Higher-order Laue Zone Research Articles

Dynamical diffraction of double-inelastically scattered electrons

Abstract Dynamical theories are developed to calculate the diffraction intensities of double-inelastically scattered high-energy electrons (i.e. the electrons that have been inelastically scattered twice as the result of exciting two distinct crystal states) in crystals within the framework of quantum mechanics. These theories are needed to quantify the data of thermal diffusely scattered electron energy loss spectroscopy. The Bloch wave and Green's function approaches proposed here take into account the full dynamical diffraction effects of the electrons before and after each inelastic event. The Bloch wave theory gives a full three-dimensional description of double-inelastic scattering in crystals including higher-order Laue zone reflections, but it may not be convenient to calculate the intensity of high-angle thermal diffuse scattering of electrons because of the difficulty of including all of the possible multiphonon excitations. Green's function theory, as an alternative approach, is most suitable f...

Convergent Beam Electron Diffraction Analysis of Strain in Multilayer Structures: A Kinematical Approach

Convergent beam electron diffraction (CBED) performed on plan-view samples has been considered in order to analyse strained superlattices. The composition modulation along the growth axis of multilayer materials and the strain modulation at the interfaces give rise to special features in higher order Laue zone (HOLZ) reflections. A simple model based on the expansion-contraction of the lattice spacing in each layer along the growth direction is proposed to simulate rocking curves in the kinematical approximation. Applications to Cd x Zn 1-x Se/ZnSe superlattices are discussed.

Strain Distribution Measurement in Stainless Steels by Convergent-Beam Electron Diffraction

A convergent-beam electron diffraction technique was utilized to measure local strain distributions in stainless steels. Electron beams were focused into {210} crystal planes and the higher order Laue zone (HOLZ) lines diffracted from {375} and {119} planes were characterized by measuring the distances between the intersections of these HOLZ lines. Four parameters, including strains in three directions and one magnification factor, were calculated based on the least squares method. Strain distributions near a chromium carbide precipitate were measured in sensitized SUS 304 and it was confirmed that parallel tensile strain and vertical compressive strain to the edge of the precipitate existed and that they decreased with the distance from the edge of precipitate.

Determination of Bulk Mismatch Values in Heterostructures” by TEM/CBED

AbstractUsing isotropic elasticity theory it is possible to determine the bulk mismatch in thinned, cross-sectioned heterostructures, where a relaxation occurs along the thinning direction. This is accomplished by measuring, in the central disk of a single Convergent Beam Electron Diffraction (CBED) pattern, the position of the High Order Laue Zone lines, which are sensitive to lattice parameters along different crystallographic directions. The results obtained in both uniform and graded Sil-xGex/Si heterostructures are in good agreement with bulk values deduced from independent techniques.

Refinement of crystal structure parameters using convergent-beam electron diffraction: the low-temperature phase of SrTiO3

The method to refine crystal structure parameters (atom positions and Debye–Waller factors) using convergent-beam electron diffraction (CBED), which is applicable to crystal structure analysis of a small specimen area down to a few nm in diameter, is studied. The line profiles of higher-order Laue-zone reflections are recorded using imaging plates. Theoretical intensities are calculated based on the dynamical theory of electron diffraction with the aid of the generalized Bethe approximation to shorten calculation time. The structural parameters are determined by fitting the experimental profiles with the theoretical ones using a nonlinear least-squares method. The present method has been applied to the low-temperature phase of SrTiO3. The structural parameter or the rotation angle of the oxygen octahedron has been determined to be \varphi = 1.12 (4)°, which shows good agreement with that obtained from electron spin resonance experiments.

Grain-boundary constraint and oxygen deficiency in YBa2Cu3O7−δ: Application of the coincidence site lattice model to a non-cubic system

Abstract Local lattice parameters and oxygen (hole) densities in the vicinity of large-angle grain boundaries in bulk YBa2Cu3O7−δ were measured simultaneously using convergent-beay electron diffraction and electron-energy-loss spectroscopy with a less than 20 A field emission probe respectively. The oxygen (hole) contents were quantified by Gaussian fitting to the observed oxygen K-edge absorption spectra and comparing them with results for a set of standard samples with different oxygen contents. The local lattice parameters were determined by matching the simulated high-order Laue zone patterns with experimental patterns by a least-squares algorithm. We found that large-angle grain boundaries in YBa2Cu3O7−δ often exhibit volume expansion. The increase of the c-lattice parameter near the boundary core can be attributed to lattice distortion and/or to an oxygen deficiency. The lattice distortion extends less than 20 A, while the deficiency can extend more than 100 A. Comparison of the c/a ratio of the lat...

LACBED determination of structure factors and alloy composition of GeSi/Si SLS

The Taftø and Gjønnes three-beam dynamical analysis of large-angle convergent beam electron diffraction (Ultramicroscopy 17 (1985) 329) has been generalized to include high-order Laue zone corrections and has been applied to non-centrosymmetric strained layer superlattices (SLSs). The method of intersecting Kikuchi lines has been used to measure the structure factors of the Ge x Si 1− x SLS and to determine the alloy composition.

Dynamical diffraction effect on HOLZ-pattern geometry in Si-Ge alloys and determination of local lattice parameter

Higher-order Laue zone (HOLZ) patterns are observed in convergent-beam electron diffraction (CBED) patterns of Si-Ge alloys and the dynamical effect on HOLZ lines is discussed in relation to the kinematical approximation to simulate the HOLZ lines. Double lines often appear from one HOLZ reflection due to the excitation of the first and second branches of the zeroth-layer dispersion surface. The magnitude of the systematic line shift from the line positions predicted by the kinematical simulation changes sensitively with specimen thickness and alloy composition depending on which branch is dominantly excited. The kinematical simulation is applicable to accurate determination of lattice parameters with a proper correction δ E: a difference between the effective electron energy and the actual microscope operating voltage. Usefulness of the kinematical simulation is demonstrated in the investigation of the segregation in a Si-Ge alloy.

Preliminary studies of the interface strain field in an Al-Al2O3 composite by means of convergent-beam electron diffraction

Various kinds of composite materials have been extensively studied concerning their manufacture processing and applications in recent years. Among them, metallic composite materials have drawn extensive attention, because they can offer selected physical, chemical and mechanical properties which cannot be obtained with traditional alloys, ceramics or organic matrix composites. The interfaces in the composites play an important role in the properties of the composite. Up to now, the studies on the interfaces have been mainly limited in the interface morphologies, interfacial reaction products and orientation relationship between the reinforcing phase, crystalline reaction products and the matrix. However, differences in thermal expansion between the reinforced phase and the matrix, generate residual stresses which may affect the aging characteristics of the matrix and the mechanical properties of such composites. Efforts have been made to measure the local lattice strain in the vicinity of the interface, quantitatively by contrast simulation and by detecting the movements of the higher order Laue zone (HOLZ) lines in conventional convergent-beam electron diffraction (CBED). As the electron probe is very close to the interface, HOLZ lines often become obscure or even split. In such cases, it is very difficult to measure the movements of HOLZ lines. Nevertheless, itmore » is possible to measure the strain by matching a dynamical simulation to the experimental splitting of HOLZ lines. Large-angle CBED (LACBED) combines real space and reciprocal space information. A single LACBED pattern shows directly the spatial variation of the lattice strain. The purpose of the present work is to explore the possibility of studying the interface strain field in composite materials according to the shifts and splittings of HOLZ lines by means of improved LACBED technique.« less

Application of microdiffraction to crystal structure identification

In a previous paper by J.P. Morniroli and J.W. Steeds [Ultramicroscopy 45 (1992) 219] it was shown that microdiffraction patterns obtained with a nearly parallel electron beam and a small spot size can be used to identify and to characterize crystal structures. A systematic method was proposed to deduce, via the Buerger extinction symbol, the space group or a set of consistent space groups. In order to illustrate how this method can be applied experimentally, three examples concerning the σ, X and R intermetallic phases are detailed. Experimental solutions are given to overcome difficulties connected with the frequent absence of high-order Laue zone reflections on high-symmetry zone axis patterns. The possibility to identify the Buerger extinction symbol from a unique zone axis pattern is also explored.

Polarity-driven intensity asymmetries in higher-order Laue zone electron diffraction

Abstract The theory of high-energy electron diffraction into higher-order Laue zones (HOLZs) is extended to deal with absorption of, as well as coupling between, different states of the zeroth-order zone. The cross-terms generated give rise to diffracted intensities which are sensitive to the polarity of a HOLZ reflection. Limiting cases of the theory are taken to explain the various forms of HOLZ asymmetry observed at 〈001〉 axes of the common III–V and II–VI semiconductors. The asymmetries are found to be a remarkably stable measure of polarity for these materials.

Measurement of Debye-Waller factors of silicon as a function of temperature using energy-filtered CBED rocking-curve technique

The advantages of quantitative Convergent Beam Electron Diffraction (CBED) method for x-ray structure factor determination have been reviewed by Spence. The CBED method requires accurate values of Debye-Waller (D-W) factors for the estimation of the coefficients of crystal potential of the higher order beams, Vg, the calculation of the absorption potential, V′gusing the Einstein model for phonons, and finally the conversion of the fitted values of the coefficients of crystal potential, V″, to x-ray structure factors. Debye-Waller factors are conventionally determined by neutron or x-ray diffraction methods. Because of the difficulties in conducting high temperature neutron and x-ray diffraction experiments, D-W factors are rarely measured at temperatures above room temperature. Debye-Waller factors at high temperatures can be determined by Convergent Beam Electron diffraction (CBED) method using Transmission Electron Microscopy (TEM) employed with a hot stage attachment. Recently Holmestad et al. have attempted to measure the D-W factors by matching the energy-filtered Higher Order Laue Zone (HOLZ) line intensities near liquid nitrogen temperature.

Dynamical treatment of the splitting of higher-order-Laue-zone lines induced by dislocations in an icosahedral quasicrystal

The dynamical theory of electron diffraction for quasicrystals (QCs) was used to treat the splitting behaviour of higher-order-Laue-zone (HOLZ) lines induced by dislocations in icosahedral quasicrystals (IQCs). The influences of some parameters on the splitting of HOLZ lines were calculated. On the basis of this calculation, several experimental convergent-beam electron diffraction patterns from the aluminium–copper–iron IQC were simulated. Good agreement between the experiment and the simulation confirms the correctness of the dynamical theory of electron diffraction for QCs.

A Cbed Procedure for Determining Local Residual Stresses from Nanoscale Areas in Cermets

ABSTRACTA convergent beam electron diffraction method was used in a transmission electron microscope to determine residual thermal stresses from nanometer-scale areas in a ceramic-metal (cermet) composite material. It is demonstrated that the method is simple, but requires a standard sample for each of the phases in the composite. The principle of the technique is that the stresses are determined by comparing higher order Laue zone line shifts, due to lattice strain, in high-symmetry zone-axis electron diffraction patterns recorded under the same experimental conditions from the unknown phases and the standards. The application of the method to composite systems containing dissimilar phases, such as ceramic-metal composite systems, demonstrates the universality of the technique in obtaining local structural information of anisotropic strain unambiguously with a high accuracy. The procedure involved is described in detail with application to B4C-AI, a three-dimensional intertwined ceramic-metal composite.

New method of Bravais lattice determination

A new method of Bravais lattice determination from a single convergent beam electron diffraction (CBED) pattern is proposed. The method is based on the principle that a primitive reciprocal lattice cell can be measured accurately from the deficiency line of medium or high order reflections or high order Laue zone (HOLZ) line positions within the central disk of a single CBED pattern taken near an arbitrary zone axis. This primitive cell can be reduced to the unique Niggli cell and thus the Bravais cell by Krivy and Gruber reduction procedures. This method could be very useful in identifying phase of mixed phase materials or as the first step toward determination of a new structure or as an automated procedure of indexing diffraction patterns of known structure.

Convergent beam electron diffraction zone axis pattern maps from a copper single crystal

Two zone axis pattern (ZAP) maps were obtained from a copper single crystal, using convergent beam electron diffraction in a transmission electron microscope: the Kossel pattern (KP) and the convergent beam electron diffraction (CBED) pattern ZAP map. A computer simulation of the KP map was used to aid its interpretation. The CBED ZAP map of copper consists of zero layer disks showing an intersecting set of higher-order Laue zone (HOLZ) deficiency lines. From the computer simulations of these disks, an operating high voltage of 84kV was determined. Both kinds of map are useful for subsequent “fingerprint” identification of copper in precipitates, alloy crystalline phases, and crystallographic direction determination.

Are HOLZ lines kinematic in off-zone-axis orientation?

The application of high order reflections in a weak diffraction condition off the zone axis center, including those in high order laue zones (HOLZ), holds great promise for structure determination using convergent beam electron diffraction (CBED). It is believed that in this case the intensities of high order reflections are kinematic or two-beam like. Hence, the measured intensity can be related to the structure factor amplitude. Then the standard procedure of structure determination in crystallography may be used for solving unknown structures. The dynamic effect on HOLZ line position and intensity in a strongly diffracting zone axis is well known. In a weak diffraction condition, the HOLZ line position may be approximated by the kinematic position, however, it is not clear whether this is also true for HOLZ intensities. The HOLZ lines, as they appear in CBED patterns, do show strong intensity variations along the line especially near the crossing of two lines, rather than constant intensity along the Bragg condition as predicted by kinematic or two beam theory.

Accurate methods for crystal structure determination using aspects of dynamical diffraction theory

An approach to the solution of crystal structures has been developed over the past ten years which is based on the intensities of high order Laue zone (HOLZ) fine structure lines present in zone axis patterns. It is a common observation that when the individual reflections in a HOLZ ring are studied in detail, several individual excess lines are observed. These individual lines are directly related to the excited branches of the dispersion surface, created by the two dimensional dynamical diffraction which occurs in the zero order Laue zone (ZOLZ). They may either directly correspond to a single branch of the dispersion surface (a unique Bloch state) or else to a cluster of close branches (a small number of distinct Bloch states). It is often the case that the intensities of these lines may be derived from a pseudo-kinematical theory, which greatly simplifies the process of deducing atomic positions from the experimental data.

Simulating electron microscope diffraction mode with a Macintosh-based program

With the recent trend towards to the quantification of electron diffraction patterns, there is an increasing need for simulating the geometry of convergent beam electron diffraction patterns, and especially the high order Laue zone (HOLZ) lines in such patterns. The simulation program is useful in the way that the simulated and the experimental pattern can be compared, and then the important diffraction parameters such as reflection indices, beam directions and lattice constant could be found and used. Here we describe a Macintosh based program, which simulates electron diffraction pattern in the same way as the operation of electron microscope diffraction mode. The program has a control panel with the ‘scroll bar’ control devices for x and y tilt of specimen stage, x and y deflection of diffraction pattern and camera length (see figure 1). The user can change the simulated diffraction pattern by changing the ‘control devices’ with a pointing device such as a mouse.

Bravais lattice determination from a single CBED pattern

Traditional methods for Bravais lattice determination of unknown materials using electron diffraction involve symmetry determination from several low index zone axis diffraction patterns, measuring the distances between spots or disks and indexing them by trial and error method. Such methods are ofte repetitive, error prone and difficult, if not impossible, for low-symmetry crystals. Thus a simpler, accurate, and automated method for lattice determination is highly desirable in electron diffraction. Here new method of Bravais lattice determination using an arbitrary single convergent beam diffraction (CBED pattern is summarized. A full description of the method has been submitted for elsewhere. The method uses the deficiency lines of high order and high order Laue zone (HOLZ) reflections to measure the primitive cell of the zone axis, and use the standard Krivy and Gruber procedure to reduce this arbitrary primitive cell to a unique Niggli cell, and thus the Bravais cell.