Interpretation of Electron Channeling by the Dynamical Theory of Electron Diffraction

1974 ◽  
Vol 29 (7) ◽  
pp. 1034-1044 ◽  
Author(s):  
K. Kambe ◽  
G. Lehmpfuhl ◽  
F. Fujimoto
Keyword(s):  
Electron Diffraction ◽  
Tight Binding ◽  
Diffraction Theory ◽  
Bloch Wave ◽  
Dynamical Theory ◽  
Two Dimensions ◽  
Band Theory ◽  
Bloch Waves ◽  
Electron Channeling ◽  
The Many

The connection between electron channeling and electron diffraction is discussed on the basis of the dynamical theory. Results of the many-beam calculations for 50 keV to 2 MeV electrons incident almost parallel to a [110] axis of a MgO crystal are used as examples. Bloch waves with a marked concentration of electron density at rows of atoms are obtained, and interpreted as states of electrons bound to the rows of atoms, corresponding to the classical picture of channeling. This can be shown properly by applying the tight-binding method of band theory in the two dimensions perpendicular to the axis. In this picture the "rosette motions"' in the classical theory are interpreted as p-tvpe, d-type, etc. Bloch waves, and the "weavons" as loosely-bound s-type Bloch waves. They are connected to the pictures of the Borrmann effect and the Bloch-wave channeling in the diffraction theory.

Theory of transmission low-energy electron diffraction

1993 ◽  
Vol 51 ◽  
pp. 696-697
Author(s):  
W. Qian ◽  
J.C.H. Spence
Keyword(s):  
Electron Diffraction ◽  
Diffraction Theory ◽  
Bloch Wave ◽  
Low Energy ◽  
Energy Electron ◽  
Band Theory ◽  
Total Electron ◽  
Dynamical Diffraction ◽  
Low Energy Electron Diffraction ◽  
Low Energy Electron

Interpretation of the images from a point source electron microscope requires a detailed analysis of transmission low energy electron diffraction. Here we present a general approach for solutions to the mixed Bragg-Laue case in transmission LEED (100-1000eV), based on the dynamical diffraction theory of Bethe. However, the validity of the dynamical diffraction theory to low energy electrons can be justified by its connection to the band theory for low energy crystal electrons.Assume that the incident beam forms a plane wave and the crystal is a thin slab. According to Bethe, the total electron wavefield within crystal can be written as a linear combination of Bloch waves (equation 1). The Bloch wave excitation coefficients b(j) can be determined by matching the boundary conditions, the wave amplitudes Cg(j) and the wave vectors k(j) for each Bloch wave can be obtained by solving the time independent Schrodinger equations (equation 2).

Axial Channeling in Electron Diffraction

1978 ◽  
Vol 33 (3) ◽  
pp. 269-281 ◽  
Author(s):  
A. Ichimiya ◽  
G. Lehmpfuhl
Keyword(s):  
Electron Beam ◽  
Electron Diffraction ◽  
Intensity Distribution ◽  
Bloch Wave ◽  
Dynamical Theory ◽  
Zone Axis ◽  
Intensity Maximum ◽  
Bloch Waves ◽  
Axial Channeling

AbstractKossel patterns from Silicon and Niobium were obtained with a convergent electron beam. An intensity maximum in the direction of the zone axes [001] and [111] of Nb was interpreted as axial channeling. The intensity distribution in Kossel patterns was calculated by means of the Bloch wave picture of the dynamical theory of electron diffraction. Particularly zone axis patterns were calculated for different substance-energy combinations and they were compared with experimental observations. The intensity distribution in the calculated Kossel patterns was very sensitive to the model of absorption and it was found that a treatment of the absorption close to the model of Humphreys and Hirsch [Phil. Mag. 18, 115 (1968)] gave the best agreement with the experimental observations. Furthermore it is shown which Bloch waves are important for the intensity distribution in the Kossel patterns, how they are absorbed and how they change with energy.

Theory of bulk resonance diffraction in THEED

1993 ◽  
Vol 440 (1908) ◽  
pp. 95-115 ◽  
Keyword(s):  
Elastic Scattering ◽  
Incident Beam ◽  
Bloch Wave ◽  
Dynamical Theory ◽  
Zone Axis ◽  
Bloch Waves ◽  
Diffraction Geometry ◽  
Ewald Sphere ◽  
Lattice Images ◽  
New Type

A full dynamical theory has been developed for an off-axis diffraction geometry. A new type of resonance elastic scattering is found and discussed. This occurs when the Ewald sphere is almost tangential to one of the minus high order Laue zones, and is termed bulk resonance diffraction. It is shown that under certain diffraction conditions, i. e. bulk resonance diffraction conditions, effectively only a single distinct tightly bound Bloch wave localized around atom strings is excited within the crystal, and selection can be made of the particular bound Bloch waves by appropriately tilting the incident beam or the crystal. A new scheme for imaging individual tightly bound Bloch waves is proposed. Full dynamical calculations have been made for 1T–V Se 2 single crystals. It is demonstrated that chemical lattice images of V and Se atom strings can be obtained along the [0001] zone axis of a 1T–V Se 2 crystal for angles of incidence of 109.54 and 109.90 mrad respectively.

Failure of the Conservation of Flux in the Dynamical Theory of the Electron Diffraction

1985 ◽  
Vol 43 ◽  
pp. 76-77
Author(s):  
H. S. Kim ◽  
S.S. Sheinin
Keyword(s):  
Electron Diffraction ◽  
Dynamical Theory ◽  
Dynamical Matrix ◽  
Laue Diffraction ◽  
Bloch Waves ◽  
Reflected Waves

It is widely believed that flux is conserved in the dynamical theory of electron diffraction. Conservation of flux requires thatwhere ϕg is the amplitude of a diffraction beam and the sum is over all diffracted beams. (1) is valid if the crystal is assumed to be non-absorbing and if reflected waves are neglected. It can also be shown that (1) only holds if the eigenvectors of the dynamical matrix are orthogonal. Orthogonality of these eigenvectors is, however, only obtained when certain approximations and assumptions are made including, for example, the column approximation and the assumption of symmetrical Laue diffraction conditions.In order to see the possible effects of relaxing these assumptions and approximations on conservation of flux, consider the expression for diffracted beam amplitude in non-symmetrical Laue casewhere the summation is over all Bloch waves excited in the crystal.

Energy Loss Analysis in Bloch Waves

1974 ◽  
Vol 29 (6) ◽  
pp. 955-956
Author(s):  
F. Fujimoto ◽  
G. Lehmpfuhl
Keyword(s):  
Electron Diffraction ◽  
Energy Loss ◽  
Single Crystal ◽  
Bloch Wave ◽  
Diffraction Spot ◽  
Loss Peak ◽  
Bloch Waves ◽  
Loss Analysis ◽  
Partial Waves

In electron diffraction experiments with a single-crystal wedge Bloch waves can be analyzed directly because of their separation into partial waves when leaving the crystal. In a two-beam case the diffraction spot is split into a double representing two partial waves of the two Bloch waves. The energy-loss spectrum in the 220 doublet of MgO was investigated with a Möllenstedt velocity-analyzer. Two loss peaks at about 14 and 22 eV were found in each Bloch wave. Thermal losses were identified as a background in the no-loss peak.

Diffraction Contrast of Small Vacancy Clusters in Silicon

1970 ◽  
Vol 28 ◽  
pp. 514-515 ◽  
Author(s):  
T. J. Magee ◽  
R. H. Morriss
Keyword(s):  
Electron Diffraction ◽  
Differential Equations ◽  
Vacancy Cluster ◽  
Dynamical Theory ◽  
Integration Scheme ◽  
Thin Sample ◽  
Vacancy Clusters ◽  
Diffraction Contrast ◽  
Exit Surface ◽  
The Many

Diffraction contrast profiles have been examined utilizing the two beam dynamical theory for a thin sample containing a spherical vacancy cluster. The differential equations describing electron diffraction amplitudes in an absorbing crystal have been given by Howie and Whelan and the strain parameter for a center of dilation in a semi-infinite solid has been derived by Mindlin and Cheng. Using a Runge-Kutta integration scheme, the contrast equations were solved numerically to yield the intensities of the transmitted and scattered beams at the exit surface of the foil. In all computations, the many beam 220 extinction distance of 723 Å was used in predicting contrast profiles.

Structure determination using convergent-beam electron diffraction

1992 ◽  
Vol 50 (2) ◽  
pp. 1162-1163
Author(s):  
J W Steeds ◽  
R Vincent
Keyword(s):  
Electron Diffraction ◽  
Model Building ◽  
Diffraction Theory ◽  
Bloch Wave ◽  
X Ray ◽  
X Ray Crystallography ◽  
Local Perturbations ◽  
Fitting Procedures ◽  
Convergent Beam ◽  
Kinematical Approach

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.

Bloch-wave interpretations of Gjønnes–Moodie lines in convergent-beam electron diffraction for non-symmorphic space-group crystals

2021 ◽  
Vol 77 (3) ◽  
pp. 222-231
Author(s):  
Hirofumi Matsuhata
Keyword(s):  
Electron Diffraction ◽  
Dimensional Space ◽  
High Energy ◽  
Bloch Wave ◽  
Convergent Beam Electron Diffraction ◽  
Bloch Waves ◽  
Beam Electron ◽  
Space Group ◽  
Space Groups ◽  
Convergent Beam

The contrast of Gjønnes–Moodie (GM) lines which appear in convergent-beam electron diffraction patterns for non-symmorphic space-group crystals is explained using Bloch waves. In the two-dimensional space groups p2mg and pg the Bloch waves for electron diffraction are described. In both space groups along the Δ line, Bloch waves are arranged as two different types, and it is shown that the two types of Bloch waves do not contribute to the intensity of forbidden reflections. Along the position where the forbidden reflection satisfies the Bragg condition, degeneracies of two Bloch waves are found and it is shown that the degenerated pair of Bloch waves do not contribute to the intensity. These Bloch-wave results provide a new perspective in the understanding of the contrast mechanism of GM lines previously described using scattering polynomials. They also advance the understanding of Bloch-wave behaviour in high-energy electron diffraction.

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