leads to broadened diffraction peaks and, in the case of coherent electron diffraction, the broadening gives additional diffraction information and under not so restricted conditions can lead to inversion of diffraction patterns [18.133].

Here, we introduce coherent electron diffraction techniques for nanometer sized objects. We start with a description of the noncrystallographic phase problem, which is followed by a discussion on the different iterative transformation algorithms for solving the noncrystallographic phase problem and their requirements. We then use quantum dots as an example to demonstrate the phasing of experimental electron diffraction patterns and introduce techniques for achieving this.

#### The Noncrystallographic Phase Problem

The exit wave function \(\psi_{\text{exit}}(\boldsymbol{r})\) can be reconstructed by inverse Fourier transform of \(\Psi(\boldsymbol{k})\). Experimentally, however, one can only measure the length of the complex vector (\(|\Psi(\boldsymbol{k})|\)), while the phase angle \(\varphi(\boldsymbol{k})\) cannot be measured directly from the diffraction pattern. This is known as the phase problem. The missing phase that is commonly referred to in crystallography is the phase of structure factors. The phase \(\varphi(\boldsymbol{k})\) here is more general, including the effects of electron multiple scattering. The phase problem thus prevents one from direct inversion of diffraction using Fourier transformation.

The exit wave phase, in principle, can be measured by holography. A reference wave is used in holography to interfere with the object exit wave, which gives a set of interference fringes. The maxima of the fringes are locations where the phase of the scattered wave matches that of the reference wave, therefore the phase of the scattered wave can be measured from the intensities recorded in the interference pattern. When the hologram is illuminated with the same reference wave, one can reconstruct the object exit wave by a backward propagation. The re-illumination stage is equivalent to the inverse Fourier transformation of the hologram. *Lichte* et al showed experimentally that in off-axis electron holography, the complete information about amplitude and phase of the electron exit wave can be reconstructed numerically from a single hologram [18.134, 18.135]. The reference wave is created by splitting the illumination using an electron briprism in imaging. This is performed in the imaging mode, and because of this, the measured wave function also contains the phases due to the lens aberrations, which is the a major limiting factor to the ultimate information that can be obtained by electron holography [18.56].

The same holographic experiment in diffraction requires diffractive waves from an aperture. This has been demonstrated in the case of soft x-ray diffraction [18.136].

There are a number of established crystallographic methods to solve the crystallographic phase problem. For inorganic crystals and organic molecules with a small number of atoms, direct methods [18.137, 18.138] are widely used. Direct methods are a group of ab initio phase determination techniques based on mathematical procedures that compare structure factor amplitudes derived from a single crystal. For example, by using the statistically correct phase relation proposed by *Karle* et al [18.137, 18.138], \(\varphi(\boldsymbol{h})\approx\varphi(\boldsymbol{k})+\varphi(\boldsymbol{h}-\boldsymbol{k})\), one can obtain the phase of \(\boldsymbol{h}\) from the phases of \(\boldsymbol{k}\) and \(\boldsymbol{h}-\boldsymbol{k}\). For macromolecules such as proteins, the large number of atoms (of the order of \(E2{-}E5\)) makes deriving structures using direct methods computationally prohibitive. Alternative methods have been developed based on atomic replacement using chemically modified molecules, which are more efficient for macromolecular phasing.

Solving crystal structure by direct methods or other crystallographic methods requires the preparation of a crystalline specimen. Many biologically important macromolecules, such as viruses and cells, cannot be crystallized. In materials science, diffraction of noncrystalline materials gives broad peaks and continuous background. The difficulty of crystallizing nonperiodic structures prevents structure determination at the atomic resolution through conventional crystallography methods. Overcoming this difficulty requires the solution of the phase problem for nonperiodic structures or the so-called noncrystallographic phase problem.

#### Coherent Diffractive Imagining of Finite Objects

A number of experimental methods have been proposed to image nonperiodic structures from diffraction patterns. *Gabor* first proposed a two-stage imaging process [18.139]. In the first stage, a diffraction pattern of the specimen is recorded on a photographic plate using a divergent electron beam emerging from a point source. The diffraction pattern recorded this way is essentially a hologram (since scattered and unscattered beams overlap and interfere) and thus carries both the phase and amplitude of the electron wavefront. In the second stage, the plate is illuminated using visible light, and the electron wavefront emerged from the specimen is reconstructed using optical lenses. Since the spherical aberrations of the optical lenses are easier to correct than electron lenses, the image resolution could be improved in the second stage. However, atomic resolution was never achieved using Gabor's idea due to a number of technical difficulties. Firstly, it is very difficult to realize an ideal point source. Secondly, the electron beam that Gabor used had limited coherence, and, therefore, not enough inference was formed to carry the phases. More recently, *Spence* and his co-workers experimented with a field emission point source for point projection microscopy [18.140, 18.141].

Recent breakthroughs in coherent diffractive imaging come from the convergence of several ideas that has led to a working solution of the noncrystallographic phase problem.

##### Oversampling

*Harry Nyquist*in 1928 [18.142] and further developed by

*Claude E. Shannon*[18.143] in 1949. The Nyquist–Shannon theorem states that [18.143]

The minimum sampling frequency of \(1/a\) is called theif a function \(f(x)\) vanishes outside the points \(x=\pm a/2\), then its Fourier transform \(F(k)\) is completely specified by the values which it assumes at the points \(k=0,\pm 1/a,\pm 2/a,\dots\)

*Nyquist frequency*.

*Miao*et al [18.144] suggests that oversampling diffraction experiment gives the extra information which can be used to solve the phase problem. The inverse problem can be phrased as the following equation

*the other half of*pixels are known to be zero. Therefore, by oversampling, the inverse problem becomes mathematically over-determined. The above argument also links the degree of oversampling with the over-determination. Therefore we can define an oversampling ratio \(\sigma\)

##### Sampling Experimental Diffraction Patterns and the Field of View

The minimum oversampling ratio required to solve the phase problem according to *Miao* et al [18.144] is \(> 2\) for a 1-D object, \(> 2^{1/2}\) in each dimension for a 2-D square object, and \(> 2^{1/3}\) in each dimension for a 3-D cubic object.

##### Requirements on Beam Coherence

Partial coherence leads to a reduction of the oversampling ratio. In the ideal case, this has no effect as long as the oversampling ratio meets the minimum requirement. In practice, the combination of a reduced oversampling ratio and the noise in recorded experimental diffraction patterns leads to a loss of information in the reconstructed object function. This effect has been demonstrated by *Huang* et al in a simulation study [18.146].

##### Phase Retrieval Algorithms

Given the phase problem is overdetermined by oversampling, the question becomes how to retrieve the phase from the recorded diffraction patterns and solve the inverse problem numerically. Several iterative transformation algorithms ( ) [18.147, 18.148, 18.149, 18.150, 18.151] have been proposed to retrieve the missing phases from diffraction data. Common to all of these algorithms is iteration between two domains, typically the real space and the reciprocal space, and the iterant is forced to satisfy what is known in each domain, called constraints in each domain. This iterative algorithm was first proposed by Gerchberg and Saxton. The iteration continues until some error metric reaches a certain level.

- 1.
*Gerchberg–Saxton Algorithm ()*[18.150]In the GS algorithm, what is aimed to reconstruct is a complex wave field, and the algorithm assumes that the amplitudes are known in both the object and the Fourier domains. Therefore, the last step in the loop iswhere \(|A(x)|\) is the amplitude measured in the object domain.$$g_{\text{new}}=g^{\prime}(x)\cdot\frac{|A(x)|}{|g^{\prime}(x)|}\;,$$(18.69) - 2.
*Error Reduction Algorithm (ER)*[18.148]*Fienup*[18.148] modified the GS algorithm to extend its application to situations where only the amplitudes in the reciprocal space are measured, such as in x-ray diffraction. Instead of using the amplitude constraint in the object domain as the GS algorithm requires, he suggested using a support constraintwhere \(S\) denotes the support of the object, which is the region where the object has nonzero density. Equation (18.70) essentially applies the information gained by oversampling, which is the zero-valued region surrounding the support. For a real-valued object under kinematical diffraction, for example, two more constraints can be applied on top of the support constraint. They are the real constraint and the positivity constraint$$g_{\text{new}}(x)=\begin{cases}g^{\prime}(x)\;,&x\in S\quad\text{ and}\\ 0\;,&x\notin S\;,\end{cases}$$(18.70)$$g_{\text{new}}(x) =\mathfrak{Re}\{g^{\prime}(x)\}\;,$$(18.71)For kinematical x-ray diffraction without the absorption and refraction effects, the electron density is nonnegative everywhere. Therefore, the conditions of (18.71) and (18.72) generally hold. For kinematical electron diffraction, the atomic potential of ionic materials can have both signs simultaneously.$$g_{\text{new}}(x) =0\;,\quad\text{ if }\mathrm{g}^{\prime}(x)<0\;.$$(18.72)*Fieunp*[18.149] showed that the ER algorithm can work with a complex-valued object by removing the positivity constraint. The object function becomes complex as a result of dynamical scattering, or in the case of kinematical diffraction, the projection of three-dimensional diffraction, or absorption or refraction. - 3.
*Hybrid-Input-Output Algorithm (HIO)*[18.148]The ER algorithm suffers from slow convergence and a tendency to stagnate at local minima in the solution space [18.148]. To solve this problem, Fienup introduced a feedback mechanism into the ER as illustrated in Fig. 18.46a,b. The input and output of the Fourier domain modification are \(g(x)\) and \(g^{\prime}(x)\), respectively. This operation produces the function, \(g^{\prime}(x)\), which satisfies the amplitude constraint in the Fourier domain. The algorithm ultimately seeks a solution that satisfies the constraints in both the Fourier and the object (support) domains. To take account of this, Fienup treated the \(g(x)\) as a driving function, rather than a solution, to drive the \(g^{\prime}(x)\) toward a solution that satisfies the object domain constraints. Although, in general, the overall procedure is nonlinear, for a small change in \(g(x)\), say \(\Updelta g(x)\), it produces approximately a linear response (Fig. 18.46a,b), \(g^{\prime}(x)+\alpha\,\Updelta g(x)\). To satisfy the support constraint, the property of the small change should haveThat is, the operation drives the output toward zero outside the support. Therefore, the desired input for the next iteration should take the following form$$\alpha\,\Updelta g(x)=\begin{cases}0\;,&x\in S\text{ and }\\ -g^{\prime}(x)\;,&x\notin S\;.\end{cases}$$(18.73)where \(\beta\) is \(\alpha^{-1}\). Since the new input mixes the previous input and output of the operation in a linear combination, this algorithm was named the hybrid-input-output (HIO) algorithm.$$g_{\text{new}}(x)=\begin{cases}0\;,&x\in S\text{ and }\\ g(x)-\beta g^{\prime}(x)\;,&x\notin S\;,\end{cases}$$(18.74) - 4.
*Charge Flipping Algorithm ()*[18.151]:*Oszlanyi*and*Suto*[18.151] proposed a very simple phase retrieval scheme for x-ray crystallography, based on the fact that electron density is positive everywherewhere \(\delta\) is a positive threshold for flipping. Note that in (18.75), no support constraint is needed. Therefore, the CF algorithm can also be applied without oversampling.$$g_{\text{new}}(x)=-g^{\prime}(x)\,\text{ if }\mathrm{g}^{\prime}(x)<\delta\;,$$(18.75)*Wu*and*Spence*[18.152] extended the CF algorithm to phase a complex-valued object using a support constraint: for pixels outside the support, the algorithm flips the signs of their real parts.

##### Use of Image Information

In a TEM, an image of the diffraction object can be recorded directly up to microscope resolution. At the minimum, the electron image provides accurate support information. At the maximum, the electron images can be used to obtain both the amplitude and phase of the exit wave function. Imaging, both the amplitude and the phase are affected by the microscope contrast transfer function (CTF ). In general, the phase is more reliably recorded up to the frequency where the CTF changes sign.

In electron diffraction, the recorded diffraction patterns are also far from ideal and contain only limited information. For example, weak intensities between diffraction spots are often lost because of detector noise. The central peak is often missing or saturated in experimentally recorded diffraction patterns because of its strong intensities. Even if the central peak is recorded, its intensity is mixed with other small angle scatterings, such as inelastic scattering from apertures, which makes small angle scattering intensities less reliable for diffractive imaging. The effect of noise is twofold: it limits the amount information that can be recorded about the shape factors of the particles in the diffraction pattern, and it also limits the amount of information about the weak interference originating from local defects.

- 1.
The image provides an accurate determination of the object boundary as support.

- 2.
At a reduced resolution, the contrast of the reconstructed object should agree with that of the direct image.

The approach is to take limited information in an electron image and diffraction pattern and reconstruct the object function at resolution of, or near, the diffraction limit based on the additional information obtained by oversampling. The authors of this chapter have developed techniques for achieving this. The information used for resolution improvement mainly comes from the diffraction intensity. The electron image recorded at the resolution available from the instrument is used 1) to provide an initial set of phases for low frequency diffraction intensities and 2) to estimate the object boundary for real space constraints or support. To use image and diffraction information effectively, the electron image and the diffraction pattern are recorded from the same area of the sample. As the recorded electron diffraction pattern and the image are then aligned and scaled to match each other using image processing techniques and used for image reconstruction with iterative phase retrieval techniques. Details about these procedures can be found in [18.153] and below.

#### Phasing CdS Quantum Dots

Cadmium selenide quantum dots were prepared by solution based chemical methods by Dr. K.W. Kwon of Professor Moonsub Shim's group at the University of Illinois. A solution containing \({\mathrm{300}}\,{\mathrm{mg}}\) of trioctylphosphine oxide, \({\mathrm{315}}\,{\mathrm{mg}}\) 1,2-hexadecanediol, and \({\mathrm{10}}\,{\mathrm{mL}}\) of octyl ether was vacuum degassed at \({\mathrm{100}}\,{\mathrm{{}^{\circ}\mathrm{C}}}\) for \({\mathrm{30}}\,{\mathrm{min}}\). Sulfur powder (\({\mathrm{15}}\,{\mathrm{mg}}\)) was added at \({\mathrm{100}}\,{\mathrm{{}^{\circ}\mathrm{C}}}\) under \(\mathrm{N_{2}}\) and stirred for \({\mathrm{5}}\,{\mathrm{min}}\). After cooling to \({\mathrm{80}}\,{\mathrm{{}^{\circ}\mathrm{C}}}\), cadmium acetylacetonate (\({\mathrm{150}}\,{\mathrm{mg}}\)) was added and stirred for \({\mathrm{10}}\,{\mathrm{min}}\). The reaction mixture was heated to \({\mathrm{280}}\,{\mathrm{{}^{\circ}\mathrm{C}}}\) and annealed for \({\mathrm{30}}\,{\mathrm{min}}\). The final CdS nanocrystals were precipitated with ethanol, centrifuged to remove excess capping molecules, and redissolved in chloroform. As-synthesized CdS nanocrystals were found by x-ray powder diffraction to be a mixture of wurzite and zinc blend structures.

The CdS quantum dots were supported on ultrathin graphene sheets or carbon nanotubes bundles. Lacey carbon films attached to a copper TEM grid were used to support the graphene and nanotubes. The TEM grid was first immersed in chloroform for \({\mathrm{10}}\,{\mathrm{s}}\) to dissolve the Formvar backing layer and then heated in argon ambient to remove the residue Formvar. Formvar was found to cause contamination problems under the nanoarea electron beam. Commercial double-walled carbon nanotubes were then dispersed onto the grid. Most of the nanotubes form bundles with each other. Graphene sheets were also dispersed onto the grid. Diluted solutions of CdS quantum dots were finally dispersed onto the grid. The quantum dot density was controlled to a very low level, about 1 quantum dot in every \(900{-}1600\,{\mathrm{nm^{2}}}\).

Figure 18.47a,b shows a single crystalline CdS quantum dot of \({\mathrm{9}}\,{\mathrm{nm}}\) in diameter. The diffraction pattern from this quantum dot is shown in Fig. 18.48a,b. By indexing the diffraction pattern, the quantum dot was identified to have a wurzite structure, and the zone axis along which the pattern was recorded was near its \(c\)-axis, or [0001]. The structural model suggests that along this orientation the image should display a honeycomb-like structure, while it now appears to be a close-packed structure instead. This is because the resolution in the direct image is not enough to resolve the pair of atoms separated by \({\mathrm{2.5}}\,{\mathrm{\AA{}}}\).

- 1.
An initial estimate of the object function is obtained from the start image. A small background noise about \(1/5\) of the maximum in the starting image is generated with a random seed number and added to the estimated object function.

- 2.
An iterative phase retrieval is performed starting with the estimated object function and using the HIO algorithm. A real object constraint is applied during this step.

- 3.
Step 2 is followed by iterations using the error reduction algorithm with the real object constraint.

- 4.
Step 3 is followed by iterations using the HIO algorithm without the real object constraint to reconstruct the complex exit wave function.

- 5.
Steps 1 to 4 are repeated using a different random seed number.

- 6.
The object functions obtained from the above steps are averaged.

Compared to the as-recorded HREM image, the reconstructed image clearly shows a honeycomb structure, due to the \({\mathrm{0.72}}\,{\mathrm{\AA{}}}\) information transfer in the recorded diffraction pattern. The resolution improvement can also be seen in the comparison between the power spectra of the as-recorded image and the reconstructed image (Fig. 18.49a,bb).

The resolution improvement of the reconstructed image to sub-Å using information from diffraction pattern over the as-recorded TEM image is further evidenced in Fig. 18.50a-d. The information in the reconstructed image extends to the cubic (\(62\overline{4}\)) reflection of \({\mathrm{0.78}}\,{\mathrm{\AA{}}}\) \(d\)-spacing. In comparison, only the \(\pm(111)\) cubic reflections of \({\mathrm{3.3}}\,{\mathrm{\AA{}}}\) \(d\)-spacing are present in the power spectrum of the starting image. Thus, the resolution improvement from the diffraction pattern is about a factor of 4. The reconstructed image from the diffraction pattern shows clearly resolved atomic columns in areas near the highlighted region. At the orientation where the diffraction pattern was recorded, the smallest separation between the Cd and S atomic columns is \({\mathrm{0.84}}\,{\mathrm{\AA{}}}\). This is clearly resolved in Fig. 18.50a-d. Asymmetric peaks are also seen in the reconstructed image from the intensity profile, taken across a pair of atoms, as shown in Fig. 18.50a-d. In the weak-phase-object approximation, the real part of the complex object is proportional to the object potential and, therefore, a larger peak is expected for Cd than S. Using this information, the crystal polarity can be directly determined from the reconstructed image. Interestingly, both the low-resolution starting image and the reconstructed image indicate an interface present within the particle. The lattices on the two sides of the boundary have a very small misorientation, as can be seen from the change in the contrast of the atomic columns. The lattice is also shifted across the boundary. The shift, measured directly from the reconstructed image, is \({\mathrm{1.6}}\,{\mathrm{\AA{}}}\). The small-angle boundary is also consistent with split Bragg peaks for some reflections, most notably \(\pm(111)\) and \(\pm(222)\), but not others such as (\(2\overline{2}0\)) in the diffraction pattern, as is shown in Fig. 18.51, because the lattice shift occurs only for the \(\pm(111)\) lattice. The reconstructed image thus provides a direct interpretation of the complex diffraction pattern here.

A major difficulty of object reconstruction without image information is that the results of ITA are quite often trapped in local minima, or iterations starting with different set of random noises produce different reconstructions. Often, these constructions are similar to those measured by merit metrics designed to monitor the progress of ITA, such as examining the difference between the calculated intensities and the experimental ones. Thus, none are accurate enough to be selected as the *right* reconstruction [18.155]. To overcome this problem, the guided HIO method was developed based on the idea of a set of images close to the sought-after solution can be used efficiently to guide ITA towards to the solution. In the proposal by *Chen* et al, the images used to guide the search are selected from a set of images reconstructed using different random started phases based on the level of fit to experimental diffraction intensities. The guided HIO method has been used successfully for the reconstruction of crystalline particles and cells without starting image information [18.156, 18.157, 18.158]. The role of the starting image is thus similar in guiding the search of ITA closer to the final solution. However, unlike guided HIO, where the initial selection of starting images can be subjective, there is no ambiguity in the starting image obtained experimentally with its phase and amplitude information, albeit limited in resolution. Further, random noises can be added to the object function estimated from the starting image. The effect of adding noise perturbs the starting phases for the spatial frequencies that are recorded in the image. For high spatial frequencies beyond the image information limit, the random noises introduce new starting phases. Results obtained from different starting random noises can be averaged to provide a robust reconstruction.

#### Atomic Resolution Tomography

Three-dimensional reconstruction at atomic resolution is a major challenge in the characterization of the structure of materials. Recent advances in electron optics have enabled a direct determination of the atomic structure at sub-ångström resolution in 2-D projection [18.159, 18.160, 18.161, 18.162]. However, imaging atoms inside, and at the surface of, a nanoparticle or determining the structure of a 3-D defect requires information of the 3-D atomic structure. Tomographic reconstruction at atomic resolution provides a way forward to 3-D structure determination for any objects.

At atomic resolution, several groups have reported 3-D reconstruction using Z-contrast images obtained in a scanning transmission electron microscope (STEM) equipped with an annular dark field ( ) detector. *Van Aert* et al reported a successful reconstruction of the 3-D atomic structure of Ag precipitates in the Al matrix using discrete tomography [18.163]. This method only requires electron images recorded in a few zone-axis projections, but a prior knowledge of the structure is necessary for the reconstruction. The other approach is to detect atomic position in 3-D using the STEM depth sectioning method [18.164, 18.165, 18.166, 18.77] or using a combination of quantitative STEM and multislice simulation [18.167]. The resolution of these methods, however, is limited along the beam direction by the probe elongation effect resulting from the small electron beam convergence angle and by electron multiple scattering along atomic columns [18.164, 18.168]. Recently, *Jianwei Miao's* group at UCLA demonstrated the 3-D reconstruction of an Au nanoparticle at \({\mathrm{2.4}}\,{\mathrm{\AA{}}}\) resolution by using tomographic reconstruction based on the so-called equal-sloped fast Fourier transform [18.169]. The reconstruction is based on the \(Z\)-contrast image data recorded in a tilt series from \(\mathrm{-72.6}\) to \(72.6^{\circ}\) in equal-slope increment. This method has been further applied to image dislocations in an Au nanoparticle in 3-D [18.170].

In what follows, we describe a 3-D tomographic reconstruction algorithm based on the hybrid input-output (HIO) algorithm developed by *Fienup* [18.148] and polar Fourier fast transform ( ). Only coherent diffraction data obtained in a tilt series are needed for reconstruction. Because information recorded in a diffraction pattern is limited only by scattering, the method described here has the potential to achieve the highest resolution under the kinematical diffraction conditions.

Iteration in HIO requires forward and backward fast Fourier transformation (FFT). However, conventional FFT uses equispaced rectilinear sampling, which cannot be directly extended to tomography reconstruction of diffraction data recorded in a tilt series. To overcome this issue, polar FFT as implemented in the general category of non-equispaced FFT ( ) methods is used. This avoids resampling in the diffraction space and its related issues [18.172]. The basic concept of NFFT computation is to use conventional FFT in a Cartesian grid, while the Fourier frequencies are oversampled, and a window function is used to interpolate between equally and nonequally sampled space. The inverse FT is obtained through a least squares minimization [18.173]. In the 3-D HIO algorithm using polar FFT, the object function is sampled in a Cartesian grid, while the Fourier space is sampled in the cylindrical grid, where in the \(xy\)-plane the sampling is in the polar grid with points equally spaced on concentric circles (Fig. 18.52a,b).

For 3-D reconstruction, the computational cost increases dramatically in order to achieve high resolution. Furthermore, because of oversampling, polar FFT requires a significant increase in the data size, which makes it challenging to implement the HIO algorithm not only in Fourier space but also in object space, while maintaining computational efficiency.

Reconstruction of the 3-D object using the HIO algorithm is performed in both Fourier and object spaces. A set of 2-D diffraction patterns are recorded in a tilt series. The diffraction patterns are combined in 3-D by taking the rotation axis as the \(z\)-axis in the cylindrical coordinate, as illustrated in Fig. 18.52a,ba. For each \(xy\)-plane of constant \(z\), the data is sampled in the polar coordinate of \(\boldsymbol{k}=(k,\theta)\) with \(k_{n}=n\Updelta k\) and \(\theta_{m}=m\Updelta\theta\). Along \(\boldsymbol{z}\), the sampling is performed in equal space with \(z_{l}=l\Updelta z\). Together, the sampling points of \((n,m,l)\) constitute a cylindrical grid. In the object space, a 3-D object is sampled in the equispaced Cartesian coordinate, as shown in Fig. 18.52a,bb.

The 3-D forward FT is achieved by first performing 2-D polar FFT in the \(xy\)-plane, followed by 1-D FFT along \(z\). We call this approach forward cylindrical FT ( ) in what follows. The 2-D polar FFT is applied in each slice of 3-D object data (\(x\) and \(y\)-planes) as step I. Step II uses regular 1-D FFT in the \(z\)-dimension of the 3-D object data. The 3-D inverse FT is carried out by performing 2-D inverse polar FT in the \(xy\)-plane first, and then 1-D inverse FFT along \(z\). We call this inverse CFT (CFT\({}^{-1}\)) in what follows.

- 1.
Selection and precomputation of the FT of a window function

- 2.
FFT of \(\rho(\boldsymbol{r})\) on an oversampled Cartesian grid

- 3.
Calculation of the Fourier frequency on the polar grid by interpolation using convolution between the selected window function and the oversampled Fourier frequencies.

*Lu* et al describe a 3-D reconstruction algorithm based on the cylindrical Fourier transform computed by the nonequispaced fast Fourier transform [18.171]. The algorithm is accelerated and implemented using computer unified device architecture (CUDA ) on a graphics processing unit (GPU ). The algorithm was used to test 3-D atomic resolution reconstruction for a 309-atom Au nanoparticle using calculated diffraction patterns based on kinematical approximation. Results using simulated test data show that the algorithm is capable of reconstructing the 3-D structure at atomic resolution. The reconstructed 3-D object is highly dependent on the quality of the input 2-D diffraction patterns as described by resolution, noise, and missing wedges in the simulated image tilt series (see Fig. 18.53a-i for an example). Without the missing wedge, the algorithm reconstructs the full atomic structure information up to the resolution limit in the simulated diffraction patterns. The missing wedge introduces additional degradation in resolution in the direction normal to the missing wedge. Overall, for the nanoparticle with 309 atoms, the algorithm was capable of reconstructing its 3-D structure by resolving individual atoms using simulated data, provided that the experimental condition is controlled within the tolerance of a \({\mathrm{25}}\%\) noise level, \({\mathrm{1}}\,{\mathrm{\AA{}}}\) information transfer, and a sample tilting range of \(150^{\circ}\) with a \(2^{\circ}\) increment.

When applied to real experimental results, in principle, the utilization of diffraction patterns should be able to promote the image quality, because of the higher signal/noise ratio, the absence of both objective lens aberration and scanning noise. However, significant challenges remain for experimental realization of a truly universal atomic resolution tomography. The unanswered questions include: could the recent success of atomic resolution tomography based on STEM be extended to nonperiodic structures, such as amorphous glass? Is there a limit to the electron multiple scattering effect in a tilt series 3-D reconstruction? How do we interpret reconstructed data in the presence of varying electron multiple scattering effects? Most importantly, 3-D diffraction data collection from a single nanoparticle remains an unsolved experimental challenge, especially for radiation sensitive materials.

## 18.5 Conclusions

The advantages of electron nanodiffraction are the small probes, the strong elastic scattering cross-sections of the high-energy electrons, and the high information content. These advantages allow the recording of diffraction patterns from very small nanostructures, for example, a single-walled carbon nanotube. The challenge in electron nanodiffraction is to relate diffraction information to the atomic structure. Since electron diffraction is not affected by lens aberrations (except geometric distortions at large diffraction angles from the projector lens of the electron microscope), the relationship between the electron diffraction pattern and the structure is simpler than electron imaging. In this chapter, we have demonstrated how this simpler relationship can be made to be a greater benefit for quantitative and high-resolution structure analysis. Specifically, we have outlined an electron diffraction theory based on both kinematic approximation and dynamic diffraction, which can serve as the basis for the interpretation of electron nanodiffraction patterns. We also emphasized the different electron nanoprobes that can be formed inside an electron microscope, which range from a focused beam to parallel illuminations. The flexibility of the electron illumination system for forming different probes is another advantage of electron nanodiffraction. In particular, the use of parallel beams for diffraction imaging is very promising for achieving diffraction-limited resolution. We also demonstrated the principles and applications of scanning electron nanodiffraction and diffraction imaging for orientation mapping, imaging strain, 3-D nanostructure determination, and the study of defects.

## Notes

### Acknowledgements

The writing of this chapter was made possible with the support by US Department of Energy, Grant DEFG02-01ER45923 and NSF DMR 1410596. The work described here would not have been possible without the outstanding efforts of students and postdoc students, especially work by Weijie Huang, Yifei Meng, Yu-Tsun Shao, Kyouhyun Kim, Xiangwen Lu, Wenpei Gao, and Piyush Vivek Deshpande has contributed directly to the writing of this chapter.

## References

- G. Möllenstedt: My early work on convergent-beam electron-diffraction, Phys. Status Solidi (a)
**116**, 13–22 (1989)CrossRefGoogle Scholar - C.H. MacGillavry: Examination of the dynamic theory of electron diffraction on lattice, Physica
**7**, 329–343 (1940)CrossRefGoogle Scholar - J.M. Cowley: Electron nanodiffraction, Microsc. Res. Tech.
**46**, 75–97 (1999)CrossRefGoogle Scholar - J.M. Cowley: Applications of electron nanodiffraction, Micron
**35**, 345 (2004)CrossRefGoogle Scholar - J.C.H. Spence, J.M. Zuo: Large dynamic-range, parallel detection system for electron-diffraction and imaging, Rev. Sci. Instrum.
**59**, 2102–2105 (1988)CrossRefGoogle Scholar - J.M. Zuo: Electron detection characteristics of a slow-scan CCD camera, imaging plates and film, and electron image restoration, Microsc. Res. Tech.
**49**, 245–268 (2000)CrossRefGoogle Scholar - L. Reimer (Ed.):
*Energy-Filtering Transmission Electron Microscopy*(Springer, New York 1995)Google Scholar - J.C.H. Spence, J.M. Zuo:
*Electron Microdiffraction*(Plenum, New York 1992)CrossRefGoogle Scholar - H.E. Elsayed-Ali, P.M. Weber: Time-resolved surface electron diffraction. In:
*Time-Resolved Diffraction*, ed. by J.R. Helliwell, P.M. Rentzepis (Oxford Univ. Press, New York 1997) pp. 284–322Google Scholar - W.E. King, G.H. Campbell, A. Frank, B. Reed, J.F. Schmerge, B.J. Siwick, B.C. Stuart, P.M. Weber: Ultrafast electron microscopy in materials science, biology, and chemistry, J. Appl. Phys.
**97**, 111101 (2005)CrossRefGoogle Scholar - B.J. Siwick, J.R. Dwyer, R.E. Jordan, R.J.D. Miller: An atomic-level view of melting using femtosecond electron diffraction, Science
**302**, 1382–1385 (2003)CrossRefGoogle Scholar - J.M. Zuo, M. Gao, J. Tao, B.Q. Li, R. Twesten, I. Petrov: Coherent nano-area electron diffraction, Microsc. Res. Tech.
**64**, 347–355 (2004)CrossRefGoogle Scholar - G. Deptuch, A. Besson, P. Rehak, M. Szelezniak, J. Wall, M. Winter, Y. Zhu: Direct electron imaging in electron microscopy with monolithic active pixel sensors, Ultramicroscopy
**107**, 674–684 (2007)CrossRefGoogle Scholar - D. Contarato, P. Denes, D. Doering, J. Joseph, B. Krieger: Direct detection in transmission electron microscopy with a 5 μm pitch CMOS pixel sensor, Nucl. Instrum. Methods Phys. Res. A
**635**, 69–73 (2011)CrossRefGoogle Scholar - J.M. Cowley (Ed.):
*Electron Diffraction Techniques*, Vol. I, II (Oxford Univ. Press, Oxford 1992)Google Scholar - L.M. Peng, S.L. Dudarev, M.J. Whelan:
*High-Energy Electron Diffraction and Microscopy*(Oxford Univ. Press, Oxford 2004)Google Scholar - J.P. Morniroli:
*Large-Angle Convergent Beam Electron Diffraction*(Society of French Microscopists, Paris 2002), English VersionGoogle Scholar - J.M. Zuo, J.C.H. Spence:
*Advanced Transmission Electron Microscopy, Imaging and Diffraction in Nanoscience*(Springer, New York 2017)CrossRefGoogle Scholar - S. Morishita, J. Yamasaki, K. Nakamura, T. Kato, N. Tanaka: Diffractive imaging of the dumbbell structure in silicon by spherical-aberration-corrected electron diffraction, Appl. Phys. Lett.
**93**, 183103 (2008)CrossRefGoogle Scholar - K.D. Van der Mast, C.J. Rakels, J.B. Le Poole: A high quality multipurpose objective lens. In:
*Proc. Eur. Congr. Electron Microsc.*(1980) pp. 72–73Google Scholar - K. Ran, X. Mi, Z.J. Shi, Q. Chen, Y.F. Shi, J.M. Zuo: Molecular packing of fullerenes inside single-walled carbon nanotubes, Carbon
**50**, 5450–5457 (2012)CrossRefGoogle Scholar - J.T. McKeown, J.C.H. Spence: The kinematic convergent-beam electron diffraction method for nanocrystal structure determination, J. Appl. Phys.
**106**, 074309 (2009)CrossRefGoogle Scholar - R. Vincent: Techniques of convergent beam electron-diffraction, J. Electron Microsc. Tech.
**13**, 40–50 (1989)CrossRefGoogle Scholar - M. Tanaka, R. Saito, K. Ueno, Y. Harada: Large-angle convergent-beam electron-diffraction, J. Electron Microsc.
**29**, 408–412 (1980)Google Scholar - J.A. Eades: Zone-axis diffraction patterns by the Tanaka method, J. Electron Microsc. Tech.
**1**, 279–284 (1984)CrossRefGoogle Scholar - I.K. Jordan, C.J. Rossouw, R. Vincent: Effects of energy filtering in LACBED patterns, Ultramicroscopy
**35**, 237–243 (1991)CrossRefGoogle Scholar - K.K. Fung: Large-angle convergent-beam zone axis patterns, Ultramicroscopy
**12**, 243–246 (1984)CrossRefGoogle Scholar - M. Terauchi, M. Tanaka: Simultaneous observation of zone-axis pattern and ±G-dark-field pattern in convergent-beam electron-diffraction, J. Electron Microsc.
**34**, 347–356 (1985)Google Scholar - M. Tanaka, M. Terauchi, T. Kaneyama:
*Convergent Beam Electron Diffraction II*(JEOL, Tokyo 1988)Google Scholar - J.P. Morniroli: CBED and LACBED analysis of stacking faults and antiphase boundaries, Mater. Chem. Phys.
**81**, 209–213 (2003)CrossRefGoogle Scholar - J.P. Morniroli, F. Gaillot: Trace analyses from LACBED patterns, Ultramicroscopy
**83**, 227–243 (2000)CrossRefGoogle Scholar - J.P. Morniroli, R.K.W. Marceau, S.P. Ringerz, L. Boulanger: LACBED characterization of dislocation loops, Philos. Mag.
**86**, 4883–4900 (2006)CrossRefGoogle Scholar - C.T. Koch: Aberration-compensated large-angle rocking-beam electron diffraction, Ultramicroscopy
**111**, 828–840 (2011)CrossRefGoogle Scholar - W. Krakow, L.A. Howland: A method for producing hollow cone illumination electronically in the conventional transmission microscope, Ultramicroscopy
**2**, 53–67 (1976)CrossRefGoogle Scholar - J.A. Eades: Zone-axis patterns formed by a new double-rocking technique, Ultramicroscopy
**5**, 71–74 (1980)CrossRefGoogle Scholar - R. Vincent, P.A. Midgley: Double conical beam-rocking system for measurement of integrated electron-diffraction intensities, Ultramicroscopy
**53**, 271–282 (1994)CrossRefGoogle Scholar - J.M. Zuo, J. Tao: Scanning electron nanodiffraction and diffraction imaging. In:
*Scanning Transmission Electron Microscopy*, ed. by S. Pennycook, P. Nellist (Springer, New York 2011)Google Scholar - K.H. Kim, H. Xing, J.M. Zuo, P. Zhang, H.F. Wang: TEM based high resolution and low-dose scanning electron nanodiffraction technique for nanostructure imaging and analysis, Micron
**71**, 39–45 (2015)CrossRefGoogle Scholar - K.H. Downing, R.M. Glaeser: Improvement in high-resolution image quality of radiation-sensitive specimens achieved with reduced spot size of the electron-beam, Ultramicroscopy
**20**, 269–278 (1986)CrossRefGoogle Scholar - C.S. Own, L.D. Marks, W. Sinkler: Electron precession: A guide for implementation, Rev. Sci. Instrum.
**76**, 033703 (2005)CrossRefGoogle Scholar - D. Jacob, P. Cordier, J.P. Morniroli, H.P. Schertl: Precession electron diffraction for the characterization of twinning in pseudo-symmetrical crystals: Case of coesite. In:
*Proc. EMC 2008 14th Eur. Microsc. Congr*, ed. by M. Luysberg, K. Tillmann, T. Weirich (Springer, Berlin, Heidelberg 2008) pp. 193–194Google Scholar - M. Blackman: On the intensities of electron diffraction rings, Proc. R. Soc. A
**173**, 68–82 (1939)CrossRefGoogle Scholar - M. Horstmann, G. Meyer: Messung der Elektronenbeugungsintensitäten polykristalliner Aluminiumschichten bei tiefer Temperatur und Vergleich mit der dynamischen Theorie, Z. Phys.
**182**, 380–397 (1965)CrossRefGoogle Scholar - K. Gjonnes: On the integration of electron diffraction intensities in the Vincent-Midgley precession technique, Ultramicroscopy
**69**, 1–11 (1997)CrossRefGoogle Scholar - J. Hwang, J.Y. Zhang, J. Son, S. Stemmer: Nanoscale quantification of octahedral tilts in perovskite films, Appl. Phys. Lett.
**100**, 191909 (2012)CrossRefGoogle Scholar - J.C.H. Spence, J.M. Cowley: Lattice imaging in STEM, Optik
**50**, 129–142 (1978)Google Scholar - J.C.H. Spence, J. Lynch: STEM microanalysis by transmission electron-energy loss spectroscopy in crystals, Ultramicroscopy
**9**, 267–276 (1982)CrossRefGoogle Scholar - J. Zhu, J.M. Cowley: Micro-diffraction from stacking-faults and twin boundaries in fcc crystals, J. Appl. Crystallogr.
**16**, 171–175 (1983)CrossRefGoogle Scholar - J.M. Cowley, J.C.H. Spence: Convergent beam electron microdiffraction from small crystals, Ultramicroscopy
**6**, 359–366 (1981)CrossRefGoogle Scholar - J.M. LeBeau, S.D. Findlay, L.J. Allen, S. Stemmer: Position averaged convergent beam electron diffraction: Theory and applications, Ultramicroscopy
**110**, 118–125 (2010)CrossRefGoogle Scholar - C. Mory, C. Colliex, J.M. Cowley: Optimum defocus for STEM imaging and microanalysis, Ultramicroscopy
**21**, 171–177 (1987)CrossRefGoogle Scholar - S.D. Berger, I.G. Salisbury, R.H. Milne, D. Imeson, C.J. Humphreys: Electron energy-loss spectroscopy studies of nanometer-scale structures in alumina produced by intense electron-beam irradiation, Philos. Mag. B
**55**, 341–358 (1987)CrossRefGoogle Scholar - J.M. Zuo, I. Vartanyants, M. Gao, R. Zhang, L.A. Nagahara: Atomic resolution imaging of a carbon nanotube from diffraction intensities, Science
**300**, 1419–1421 (2003)CrossRefGoogle Scholar - A. Beche, J.L. Rouviere, L. Clement, J.M. Hartmann: Improved precision in strain measurement using nanobeam electron diffraction, Appl. Phys. Lett.
**95**, 123114 (2009)CrossRefGoogle Scholar - G. Botton: Analytical Electron Microscopy. In:
*Science of Microscopy*, Vol. I, ed. by P. Hawkes, J.C.H. Spence (Springer, New York 2007)Google Scholar - H. Lichte, M. Lehmann: Electron holography—Basics and applications, Rep. Prog. Phys.
**71**, 016102 (2008)CrossRefGoogle Scholar - R.F. Egerton:
*Electron Energy-Loss Spectroscopy in the Electron Microscope*, 2nd edn. (Springer, New York 2011)CrossRefGoogle Scholar - P. Duval, N. Hoan, J. Brian, L. Henry: Réalisation d'un dispositif de filtrage en énergie des images de microdiffraction électronique, Nouv. Rev. Opt. Appl.
**1**, 221–228 (1970)CrossRefGoogle Scholar - M.M.J. Treacy, J.M. Gibson: The effects of elastic relaxation on transmission electron-microscopy studies of thinned composition-modulated materials, J. Vac. Sci. Technol. B
**4**, 1458–1466 (1986)CrossRefGoogle Scholar - P. Hirsch, A. Howie, R.B. Nicolson, D.W. Pashley, M.J. Whelan:
*Electron Microscopy of Thin Crystals*(Krieger, Malabar 1977)Google Scholar - J.M. Zuo, A.L. Weickenmeier: On the beam selection and convergence in the Bloch-wave method, Ultramicroscopy
**57**, 375–383 (1995)CrossRefGoogle Scholar - I.A. Sheremetyev, A.V. Turbal, Y.M. Litvinov, M.A. Mikhailov: Computer deciphering of Laue patterns: Application to white synchrotron x-ray topography, Nucl. Instrum. Methods Phys. Res. A
**308**, 451–455 (1991)CrossRefGoogle Scholar - H.R. Wenk, F. Heidelbach, D. Chateigner, F. Zontone: Laue orientation imaging, J. Synchrotron Radiat.
**4**, 95–101 (1997)CrossRefGoogle Scholar - S. Zaefferer: New developments of computer-aided crystallographic analysis in transmission electron microscopy, J. Appl. Crystallogr.
**33**, 10–25 (2000)CrossRefGoogle Scholar - E.F. Rauch, L. Dupuy: Rapid spot diffraction patterns identification through template matching, Arch. Metall. Mater.
**50**, 87–99 (2005)Google Scholar - E.F. Rauch, A. Duft: Orientation maps derived from TEM diffraction patterns collected with an external CCD camera, Mater. Sci. Forum
**495–497**, 197–202 (2005)CrossRefGoogle Scholar - E.F. Rauch, M. Veron: Coupled microstructural observations and local texture measurements with an automated crystallographic orientation mapping tool attached to a TEM, Materialwiss. Werkstofftech.
**36**, 552–556 (2005)CrossRefGoogle Scholar - G. Wu, S. Zaefferer: Advances in TEM orientation microscopy by combination of dark-field conical scanning and improved image matching, Ultramicroscopy
**109**, 1317–1325 (2009)CrossRefGoogle Scholar - E.F. Rauch, J. Portillo, S. Nicolopoulos, D. Bultreys, S. Rouvimov, P. Moeck: Automated nanocrystal orientation and phase mapping in the transmission electron microscope on the basis of precession electron diffraction, Z. Kristallogr.
**225**, 103–109 (2010)CrossRefGoogle Scholar - Y. Meng, J.-M. Zuo: Improvements in electron diffraction pattern automatic indexing algorithms, Eur. Phys. J. Appl. Phys.
**80**, 10701 (2017)CrossRefGoogle Scholar - J.P. Lewis: Fast template matching, Vis. Interface
**95**, 120–123 (1995)Google Scholar - D. Dingley: Progressive steps in the development of electron backscatter diffraction and orientation imaging microscopy, J. Microsc.
**213**, 214–224 (2004)CrossRefGoogle Scholar - R. van Bremen, D. Ribas Gomes, L.T.H. de Jeer, V. Ocelík, J.T.M. De Hosson: On the optimum resolution of transmission-electron backscattered diffraction (t-EBSD), Ultramicroscopy
**160**, 256–264 (2016)CrossRefGoogle Scholar - K.J. Ganesh, A.D. Darbal, S. Rajasekhara, G.S. Rohrer, K. Barmak, P.J. Ferreira: Effect of downscaling nano-copper interconnects on the microstructure revealed by high resolution TEM-orientation-mapping, Nanotechnology
**23**, 135702 (2012)CrossRefGoogle Scholar - E.F. Rauch, M. Véron: Automated crystal orientation and phase mapping in TEM, Mater. Charact.
**98**, 1–9 (2014)CrossRefGoogle Scholar - A.D. Darbal, K.J. Ganesh, X. Liu, S.B. Lee, J. Ledonne, T. Sun, B. Yao, A.P. Warren, G.S. Rohrer, A.D. Rollett, P.J. Ferreira, K.R. Coffey, K. Barmak: Grain boundary character distribution of nanocrystalline Cu thin films using stereological analysis of transmission electron microscope orientation maps, Microsc. Microanal.
**19**, 111–119 (2013)CrossRefGoogle Scholar - Y. Hu, J.H. Huang, J.M. Zuo: In situ characterization of fracture toughness and dynamics of nanocrystalline titanium nitride films, J. Mater. Res.
**31**, 370–379 (2016)CrossRefGoogle Scholar - J.L. Rouviere, A. Beche, Y. Martin, T. Denneulin, D. Cooper: Improved strain precision with high spatial resolution using nanobeam precession electron diffraction, Appl. Phys. Lett.
**103**, 241913 (2013)CrossRefGoogle Scholar - H.N. Chapman, A. Barty, S. Marchesini, A. Noy, S.R. Hau-Riege, C. Cui, M.R. Howells, R. Rosen, H. He, J.C.H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, D. Shapiro: High-resolution ab initio three-dimensional x-ray diffraction microscopy, J. Opt. Soc. Am. A
**23**, 1179–1200 (2006)CrossRefGoogle Scholar - H. Poulsen: An introduction to three-dimensional x-ray diffraction microscopy, J. Appl. Crystallogr.
**45**, 1084–1097 (2012)CrossRefGoogle Scholar - B.C. Larson, W. Yang, G.E. Ice, J.D. Budai, J.Z. Tischler: Three-dimensional x-ray structural microscopy with submicrometre resolution, Nature
**415**, 887–890 (2002)CrossRefGoogle Scholar - W. Ludwig, S. Schmidt, E.M. Lauridsen, H.F. Poulsen: X-ray diffraction contrast tomography: A novel technique for three-dimensional grain mapping of polycrystals. I. Direct beam case, J. Appl. Crystallogr.
**41**, 302–309 (2008)CrossRefGoogle Scholar - A.D. Rollett, S.B. Lee, R. Campman, G.S. Rohrer: Three-dimensional characterization of microstructure by electron back-scatter diffraction, Annu. Rev. Mater. Res.
**37**, 627–658 (2007)CrossRefGoogle Scholar - H.H. Liu, S. Schmidt, H.F. Poulsen, A. Godfrey, Z.Q. Liu, J.A. Sharon, X. Huang: Three-dimensional orientation mapping in the transmission electron microscope, Science
**332**, 833–834 (2011)CrossRefGoogle Scholar - A.S. Eggeman, R. Krakow, P.A. Midgley: Scanning precession electron tomography for three-dimensional nanoscale orientation imaging and crystallographic analysis, Nat. Commun.
**6**, 7267 (2015)CrossRefGoogle Scholar - P.A. Midgley, R.E. Dunin-Borkowski: Electron tomography and holography in materials science, Nat. Mater.
**8**, 271–280 (2009)CrossRefGoogle Scholar - Y. Meng, J.-M. Zuo: Three-dimensional nanostructure determination from a large diffraction data set recorded using scanning electron nanodiffraction, IUCrJ
**3**, 300–308 (2016)CrossRefGoogle Scholar - J.M. Zuo, A.B. Shah, H. Kim, Y.F. Meng, W.P. Gao, J.L. Rouviere: Lattice and strain analysis of atomic resolution Z-contrast images based on template matching, Ultramicroscopy
**136**, 50–60 (2014)CrossRefGoogle Scholar - S. Zaefferer: A critical review of orientation microscopy in SEM and TEM, Cryst. Res. Technol.
**46**, 607–628 (2011)CrossRefGoogle Scholar - G.T. Herman:
*Fundamentals of Computerized Tomography: Image Reconstruction from Projections*, Advances in Pattern Recognition (Springer, London 2009)CrossRefGoogle Scholar - J. Amanatides, A. Woo: A fast voxel traversal algorithm for ray tracing, Eurographics
**87**, 3–10 (1987)Google Scholar - S. Kaczmarz: Angenäherte Auflösung von Systemen linearer Gleichungen, Bull. Intern. Acad. Pol. Sci. Lett., Cl. Sci. Math. Nat. A
**35**, 335–357 (1937)Google Scholar - F.L. Markley: Attitude determination using vector observations and the singular value decomposition, J. Astronaut. Sci.
**38**, 245–258 (1988)Google Scholar - S.V. Fortuna, Y.P. Sharkeev, A.J. Perry, J.N. Matossian, I.A. Shulepov: Microstructural features of wear-resistant titanium nitride coatings deposited by different methods, Thin Solid Films
**377/378**, 512–517 (2000)CrossRefGoogle Scholar - W.-L. Pan, G.-P. Yu, J.-H. Huang: Mechanical properties of ion-plated tin films on AISI D-2 steel, Surf. Coat. Technol.
**110**, 111–119 (1998)CrossRefGoogle Scholar - A.-N. Wang, G.P. Yu, J.-H. Huang: Fracture toughness measurement on tin hard coatings using internal energy induced cracking, Surf. Coat. Technol.
**239**, 20–27 (2014)CrossRefGoogle Scholar - C.H. Ma, J.-H. Huang, H. Chen: Nanohardness of nanocrystalline tin thin films, Surf. Coat. Technol.
**200**, 3868–3875 (2006)CrossRefGoogle Scholar - P.H. Mayrhofer, C. Mitterer, J. Musil: Structure–property relationships in single- and dual-phase nanocrystalline hard coatings, Surf. Coat. Technol.
**174/175**, 725–731 (2003)CrossRefGoogle Scholar - P.H. Mayrhofer, F. Kunc, J. Musil, C. Mitterer: A Comparative study on reactive and non-reactive unbalanced magnetron sputter deposition of tin coatings, Thin Solid Films
**415**, 151–159 (2002)CrossRefGoogle Scholar - L.M. Peng, S.L. Dudarev, M.J. Whelan:
*High Energy Electron Diffraction and Microscopy*(Oxford Univ. Press, Oxford 2004)Google Scholar - D.M. Bird, Q.A. King: Absorptive form-factors for high-energy electron-diffraction, Acta Crystallogr. A
**46**, 202–208 (1990)CrossRefGoogle Scholar - A. Weickenmeier, H. Kohl: Computation of absorptive form-factors for high-energy electron-diffraction, Acta Crystallogr. A
**47**, 590–597 (1991)CrossRefGoogle Scholar - L.M. Peng: Anisotropic thermal vibrations and dynamical electron diffraction by crystals, Acta Crystallogr. A
**53**, 663–672 (1997)CrossRefGoogle Scholar - L. Sturkey: The use of electron-diffraction intensities in structure determination, Acta Crystallogr.
**10**, 858 (1957)Google Scholar - D. Jacob, J.M. Zuo, A. Lefebvre, Y. Cordier: Composition analysis of semiconductor quantum wells by energy filtered convergent-beam electron diffraction, Ultramicroscopy
**108**, 358–366 (2008)CrossRefGoogle Scholar - C.J. Rossouw, M. Alkhafaji, D. Cherns, J.W. Steeds, R. Touaitia: A treatment of dynamic diffraction for multiply layered structures, Ultramicroscopy
**35**, 229–236 (1991)CrossRefGoogle Scholar - J.M. Cowley, A.F. Moodie: The scattering of electrons by atoms and crystals. I. A new theoretical approach, Acta Crystallographica
**10**(10), 609–619 (1957)CrossRefGoogle Scholar - K. Ishizuka: Multislice formula for inclined illumination, Acta Crystallogr. A
**38**, 773–779 (1982)CrossRefGoogle Scholar - B.F. Buxton, J.A. Eades, J.W. Steeds, G.M. Rackham: Symmetry of electron-diffraction zone axis patterns, Philos. Trans. R. Soc. A
**281**, 171 (1976)CrossRefGoogle Scholar - M. Tanaka, R. Saito, H. Sekii: Point-group determination by convergent-beam electron-diffraction, Acta Crystallogr. A
**39**, 357–368 (1983)CrossRefGoogle Scholar - M. Tanaka, H. Sekii, T. Nagasawa: Space-group determination by dynamic extinction in convergent-beam electron-diffraction, Acta Crystallogr. A
**39**, 825–837 (1983)CrossRefGoogle Scholar - G.B. Hu, L.M. Peng, Q.F. Yu, H.Q. Lu: Automated identification of symmetry in CBED patterns: A genetic approach, Ultramicroscopy
**84**, 47–56 (2000)CrossRefGoogle Scholar - R. Vincent, T.D. Walsh: Quantitative assessment of symmetry in CBED patterns, Ultramicroscopy
**70**, 83–94 (1997)CrossRefGoogle Scholar - J.F. Mansfield: Error bars in CBED symmetry?, Ultramicroscopy
**18**, 91–96 (1985)CrossRefGoogle Scholar - K.H. Kim, J.M. Zuo: Symmetry quantification and mapping using convergent beam electron diffraction, Ultramicroscopy
**124**, 71–76 (2013)CrossRefGoogle Scholar - J.M. Kiat, Y. Uesu, B. Dkhil, M. Matsuda, C. Malibert, G. Calvarin: Monoclinic structure of unpoled morphotropic high piezoelectric PMN-PT and PZN-PT compounds, Phys. Rev. B
**65**, 064106 (2002)CrossRefGoogle Scholar - Y.-T. Shao, J.-M. Zuo: Fundamental symmetry of barium titanate single crystal determined using energy-filtered scanning convergent beam electron diffraction, Microsc. Microanal.
**22**, 516–517 (2016)CrossRefGoogle Scholar - J.M. Zuo, J.C.H. Spence: Automated structure factor refinement from convergent-beam patterns, Ultramicroscopy
**35**, 185–196 (1991)CrossRefGoogle Scholar - J.M. Zuo: Accurate structure refinement and measurement of crystal charge distribution using convergent beam electron diffraction, Microsc. Res. Tech.
**46**, 220–233 (1999)CrossRefGoogle Scholar - Y. Ogata, K. Tsuda, M. Tanaka: Determination of the electrostatic potential and electron density of silicon using convergent-beam electron diffraction, Acta Crystallogr. A
**64**, 587–597 (2008)CrossRefGoogle Scholar - P.N.H. Nakashima, B.C. Muddle: Differential convergent beam electron diffraction: Experiment and theory, Phys. Rev. B
**81**, 115135 (2010)CrossRefGoogle Scholar - J.M. Zuo: Quantitative convergent beam electron diffraction, Mater. Trans. JIM
**39**, 938–946 (1998)CrossRefGoogle Scholar - J. Friis, B. Jiang, J.C.H. Spence, R. Holmestad: Quantitative convergent beam electron diffraction measurements of low-order structure factors in copper, Microsc. Microanal.
**9**, 379–389 (2003)CrossRefGoogle Scholar - P.N.H. Nakashima: Improved quantitative CBED structure-factor measurement by refinement of nonlinear geometric distortion corrections, J. Appl. Crystallogr.
**38**, 374–376 (2005)CrossRefGoogle Scholar - K. Tsuda, M. Tanaka: Refinement of crystal structural parameters using two-dimensional energy-filtered CBED patterns, Acta Crystallogr. A
**55**, 939–954 (1999)CrossRefGoogle Scholar - B. Jiang, J.M. Zuo, J. Friis, J.C.H. Spence: On the consistency of QCBED structure factor measurements for TiO
_{2}(Rutile), Microsc. Microanal.**9**, 457–467 (2003)CrossRefGoogle Scholar - M. Saunders, D.M. Bird, N.J. Zaluzec, W.G. Burgess, A.R. Preston, C.J. Humphreys: Measurement of low-order structure factors for silicon from zone-axis CBED patterns, Ultramicroscopy
**60**, 311–323 (1995)CrossRefGoogle Scholar - B.T.M. Willis, A.W. Pryor:
*Thermal Vibrations in Crystallography*(Cambridge Univ. Press, Cambridge 1975)Google Scholar - G. Ren, J.M. Zuo, L.M. Peng: Accurate measurements of crystal structure factors using a FEG electron microscope, Micron
**28**, 459–467 (1997)CrossRefGoogle Scholar - J.M. Zuo: Measurements of electron densities in solids: A real-space view of electronic structure and bonding in inorganic crystals, Rep. Prog. Phys.
**67**, 2053–2103 (2004)CrossRefGoogle Scholar - J.M. Rodenburg: Ptychography and related diffractive imaging methods, Adv. Imaging Electron Phys.
**150**, 87–184 (2008)CrossRefGoogle Scholar - W. Hoppe: Trace structure-analysis, ptychography, phase tomography, Ultramicroscopy
**10**, 187–198 (1982)CrossRefGoogle Scholar - J.C.H. Spence, U. Weierstall, M. Howells: Phase recovery and lensless imaging by iterative methods in optical, x-ray and electron diffraction, Philos. Trans. R. Soc. A
**360**, 875–895 (2002)CrossRefGoogle Scholar - H. Lichte: Electron holography approaching atomic resolution, Ultramicroscopy
**20**, 293–304 (1986)CrossRefGoogle Scholar - A. Orchowski, W.D. Rau, H. Lichte: Electron holography surmounts resolution limit of electron-microscopy, Phys. Rev. Lett.
**74**, 399–402 (1995)CrossRefGoogle Scholar - S.G. Podorov, K.M. Pavlov, D.M. Paganin: A non-iterative reconstruction method for direct and unambiguous coherent diffractive imaging, Opt. Express
**15**, 9954–9962 (2007)CrossRefGoogle Scholar - I.L. Karle, J. Karle: The crystal and molecular structure of the alkaloid jamine from Ormosia jamaicensis, Acta Crystallogr.
**17**, 1356 (1964)CrossRefGoogle Scholar - J. Karle, I.L. Karle: The symbolic addition procedure for phase determination for centrosymmetric and noncentrosymmetric crystals, Acta Crystallogr.
**21**, 849 (1966)CrossRefGoogle Scholar - D. Gabor: A new microscopic principle, Nature
**161**, 777–778 (1948)CrossRefGoogle Scholar - J.C.H. Spence: Stem and shadow-imaging of biomolecules at 6 eV beam energy, Micron
**28**, 101–116 (1997)CrossRefGoogle Scholar - J.C.H. Spence, T. Vecchione, U. Weierstall: A coherent photofield electron source for fast diffractive and point-projection imaging, Philos. Mag.
**90**, 4691–4702 (2010)CrossRefGoogle Scholar - H. Nyquist: Certain topics in telegraph transmission theory, Trans. Am. Inst. Electr. Eng.
**47**, 617 (1928)CrossRefGoogle Scholar - C.E. Shannon: Communication in the presence of noise, Inst. Radio Eng.
**37**, 10 (1949)Google Scholar - D. Sayre, H.N. Chapman, J. Miao: On the extendibility of x-ray crystallography to noncrystals, Acta Crystallogr. A
**54**, 232–239 (1998)CrossRefGoogle Scholar - J.C.H. Spence, U. Weierstall, M. Howells: Coherence and sampling requirements for diffraction imaging, Ultramicroscopy
**101**, 149–152 (2004)CrossRefGoogle Scholar - W.J. Huang, B. Jiang, R.S. Sun, J.M. Zuo: Towards sub-Å atomic resolution electron diffraction imaging of metallic nanoclusters: A simulation study of experimental parameters and reconstruction algorithms, Ultramicroscopy
**107**, 1159–1170 (2007)CrossRefGoogle Scholar - V. Elser: Phase retrieval by iterated projections, J. Opt. Soc. Am. A
**20**, 40 (2003)CrossRefGoogle Scholar - J.R. Fienup: Phase retrieval algorithms—A comparison, Appl. Opt.
**21**, 2758–2769 (1982)CrossRefGoogle Scholar - J.R. Fienup: Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint, J. Opt. Soc. Am.
**6**, 118 (1987)CrossRefGoogle Scholar - R.W. Gerchberg, W.O. Saxton: Practical algorithm for determination of phase from image and diffraction plane pictures, Optik
**35**, 237 (1972)Google Scholar - G. Oszlanyi, A. Suto: Ab initio structure solution by charge flipping, Acta Crystallogr. A
**60**, 134–141 (2004)CrossRefGoogle Scholar - J.S. Wu, J.C.H. Spence: Reconstruction of complex single-particle images using charge-flipping algorithm, Acta Crystallogr. A
**61**, 194–200 (2005)CrossRefGoogle Scholar - J.M. Zuo, J. Zhang, W.J. Huang, K. Ran, B. Jiang: Combining real and reciprocal space information for aberration free coherent electron diffractive imaging, Ultramicroscopy
**111**, 817–823 (2011)CrossRefGoogle Scholar - R.P. Millane, W.J. Stroud: Reconstructing symmetric images from their undersampled Fourier intensities, J. Opt. Soc. Am. A
**14**, 568–579 (1997)CrossRefGoogle Scholar - C.C. Chen, J. Miao, C.W. Wang, T.K. Lee: Application of optimization technique to noncrystalline x-ray diffraction microscopy: Guided hybrid input-output method, Phys. Rev. B
**76**, 064113 (2007)CrossRefGoogle Scholar - R. Dronyak, K.S. Liang, Y.P. Stetsko, T.K. Lee, C.K. Feng, J.S. Tsai, F.R. Chen: Electron diffractive imaging of nano-objects using a guided method with a dynamic support, Appl. Phys. Lett.
**95**, 111908 (2009)CrossRefGoogle Scholar - H.D. Jiang, C.Y. Song, C.C. Chen, R. Xu, K.S. Raines, B.P. Fahimian, C.H. Lu, T.K. Lee, A. Nakashima, J. Urano, T. Ishikawa, F. Tamanoi, J.W. Miao: Quantitative 3-D imaging of whole, unstained cells by using x-ray diffraction microscopy, Proc. Natl. Acad. Sci. U.S.A.
**107**, 11234–11239 (2010)CrossRefGoogle Scholar - J. Gulden, O.M. Yefanov, A.P. Mancuso, V.V. Abramova, J. Hilhorst, D. Byelov, I. Snigireva, A. Snigirev, A.V. Petukhov, I.A. Vartanyants: Coherent x-ray imaging of defects in colloidal crystals, Phys. Rev. B
**81**, 224105 (2010)CrossRefGoogle Scholar - P.E. Batson, N. Dellby, O.L. Krivanek: Sub-Angstrom resolution using aberration corrected electron optics, Nature
**418**, 617–620 (2002)CrossRefGoogle Scholar - P.D. Nellist, M.F. Chisholm, N. Dellby, O.L. Krivanek, M.F. Murfitt, Z.S. Szilagyi, A.R. Lupini, A. Borisevich, W.H. Sides, S.J. Pennycook: Direct sub-angstrom imaging of a crystal lattice, Science
**305**, 1741–1741 (2004)CrossRefGoogle Scholar - R. Erni, M.D. Rossell, C. Kisielowski, U. Dahmen: Atomic-resolution imaging with a sub-50-pm electron probe, Phys. Rev. Lett.
**102**, 096101 (2009)CrossRefGoogle Scholar - H. Sawada, Y. Tanishiro, N. Ohashi, T. Tomita, F. Hosokawa, T. Kaneyama, Y. Kondo, K. Takayanagi: STEM imaging of 47-pm-separated atomic columns by a spherical aberration-corrected electron microscope with a 300-kV cold field emission gun, J. Electron Microsc.
**58**, 357–361 (2009)CrossRefGoogle Scholar - S. Van Aert, K.J. Batenburg, M.D. Rossell, R. Erni, G. Van Tendeloo: Three-dimensional atomic imaging of crystalline nanoparticles, Nature
**470**, 374–377 (2011)CrossRefGoogle Scholar - A.Y. Borisevich, A.R. Lupini, S.J. Pennycook: Depth sectioning with the aberration-corrected scanning transmission electron microscope, Proc. Natl. Acad. Sci. U.S.A.
**103**, 3044–3048 (2006)CrossRefGoogle Scholar - A.Y. Borisevich, A.R. Lupini, S. Travaglini, S.J. Pennycook: Depth sectioning of aligned crystals with the aberration-corrected scanning transmission electron microscope, J. Electron Microsc.
**55**, 7–12 (2006)CrossRefGoogle Scholar - H.L. Xin, D.A. Muller: Aberration-corrected ADF-STEM depth sectioning and prospects for reliable 3-D imaging in S/TEM, J. Electron Microsc.
**58**, 157–165 (2009)CrossRefGoogle Scholar - R. Ishikawa, A.R. Lupini, S.D. Findlay, S.J. Pennycook: Quantitative annular dark field electron microscopy using single electron signals, Microsc. Microanal.
**20**, 99–110 (2014)CrossRefGoogle Scholar - E.C. Cosgriff, P.D. Nellist: A Bloch wave analysis of optical sectioning in aberration-corrected STEM, Ultramicroscopy
**107**, 626–634 (2007)CrossRefGoogle Scholar - M.C. Scott, C.-C. Chen, M. Mecklenburg, C. Zhu, R. Xu, P. Ercius, U. Dahmen, B.C. Regan, J. Miao: Electron tomography at 2.4-ångström resolution, Nature
**483**, 444–491 (2012)CrossRefGoogle Scholar - C.-C. Chen, C. Zhu, E.R. White, C.-Y. Chiu, M.C. Scott, B.C. Regan, L.D. Marks, Y. Huang, J. Miao: Three-dimensional imaging of dislocations in a nanoparticle at atomic resolution, Nature
**496**, 74 (2013)CrossRefGoogle Scholar - X.W. Lu, W.P. Gao, J.M. Zuo, J.B. Yuan: Atomic resolution tomography reconstruction of tilt series based on a GPU accelerated hybrid input-output algorithm using polar fourier transform, Ultramicroscopy
**149**, 64–73 (2015)CrossRefGoogle Scholar - J. Miao, T. Ohsuna, O. Terasaki, K.O. Hodgson, M.A. O'Keefe: Atomic resolution three-dimensional electron diffraction microscopy, Phys. Rev. Lett.
**89**, 155502 (2002)CrossRefGoogle Scholar - J. Keiner, S. Kunis, D. Potts: Using NFFT 3---A software library for various nonequispaced fast Fourier transforms, ACM Trans. Math. Softw.
**36**, 19 (2009)CrossRefGoogle Scholar - M. Fenn, S. Kunis, D. Potts: On the computation of the polar FFT, Appl. Comput. Harmon. Anal.
**22**, 257–263 (2007)CrossRefGoogle Scholar - S. Kunis, S. Kunis: The nonequispaced FFT on graphics processing units, Proc. Appl. Math. Mech.
**12**, 7–10 (2012)CrossRefGoogle Scholar