Effect of electron beam parameters on simulated CBED patterns from edge-on grain boundaries.

Position averaged convergent beam electron diffraction: Theory and applications

Convergent-beam and small-area-parallel-beam electron diffraction of icosahedral quasicrystals of a melt-quenched Al-Mn alloy

The convergent beam electron diffraction (CBED) patterns of twisted bilayer samples exhibit interference patterns in their CBED spots. Such interference patterns can be treated as off-axis holograms and the phase of the scattered waves, meaning the interlayer distance can be reconstructed. A detailed protocol of the reconstruction procedure is provided in this study. In addition, we derive an exact formula for reconstructing the interlayer distance from the recovered phase distribution, which takes into account the different chemical compositions of the individual monolayers. It is shown that one interference fringe in a CBED spot is sufficient to reconstruct the distance between the layers, which can be practical for imaging samples with a relatively small twist angle or when probing small sample regions. The quality of the reconstructed interlayer distance is studied as a function of the twist angle. At smaller twist angles, the reconstructed interlayer distance distribution is more precise and artefact free. At larger twist angles, artefacts due to the moiré structure appear in the reconstruction. A method for the reconstruction of the average interlayer distance is presented. As for resolution, the interlayer distance can be reconstructed by the holographic approach at an accuracy of ±0.5 Å, which is a few hundred times better than the intrinsic z-resolution of diffraction limited resolution, as expressed through the spread of the measured k-values. Moreover, we show that holographic CBED imaging can detect variations as small as 0.1 Å in the interlayer distance, though the quantitative reconstruction of such variations suffers from large errors.

Accurate values of foil thickness are required in many materials science applications, such as for measurement of defect concentrations and for x-ray microanalysis absorption corrections. Kelly et al. demonstrated that convergent beam electron diffraction (CBED) patterns can be analyzed using a simple graphical technique to give values for foil thickness with ±2% accuracy. More recently, Allen extended the treatment to make use of both maxima and minima in the CBED disks. The technique requires a knowledge of the d-spacing of the reflection, the electron wavelength, an evaluation of the deviation parameter, si, associated with the i-th fringe in the diffracted beam disk, and the assignment of a set of constants to "index" the fringes.

The fine structure of the first-order Laue zone (FOLZ) reflections in convergent-beam electron diffraction (CBED) patterns of Gaxln1–xAsyP1–y/InP heterostructures is investigated. The FOLZ reflection fine structure consists mainly of three bright lines; the position and intensity of these bright lines were used for a quantitative interpretation, using many-beam dynamical calculations. The main factor determining the position of the lines appears to be the composition. However, HOLZ-beam coupling, absorption and thermal vibrations of atoms have a considerable effect on the position and the intensity of the lines. Values of these parameters were estimated from the CBED patterns. A good agreement between calculation and experiment could only be obtained if small static displacements of the atoms from their ideal lattice sites were included. The calculations suggest that the static displacements in the quaternary compounds should be largest on the group-V sublattice. The quasi-periodic strain fields ...

Convergent beam electron diffraction (CBED) is widely used as a microanalysis tool. By the relative position of HOLZ-lines (Higher Order Laue Zone) in CBED-patterns one can determine the unit cell parameters with a high accuracy up to 0.1%. For this purpose, maps of HOLZ-lines are simulated with the help of a computer so that the best matching of maps with experimental CBED-pattern should be reached. In maps, HOLZ-lines are approximated, as a rule, by straight lines. The actual HOLZ-lines, however, are different from the straights. If we decrease accelerating voltage, the difference is increased and, thus, the accuracy of the unit cell parameters determination by the method becomes lower.To improve the accuracy of measurements it is necessary to give up the HOLZ-lines substitution by the straights. According to the kinematical theory a HOLZ-line is merely a fragment of ellipse arc described by the parametric equationwith arc corresponding to change of β parameter from -90° to +90°, wherevector, h - the distance between Laue zones, g - the value of the reciprocal lattice vector, g‖ - the value of the reciprocal lattice vector projection on zero Laue zone.

In this paper an analysis is presented of the possible symmetries of convergent-beam electron diffraction (CBED) patterns taken from bicrystals with the incident electron beam parallel to the grain boundary. By establishing the connection between CBED symmetry and the possible point groups of a bicrystal, useful information on the grain boundary structure is obtained. This is illustrated by experiments on a vertical (111) twin boundary in silicon. It is shown that a complete symmetry determination from such a twin boundary is possible only if the double-rocking technique for Obtaining zone-axis patterns is employed.

This work presents observations of symmetry breakages in the intensity distributions of near-zone-axis convergent-beam electron diffraction (CBED) patterns that can only be explained by the symmetry of the specimen and not the symmetry of the unit cell describing the atomic structure of the material. The specimen is an aluminium-copper-tin alloy containing voids many tens of nanometres in size within continuous single crystals of the aluminium host matrix. Several CBED patterns where the incident beam enters and exits parallel void facets without the incident beam being perpendicular to these facets are examined. The symmetries in their intensity distributions are explained by the specimen morphology alone using a geometric argument based on the multislice theory. This work shows that it is possible to deduce nanoscale morphological information about the specimen in the direction of the electron beam - the elusive third dimension in transmission electron microscopy - from the inspection of CBED patterns.

A systematic method to identify the space group from PED and CBED patterns part I - theory

To accurately and easily determine the polarity of AlN using transmission electron microscopy, we compare the convergent-beam electron diffraction (CBED) patterns along the widely used <112̄0 > and <11̄00 > zone-axes. For the <112̄0 > zone-axis, the diffraction disk of g= 0002 differs from that of g= 0002̄, while for <11̄00 >, the diffraction disks of g= 0002 and 0002̄ are similar. The preferential clarity of these two disks is explained using Bloch-wave dynamical theory. To further support the explanation, we compare the results of GaN case. On the basis of our analysis, we conclude that the CBED patterns of the <112̄0 > zone-axis are more useful for accurately determining AlN polarity compared to the CBED patterns along the <11̄00 > zone-axis.

Under almost all circumstances, electron diffraction patterns contain information about the phases of structure factors, a consequence of the short wavelength of an electron and its strong Coulombic interaction with matter. However, extracting this information remains a challenge and no generic method exists. In this work, a set of simple analytical expressions is derived for the intensity distribution in convergent-beam electron diffraction (CBED) patterns recorded under three-beam conditions. It is shown that these expressions can be used to identify features in three-beam CBED patterns from which three-phase invariants can be extracted directly, without any iterative refinement processes. The octant, in which the three-phase invariant lies, can be determined simply by inspection of the indexed CBED patterns (i.e. the uncertainty of the phase measurement is ±22.5°). This approach is demonstrated with the experimental measurement of three-phase invariants in two simple test cases: centrosymmetric Si and non-centrosymmetric GaAs. This method may complement existing structure determination methods by providing direct measurements of three-phase invariants to replace 'guessed' invariants in ab initio phasing methods and hence provide more stringent constraints to the structure solution.

In recent years, significant progress has been made towards a solution for the general problem of crystal structure determination by convergent beam electron diffraction (CBED). Even if we consider only perfectly ordered, periodic crystals defined by one of the conventional space groups, diffraction methods based on a focussed sub-micron beam of electrons are applicable to several related sets of structural problems that are not accessible to conventional X-ray or neutron diffraction techniques. We assume here that the space group either is known or has been determined from CBED patterns and that phases and amplitudes for some subset of the structure factors are required. Two limiting cases have been explored in some detail. For crystals where the atomic parameters and Debye-Waller factors are known accurately from high quality X-ray data, information on the charge redistribution for bonding electrons is available from precise measurements of the low order structure factors. Following the original research of Kambe, some recent work has demonstrated that accurate structure amplitudes and three-beam phase invariants can be extracted from the dynamical intensity distribution in CBED reflections. In principle, this approach is completely general but considerable labour would be required to extract sufficient data to solve the structure of an unknown crystal, whereas a large set of kinematic intensities is acquired from a single X-ray pattern.

Many people have observed dynamic extinction lines in the {200} diffraction disks of the &lt;100&gt; zone axis CBED (convergent beam electron diffraction) patterns of the MgAl2O4 spinel. These dark lines are manifestation of the space group of the material and can easily be explained with the Gjønnes-Moodie theory. However, in addition to the extinction lines in the {200} disks, We have also been able to observe dynamic extinction lines present in the {420} diffraction disks in the same CBED pattern.The CBED work in this work was carried out with a JEOL-2000FX microscope operated at 100.15 kV. One &lt;100&gt; zone axis CBED pattern is shown in Fig. 1. The dark lines are clearly shown in the {420} disks running through the two perpendicular non-principal axes, in exactly same way as the dark lines in {200} disks.One can easily explain the dynamic extinction lines in the {200} disks with the Gjønnes-Moodie theory by drawing dynamic scattering pairs oG-Ga and oH-Ha, oK-Ka and oL-La etc. in the diagram (Fig. 2).

This paper presents a review of the developments in Convergent Beam Electron Diffraction (CBED), a technique widely used for determination of structure, symmetry details and atom positions in a crystal as small as 20A in size. The understanding of this technique is related to the rapid advancements in the field of transmission electron microscopy with respect to development of coherent, finer probes and electron optics for higher spatial resolution. Energy filtering devices enable imaging of several finer features in the CBED pattern from which useful information about a crystal can be obtained. These include (i) three-dimensional information about the reciprocal lattice, (ii) point and space group symmetry details, (iii) lattice parameter from regions as fine as 2 nm, (iv) atom positions within a unit cell and (v) defects in crystals and (vi) thickness. Due to abundant data obtained from microscopic regions, this technique is unique and finds wide application in materials characterization. It has been used for studying problems like identification of the presence of lattice strain, identification of point defects etc. in a material used often in the nuclear industry, namely 9Cr-1Mo steel. The present paper provides the current status of CBED starting from its historical development, the information that can be obtained and its use in a variety of applications.

ABSTRACTStrains in polycrystalline Al films grown on oxidized Si wafers were measured using convergent beam electron diffraction (CBED). CBED patterns were acquired on a Zeiss EM 912 TEM equipped with an imaging energy filter and CCD camera. HOLZ line positions in the (000) CBED disk were matched using an automated refinement procedure. A sensitivity to variations in lattice parameter of approximately 0.00007 nm was obtained. Strong deviations from a simple equibiaxial strain, perfect [111] texture model were observed.