Dynamical Theory Of Electron Diffraction Research Articles

On the applicability of bloch’s theorem to the total wave function in the dynamical theory of electron diffraction

The dynamical theory of electron diffraction is widely used in materials science problems such as determining the contrast in electron microscope images of crystal defects, calculations of structure images and calculations of diffracted beam intensities in electron diffraction patterns. In carrying out these calculations, the high energy approximation is normally made and it is usually assumed that the crystal is in a symmetrical Laue orientation. In practice, however, a specimen in the electron microscope will generally be oriented so that the non-symmetrical Laue case is obtained. Even in those special cases where the symmetrical Laue case is obtained for the zero order Laue zone reflections, non-symmetrical Laue effects may occur if reflections in higher order zones are important. It has been shown in the literature that the Bloch functions are not orthogonal in more general forms of the dynamical theory in which the high energy approximation is not made and the nonsymmetrical Laue case is considered,.

Structure determination using convergent-beam electron diffraction

There are many different approaches in quantitative electron diffraction which are being vigorously pursued at present. The approach we adopt is based on the insights provided by the Bloch-wave formulation of dynamical electron diffraction theory into the physics of dynamical scattering. This insight is used to select diffraction situations where a pseudo-kinematical approximation may be made. A forwards route is then possible directly from the experimental observations to the structural implications. This contrasts with the model-building, multi-parameter fitting procedures used in many other approaches where a problem of uniqueness inevitably arises.Because the pseudo-kinematical approach ignores many of the detailed dynamical interactions which occur locally over small angular ranges we do not attempt to make accurate measurements, and wherever possible average (visually at least) along Bragg lines to eliminate local perturbations. In a sense the work resembles early X-ray crystallography where reflections were put in one of six or so classes from very weak to very strong.

A Method for Measuring Film Thickness by Zone Axis Contrast of TEM Image

The thickness of a bending thin film was determined by the comparison of the observed contrast of zone axis patterns with the calculated contrast based on a dynamical electron diffraction theory. This method allows us to measure the film thickness nondestructively at the symmetry Bragg reflection position where crystal lattice images appear clearly. Therefore, it seems to be a useful technique for high resolution electron microscopy. The estimation of the film thickness can be easily performed if the contrast of the patterns has been calculated once as a function of thickness

The study of microstructure in iron- and sulphur-doped InP crystals by means of HREM and computer simulation

The microstructures in iron- and sulphur-doped InP crystals were studied using both electron microscopy and electron diffraction. A modulated structure has been found in S-doped InP crystal, where the commensurate modulations corresponded to periodicities of 0.68 nm and 0.7 nm in real space were related to the reflections of the cubic lattice in (111) and (113) directions; they were indexed as q111*=1/2(a*+b*+c*) and q*113=1/4(-a*-b*+3c*), respectively. Single atomic layers of iron precipitate were observed, with preferred orientations along which precipitates are formed. Simulated calculations by means of the dynamical theory of electron diffraction using models for the precipitate structure were in good agreement with the authors experimental results. The relation between the modulated structure and the precipitates is also discussed.

Fine details in satellite HOLZ reflection discs of CBED from a GaAs/AlAs multilayer

Fine details of satellite higher-order Laue zone (sHOLZ) reflection discs from a GaAs/AlAs multilayer in plan view are observed and investigated. Computer simulation based on the dynamical theory of electron diffraction shows good agreement with the experimental observations. Apart from the two lines reported previously, some weak lines and a few other features in sHOLZ reflection discs are found and investigated. The investigation confirms the previous two-line explanation and also reveals the origin of the additional weak lines and the other features. The weak lines are due to the interference between Bloch waves, and their location depends on specimen thickness. Some other features are determined to be caused by multiple diffraction from neighbouring sHOLZ reflections. The phenomenon of the interruption of lines in sHOLZ reflection discs, in comparison to the lines in fundamental higher-order Laue zone (fHOLZ) discs, originates mainly from sHOLZ reflection coupling. Furthermore, intensities of sHOLZ reflections are shown to depend on their structure factors rather than on the excitation of their corresponding fundamental reflections. Details of the computer simulation are given. Finally a simple method, not relying on computer simulation, to identify the lines in sHOLZ reflections is presented.

A general matrix representation of the dynamical theory of electron diffraction. I. General theory

A general matrix representation of the dynamical theory of high-energy electron diffraction by crystals is developed. The formulation is based on the plane wave expansion of Bloch waves, following the original approach of Bethe, and is applicable to both the Laue and Bragg cases with the influence of surface potential and structure incorporated.

A general matrix representation of the dynamical theory of electron diffraction. II. Application to rheed from relaxed and reconstructed surfaces

The general matrix formulation developed in a previous paper for calculating dynamical electron diffraction from an assembly of crystal slabs and reflection from surfaces of semi-infinite bulk crystals is used to investigate the effects of surface layers scattering on RHEED reflected beam amplitudes. The general computing procedure is illustrated by reference to a symmetric two-beam and a general four-beam Bragg case. Applications are made to crystal systems containing relaxed or reconstructed surface layers.

Electron diffraction and lattice image simulations with the inclusion of HOLZ reflections

The formulae that can be used to calculate wave functions for transmission high-energy electron diffraction including reflections from higher-order Laue zones (HOLZs) are summarized by using the Bloch wave formulation and the multislice formulation for the dynamical electron diffraction theory, respectively. In the multislice formulation, three different algorithms of calculating phase grating functions, fraction projection algorithm (FPA), section projection algorithm (SPA) and conditional projection algorithm (CPA) are examined. Electron diffraction patterns and lattice images for the YBa 2Cu 3O 7 crystals are simulated and compared with Bloch wave calculations. In summary, the FPA calculations do not include HOLZ reflections; the SPA and CPA give much stronger HOLZ reflection intensities than the Bloch wave algorithm. Experimental diffraction patterns support the Bloch wave results. However, there is no detectable difference found in the simulated lattice images by using the above-mentioned computing schemes, which suggests that the HOLZ reflections only play a minor role in forming lattice images.

Superhigh-Resolution Structural Image of InP Along [110] Zone Axis

During the past decade , K.IZUI and S.FURUNO [1] reported the structural image of Si along [110] direction with the bright spots representing the columns of atoms. This result remain somewhat controversial. J.L.HUTCHISON and W.G.WADDINGTON [2] elucidated the image as the lattice image, not the structural image , for the computer simulation showed that this just is the artifice due to electron optics. Mallard et.al [3] reported HREM image of GaInAs/InP interface along [110] direction and consider that it is not a true structure image because the spots in pairs on the picture were not representative of accurate atomic positions. Here we report the superhigh resolution image of InP along [110] direction. Combining with computer simulation technique, it is confirmed that the observed image is the structural image of InP along [110] direction. The black spots represent to the atomic columns. The nearest distance of In and P is 1.47Å as shown in the picture, which is smaller than the point-point resolution 1.9Å of JEOL-4000EX electromicroscope. Fig.1 is the superhigh resolution image of InP along [110] direction. The thickness of the specimen is about 350Å, and defocus is about ΔF=-470Å. The bigger bright spots correspond to the atomic channels. The different size black spots represent the columns of In and P atoms.Fig.2 shows the simulated structural image of InP along [110] direction obtained by multi-slice method. Fig.3 show that the thickness of the specimen depends on the amplitude of electron waves. When the amplitudes of transmitted beams reach nadirs and the amplitudes of diffracted 111, 002,220,004 beams are relatively strong, the contrast of the structural image is better. It is indicated that the amplitudes of transmitted beam interacts with diffracted beams according to the dynamical theory of electron diffraction and the electron waves passing through the sample have nearly the same amplitude, so that both In and P atomic columns may imaged. These results are in good agreement with the results on another paper about the explanation of structural image of InP along [001] direction[4]. The phases of the electron waves were investigated as shown in Fig.4. The transmitted beam has, the same value of phases as diffraced beams at the thickness of 350Å. This work is very similar to that of Au atomic image along [110] direction reported by prof. H.Hashimoto.[5] It is considered that the atomic images can be formed when obseved specimen is rather thin or near optimum thickness at which the phase of the image-forming waves the same values. This is very important for the explanation of the superhigh resolution image. The curve of CTF of JEOL--4000ex operated at 400kv with different defocuses were calculated. The CTF of sherzer condition is ΔF=-406Å. In our experiment, the superhigh resolution structural images were obtained with ΔF=-465Å, which is close to the scherzer condition. It is obvious that the electron beams with lower indexes were nearly included in the plateau of CTF curve where sinx ≈ 1, as shown in Fig.5, which may carries a lot of information about the structure, thus the atomic images of InP can be formed.

Critical Voltage for Non-Centro. Crystals and its Application to the Accurate Measurement of Structure Factor Phases by CBED

The application of the three-beam dynamical theory of transmission electron diffraction to centrosymmetric and non-centrosymmetric crystals shows that certain regions of three-beam CBED patterns are highly sensitive to the three-phase structure invariant By applying this method to the systematics geometry, a method has been described which allows structure factor phases to be measured with an accuracy of 0.07°. Using the Bethe potentials, we find a degeneracy in eigenvalues at the critical voltage VA for acentric crystals in the systematics orientation whereHere VA is now seen to depend both on the structure factors Vg (in volts) and on the systematics “two=phase” invariant In this paper we consider the application of this method to BeO (wurtzite structure), with g = (004) and h = (002). Then the CBED intensity is most sensitive to Y near the (004) Bragg condition at 46 kV. But since the theory contains only the product KSg, other voltages and orientations are also sensitive.

Intensity distribution of electron diffraction pattern from Ti-14Wt%Mo alloy containing ωphase plate crystal

The study of ω-phase in various alloys such as Ti-Mo, Ti-Cr, Zr-Nb etc. have been carried out so far by electron diffraction and electron microscopy. However, the shape of ω-phase particles has not been fully clarified yet. This is due to the facts that ω-phase is formed in ω-phase crystal, and rather difficult to obtain isolated crystal particles and also that w-phase is produced only by the displacement of atoms in characteristic directions which makes no contrast in the electron microscope images projected along this directions and hence the crosssectional shape can not be studied. Since the intensity of electron diffraction patterns is very sensitive to the amount of ω-phase existing in β-phase, its thickness dependence has been studied in the present paper by using multi-slice dynamical theory of electron diffraction for the various thickness combination of the ω- and β-phases so that the thickness and location of ω-phase can be seen.

Dynamical theory of electron diffraction for quasicrystals

AbstractThe dynamical theory of electron diffraction for crystals in the wave‐optical and the wave‐mechanical formulations is extended to quasicrystals. In these formulations the basic equations satisfied by the electron wave amplitudes have the same forms for both crystals and quasicrystals.

収束電子回折法

Symmetry determination of three-dimensional (usual) crystals, four-dimensional (one-dimensionally incommensurate) crystals and five- and six-dimensional crystals (quasicrystals) by convergent-beam electron diffraction is described and demonstrated using real materials. Crystal structure refinement based on the dynamical theory of electron diffraction is described for the low-temperture phase of SrTiO3.

等厚干渉縞を用いたＡｌＧａＡｓ超構造の組成分析法

A new method is proposed for the composition analysis of GaAs/Alx Ga1-x As superstructure using the equal thickness fringe in the transmission electron microscope image. The position of the equal thickness fringe observed at the edge of a cleaved chip shifts depending on the Al composition. The compositional dependence on the fringe shift can be explained by the dynamical electron diffraction theory. The compositional abruptness at the heterointerface and the compositional fluctuation in the thin layer can be successfully evaluated with a high spatial resolution of 0.4nm [Jpn. J. Appl. Phys. 26, 770 (1987) ] .

Linearization of the basic equations of the dynamical theory of electron diffraction by crystals

The dynamical theory based on the plane-wave expansion of Bloch waves is applied to problems of electron diffraction by reflection at crystal surfaces. It is mainly aimed at reflection high-energy electron diffraction (RHEED), but may be applied also to medium- and low-energy electron diffraction (MEED and LEED). It is shown that the theory leads to a quadratic matrix eigenvalue problem, which can be expanded into a linear matrix eigenvalue problem by applying two alternative standard methods of matrix theory, the Günther expansion and the diagonal expansion of Falk. These methods are shown to be equivalent to the expansions of the system of second-order ordinary differential equations, obtained by the two-dimensional Fourier expansion of the Schrödinger equation, into systems of first-order differential equations. The equivalence enables a physical interpretation of the quantities introduced in the matrix methods to be given.

Low-energy electron diffraction analysis of the Si(111)7×7 structure

We have used the dynamical theory of low-energy electron diffraction to analyze data of the Si(111)7×7 surface and determined the atomic structure. The method includes the use of symmetrized wave functions in real and reciprocal spaces. Individual atomic coordinates for the first five atomic planes (containing 200 atoms) are determined. The low-energy electron diffraction optimized structure shows an oscillatory relaxation: atomic planes with stretched bonds followed by planes with compressed bonds. Geometric displacements from the bulk dimer–adatom–stacking fault model are presented.

On the relationship between diffracted beam intensity and bloch function orthogonality in the dynamical theory of electron diffraction

Results presented show that when Bloch functions are orthogonal in the dynamical theory of electron diffraction, the total diffracted beam intensity in a non-absorbing crystal is independent of position in the crystal so that total diffracted beam intensity in the crystal is conserved. In addition, when the Bloch functions are orthogonal the intensity of an individual diffracted beam cannot be greater than the total diffracted beam intensity at the entrance surface. On the other hand, when Bloch functions are not orthogonal, total diffracted beam intensity in the crystal varies periodically with distance from the entrance surface so that total diffracted beam intensity is not conserved. In addition, in this case it is possible for the intensity of an individual diffracted beam to be greater than the total diffracted beam intensity at the entrance surface. Die dargestellten Ergebnisse zeigen, das, wenn die Blochfunktionen orthogonal sind, in der dynamischen Theorie der Elektronenbeugung die Gesamtbeugungsstrahlintensitat in einem nicht-absorbierenden Kristall von der Lage des Kristalls unabhangig ist, so das die Gesamtbeugungsstrahlintensitat im Kristall erhalten bleibt. Daruberhinaus kann, wenn die Blochwellen orthogonal sind, die Intensitat eines individuellen gebeugten Strahls nicht groser sein als die Gesamtbeugungsstrahlintensitat an der Eintrittsflache. Andererseits variiert, wenn die Blochfunktionen nicht orthogonal sind, die Gesamtbeugungsstrahlintensitat im Kristall periodisch mit dem Abstand von der Eintrittsflache, so das die Gesamtbeugungsstrahlintensitat nicht erhalten bleibt. Zusatzlich ist es in diesem Falle moglich, das die Intensitat eines individuellen gebeugten Strahls groser ist als die Gesamtbeugungsintensitat an der Eintrittsflache.

Anomalous reflected images of objects on crystal surfaces in reflection electron microscopy

A gap observed sometimes between the direct and reflected images of foreign objects lying on atomically flat crystal surfaces is interpreted as taking place by an anomalous displacement of the reflected image, resulting from the displacement of electron trajectories at reflection. This displacement is shown to be of the same kind as those which have been observed in acoustics and optics at the threshold angle of total reflection, and have been interpreted as the result of rapid variation of the phase of the reflection coefficient with the angle of incidence. It is shown that the same kind of rapid phase variation occurs at Bragg reflection of the electron beam, and that the displacement estimated by using the dynamical theory of electron diffraction is of the same order of magnitude as the observed ones.

On the Orthogonality of Bloch Functions in the Dynamical Theory of Electron Diffraction

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Dynamical Theory of Electron Diffraction and Its Application to the Rocking Curve Simluation for Quasicrystals

Dynamical Theory of Electron Diffraction and Its Application to the Rocking Curve Simluation for Quasicrystals