In contrast, Kogelnik's theory shows small discrepancies away from Bragg resonance. The new coupled wave theory may easily be extended to an N-coupled wave theory for the case of the multiplexed polychromatic grating and indeed for the purposes of analytically describing diffraction in the colour hologram.

In the simple case of a monochromatic spatially-multiplexed grating at Bragg resonance the theory is in exact agreement with the predictions of conventional N-coupled wave theory. Crossref Google Scholar. IOPscience Google Scholar. Google Scholar. This site uses cookies. By continuing to use this site you agree to our use of cookies. To find out more, see our Privacy and Cookies policy. Close this notification. Journal of Physics: Conference Series. Only by considering the full dynamical model through multislice calculations can the diffraction patterns generated by PED be simulated.

However, this requires the crystal potential to be known, and thus is most valuable in refining the crystal potentials suggested through direct methods approaches. The theory of precession electron diffraction is still an active area of research, and efforts to improve on the ability to correct measured intensities without a priori knowledge are ongoing. The first precession electron diffraction system was developed by Vincent and Midgley in Bristol, UK and published in Preliminary investigation into the Er 2 Ge 2 O 7 crystal structure demonstrated the feasibility of the technique at reducing dynamical effects and providing quasi-kinematical patterns that could be solved through direct methods to determine crystal structure.

Gjonnes Oslo , Migliori Bologna , and L. Marks Northwestern. This hardware solution enabled more widespread implementation of the technique and spurred its more mainstream adoption into the crystallography community. Software methods have also been developed to achieve the necessary scanning and descanning using the built-in electronics of the TEM.

This plug-in enables the widely used software package to collect precession electron diffraction patterns without additional modifications to the microscope. According to NanoMEGAS, as of June, , more than publications have relied on the technique to solve or corroborate crystal structures; many on materials that could not be solved by other conventional crystallography techniques like x-ray diffraction.

Their retrofit hardware system is used in more than 75 laboratories across the world. The primary goal of crystallography is to determine the three dimensional arrangement of atoms in a crystalline material.

## Journal of the Optical Society of America

While historically, x-ray crystallography has been the predominant experimental method used to solve crystal structures ab initio , the advantages of precession electron diffraction make it one of the preferred methods of electron crystallography. In a transmission electron microscope , this is accomplished by recording a diffraction pattern at a large number of points pixels over a region of the crystalline specimen.

By comparing the recorded patterns to a database of known patterns either previously indexed experimental patterns or simulated patterns , the relative orientation of grains in the field of view can be determined. Because this process is highly automated, the quality of the recorded diffraction patterns is crucial to the software's ability to accurately compare and assign orientations to each pixel.

Thus, the advantages of PED are well-suited for use with this scanning technique. Although the PED technique was initially developed for its improved diffraction applications, the advantageous properties of the technique have been found to enhance many other investigative techniques in the TEM. These include bright field and dark field imaging , electron tomography , and composition-probing techniques like energy-dispersive x-ray spectroscopy EDS and electron energy loss spectroscopy EELS.

Though many people conceptualize images and diffraction patterns separately, they contain principally the same information. In the simplest approximation, the two are simply Fourier transforms of one another. Thus, the effects of beam precession on diffraction patterns also have significant effects on the corresponding images in the TEM. Specifically, the reduced dynamical intensity transfer between beams that is associated with PED results in reduced dynamical contrast in images collected during precession of the beam.

This includes a reduction in thickness fringes, bend contours, and strain fields. In an extension of the application of PED to imaging, electron tomography can benefit from the reduction of dynamic contrast effects. Tomography entails collecting a series of images 2-D projections at various tilt angles and combining them to reconstruct the three dimensional structure of the specimen.

Because many dynamical contrast effects are highly sensitive to the orientation of the crystalline sample with respect to the incident beam, these effects can convolute the reconstruction process in tomography. Similarly to single imaging applications, by reducing dynamical contrast, interpretation of the 2-D projections and thus the 3-D reconstruction are more straightforward. Energy-dispersive x-ray spectroscopy EDS and electron energy loss spectroscopy EELS are commonly used techniques to both qualitatively and quantitatively probe the composition of samples in the TEM.

A primary challenge in the quantitative accuracy of both techniques is the phenomenon of channelling. Put simply, in a crystalline solid, the probability of interaction between an electron and ion in the lattice depends strongly on the momentum direction and velocity of the electron. When probing a sample under diffraction conditions near a zone axis, as is often the case in EDS and EELS applications, channelling can have a large impact on the effective interaction of the incident electrons with specific ions in the crystal structure.

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In practice, this can lead to erroneous measurements of composition that depend strongly on the orientation and thickness of the sample and the accelerating voltage. Since PED entails an integration over incident directions of the electron probe, and generally does not include beams parallel to the zone axis, the detrimental channeling effects outlined above can be minimized, yielding far more accurate composition measurements in both techniques. From Wikipedia, the free encyclopedia. AST-FFT modeling needs as input information a binary matrix representing the opaque shape of the particle and its location within the optical path.

The latter is defined by the distance Z from the particle to the position at which we want to compute the diffraction pattern or the distance from the object plane to the particle when using an optical system.

We performed numerical simulations of diffraction patterns recorded by a binary OAP probe, which in this study corresponds to a 2D-S probe described in Sect. Due to diffraction, a bright spot — called Poisson's spot — appears at the center of the shadow diffraction pattern.

We notice that a bright spot, called Poisson's spot, appears at the center of the diffraction pattern shadow image. As Z d increases, the image becomes less focused and the diameter of the Poisson spot increases. As Z d continues to increase, the diffraction pattern becomes more and more blurry, and at some point the light intensity no longer falls below a specific triggering level at any point of the pattern. The size of the pixels in the array direction can slightly vary from one instrument to another. Based on our own calibration using a spinning glass disc with imprinted opaque disc shapes described further below , we found a mean value of Thus, we can assume that a particle is illuminated by a monochromatic plane wave.

The receiving system consists of imaging optics and a linear photodiode array. The imaging optical system is based on a Keplerian telescope design. The photodiode array is positioned in the focal plane of the back lens the eyepiece in the image space. The object plane is the conjugate plane, that is, the focal plane of the front lens the objective in the object space.

Based on the instrument optics, the object plane is located in the middle of the laser beam between the two arms of the probe Fig. The probe consists of two pairs of arms allowing measurements in two orthogonal directions. We only consider one couple of arms here. The object plane is located in the middle of the laser beam between the two arms.

Considering opaque discs crossing the 2D-S laser beam, Fig.

**au.noqytivaly.tk**

## Precession electron diffraction - Wikipedia

The hatched area illustrates the arm limit and the photodiode array size. However, particles of that size are potentially observed with more or less important distortion in the out-of-focus domain. In the following third section, we compare theoretical diffraction patterns of different opaque shapes with experimental measurements of the 2D-S probe. Therefore, several spinning glass discs with various chrome opaque particle shapes imprinted on the glass disc surfaces were used Fig.

This has been performed for opaque disc shapes in the past e. Note that the chosen shapes represent only one particular projection of a 3-D particle columnar or capped columnar particles on a 2-D plane. In each column, on the left-hand side are shown the theoretical 2D-S records and on the right-hand side are shown the images recorded by the 2D-S experimentally. Note that images are framed according to the particle size; i. Results obtained for negative values of Z look very similar not shown. A striking result is that measurements obtained with the 2D-S probe are in really good agreement with the theoretical diffraction simulation results, not only in terms of diffraction pattern, but also with respect to the DoF limit which is illustrated by the disappearing image at approximately the same Z value in theory and the corresponding measurement.

The image resembles more a disc with Poisson's spot than a column. The evolution of the particle shape with Z with a 0. Comparison between theory and measurements again shows very good agreement. Also for this crystal geometry, diffraction can produce patterns that are very different compared to the initial shape.

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In addition, we notice that these patterns look very different than those found in Fig. Here, the diffraction pattern turns into a capped columnar shape as Z increases see, e. Once again, the very good agreement between simulated and recorded images is striking. We notice that several small patterns may be detached from the main particle as a function of distance Z from the object plane.

As a particle moves away from the object plane, we notice that its image becomes more and more roundish regardless of its initial shape. The information of the real shape of the particle ends up being lost as the diffraction patterns progressively adopt a more circular form. This is particularly striking on videos. This means that any particle shape far from the object plane produces a more and more circular diffraction pattern which no longer allows us to identify the original shape of the respective particle.

Depending on the initial shape, the binary diffraction pattern is generally broken into two, sometimes three image parts of similar sizes when approaching the DoF limit. Depending on the orientation of the particle, these two or three particles can be interpreted as different particles by the 2D-S probe. This can create significant differences when particles are not properly recorded or deleted by the shattering algorithm before reaching the theoretical DoF limit. Another interesting remark based on these results is that an out-of-focus image of a distinct particle shape can closely resemble another particle of a very different shape.

In this section, we present some diffraction simulation results for four chosen particle shapes to illustrate the evolution of the particle diameter, including uncertainty evaluation. Our purpose here is neither to present an exhaustive list of results related to each shape nor to quantify the uncertainty of the probe in an absolute manner. The size of the particle from a 2-D image has no absolute definition. Several definitions are used in the literature with different pros and cons depending on the objective of the study.

In the study presented here, we illustrate results with two commonly used particle size definitions: the surface-equivalent diameter D eq and the maximum diameter D max. D eq is defined as the diameter of a disc with the same surface as the analyzed particle image. D max is defined as the diameter of the smallest circle encompassing the particle image e. Note that all triggered pixels of the image are considered here for the computation of D eq and D max , even when the pattern is split into several fragments see Sect. Binary images from diffraction patterns added on top of four figures for several Z distances.

Blue and green shadow areas are the theoretical records by the 2D-S with the uncertainty due to the position of the particle in front of the photodiode array. Dashed lines show the true D eq and D max particle diameters. The produced binary images from diffraction simulations are presented on top of each sub-figure of Fig. Indeed, a spread for each size would appear due to the discrete pixel effect. We note that the pixels at the edge can be triggered or not as the particle is shifted, which then affects the particle size.

At first, we discuss diffraction simulation results of short and elongated columns presented in Figs. The evolution of D eq is rather smooth with varying Z , compared to D max evolution, which is much more oscillating. Still, both columnar particles have a comparable true D eq Repeated abrupt decreases in measured D max are related to changes in the outer binary pixel ensembles of the diffraction pattern which are drifting away from the particle center as Z increases. This then leads to sudden loss of outer pixel and related decrease in D max.

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This can lead to virtually large D max e. Secondly, the two capped column-type particles are discussed. With increasing Z , the size of the larger capped column Fig. Also, D max generally increases with increasing Z , however with few transient smaller diameter decreases at some distances depending on the evolution of the diffraction pattern details at the edges of the binary particle image. The smaller capped columnar particle Fig.

The apparent particle size first grows with Z , then shrinks continuously in terms of D eq more abruptly in terms of D max , followed by another phase of slight increase in D max , before the particle then completely disappears in both diameter definitions at a distance Z of roughly 1. Again, it can be noticed that in general D eq changes more gently with Z , as compared to D max. Close to the DoF limit, which is estimated from diffraction simulations for the particle in Fig. These different DoF limit estimations factor of 2.

However, our arbitrary diameter definition used in the classical DoF limit calculation remains questionable, since ice particles are primarily non-spherical. Furthermore, for both columnar and capped columnar particles, it is evident that the discrete pixel effect shadow areas is almost negligible with respect to the diameter variability along the Z distance according to the diffraction simulations. For this small particle, with a DoF limit smaller than the arm limit, the lower bound of the uncertainty has been calculated for a 1-pixel particle. The minimum and maximum relative errors with respect to true D eq and D max are shown in brackets behind minimum and maximum diameter values.

In this section, we compare particle size distributions retrieved theoretically from diffraction pattern simulations and experimentally measured by the 2D-S probe. For the measurements, a spinning disc Fig.

## Journal of the Optical Society of America

Uncertainty is then due to diffraction and to the discrete pixel effect. However, we see in Fig. The described behavior depends on particle size and shape, but has a common feature: particle D eq generally starts to increase and then decreases until it disappears when reaching the DoF limit, as shown for the small capped column in Fig.