Calculation of many-beam dynamic electron diffraction without high-energy approximation

1996 ◽  
Vol 54 ◽  
pp. 128-129
Author(s):  
B. R. Ahn ◽  
N. J. Kim
Keyword(s):  
Electron Diffraction ◽  
Flat Surface ◽  
Dynamic Theory ◽  
High Energy ◽  
Perfect Crystal ◽  
Zone Axis ◽  
Beam Dynamic ◽  
Diffraction Angle ◽  
Energy Approximation ◽  
The Many

High energy approximation in dynamic theory of electron diffraction involves some intrinsic problems. First, the loss of theoretical strictness makes it difficult to comprehend the phenomena of electron diffraction. Secondly, it is difficult to believe that the approximation is reasonable especially in the following cases: 1) when accelerating voltage is not sufficiently high, 2) when the specimen is thick, 3) when the angle between the surface normal of the specimen and zone axis is large, and 4) when diffracted beam with large diffraction angle is included in the calculation. However, until now the method to calculate the many beam dynamic electron diffraction without the high energy approximation has not been proposed. For this reason, the authors propose a method to eliminate the high energy approximation in the calculation of many beam dynamic electron diffraction. In this method, a perfect crystal with flat surface was assumed. The method was applied to the calculation of [111] zone axis CBED patterns of Si.

A Simple Real-Space Channelling Theory for Electron Diffraction and HREM

1990 ◽  
Vol 48 (1) ◽  
pp. 64-65
Author(s):  
D. Van Dyck
Keyword(s):  
High Resolution ◽  
Electron Diffraction ◽  
Direct Method ◽  
Real Space ◽  
Zone Axis ◽  
Phase Object ◽  
High Resolution Images ◽  
The Many ◽  
Analytical Expressions ◽  
Electron Wavefunction

The computation of the many beam dynamical electron diffraction amplitudes or high resolution images can only be done numerically by using rather sophisticated computer programs so that the physical insight in the diffraction progress is often lost. Furthermore, it is not likely that in this way the inverse problem can be solved exactly, i.e. to reconstruct the structure of the object from the knowledge of the wavefunction at its exit face, as is needed for a direct method [1]. For this purpose, analytical expressions for the electron wavefunction in real or reciprocal space are much more useful. However, the analytical expressions available at present are relatively poor approximations of the dynamical scattering which are only valid either for thin objects ((weak) phase object approximation, thick phase object approximation, kinematical theory) or when the number of beams is very limited (2 or 3). Both requirements are usually invalid for HREM of crystals. There is a need for an analytical expression of the dynamical electron wavefunction which applies for many beam diffraction in thicker crystals. It is well known that, when a crystal is viewed along a zone axis, i.e. parallel to the atom columns, the high resolution images often show a one-to-one correspondence with the configuration of columns provided the distance between the columns is large enough and the resolution of the instrument is sufficient. This is for instance the case in ordered alloys with a column structure [2,3]. From this, it can be suggested that, for a crystal viewed along a zone axis with sufficient separation between the columns, the wave function at the exit face does mainly depend on the projected structure, i.e. on the type of atom columns. Hence, the classical picture of electrons traversing the crystal as plane-like waves in the directions of the Bragg beams which historically stems from the X-ray diffraction picture, is in fact misleading.

Bloch-wave solution in the Bragg case

1989 ◽  
Vol 45 (2) ◽  
pp. 174-182 ◽  
Author(s):  
Y. Ma ◽  
L. D. Marks
Keyword(s):  
Electron Diffraction ◽  
Current Flow ◽  
High Energy ◽  
Bloch Wave ◽  
Zone Axis ◽  
Wave Method ◽  
High Energy Electron ◽  
Stepped Surfaces ◽  
Reflection Electron Microscopy ◽  
Wave Phenomena

The Bloch-wave method for reflection diffraction problems, primarily electron diffraction as in reflection high-energy electron diffraction (RHEED) and reflection electron microscopy (REM), is developed. The basic Bloch-wave approach for surfaces is reviewed, introducing the current flow concept which plays a major role both in understanding reflection diffraction and determining the allowed Bloch waves. This is followed by a brief description of the numerical methods for obtaining the results including specific results for GaAs near to the [010] zone axis. A number of other Bloch-wave phenomena are also discussed, namely resonance diffraction and its relationship to internal and external reflection and variations in the boundary conditions and Bloch-wave character, splitting of diffraction spots due to stepped surfaces, which can be completely explained, and the reflection equivalent of thickness fringes.

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

1992 ◽  
Vol 50 (2) ◽  
pp. 1448-1449
Author(s):  
H. S. Kim ◽  
S. S. Sheinin
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Materials Science ◽  
High Energy ◽  
Dynamical Theory ◽  
Crystal Defects ◽  
Special Cases ◽  
Energy Approximation ◽  
Diffraction Patterns ◽  
Bloch’S Theorem

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,.

STRUCTURAL ANALYSIS OF CRYSTAL SURFACES BY REFLECTION HIGH ENERGY ELECTRON DIFFRACTION

1997 ◽  
Vol 04 (03) ◽  
pp. 501-511 ◽  
Author(s):  
AYAHIKO ICHIMIYA ◽  
YUSUKE OHNO ◽  
YOSHIMI HORIO
Keyword(s):  
Electron Diffraction ◽  
Surface Structure ◽  
Forward Direction ◽  
High Energy ◽  
Zone Axis ◽  
Energy Electron ◽  
High Energy Electron ◽  
Structure Determinations ◽  
The One ◽  
Diffraction Condition

For surface structure determinations by reflection high energy electron diffraction (RHEED), intensity rocking curves are analyzed through RHEED dynamical calculations. Since fast electrons are scattered dominantly in the forward direction by atoms, dynamic diffraction mainly occurs in the forward direction. By the use of this feature, it is possible to choose a diffraction condition under which electrons are diffracted mainly by lattice planes parallel to the surface, when the incident direction is chosen at a certain azimuthal angle with respect to a crystal zone axis. This diffraction condition is called the one-beam condition. Under this condition, the RHEED intensity is a function of interlayer distances and atomic densities of the surface layers. Therefore the surface normal components of the atomic positions are determined by analysis of the one-beam rocking curve using a RHEED dynamical calculation. Then, using the result of the surface normal components of atomic positions, lateral positions of the surface atoms are determined from analysis of the rocking curves at many-beam conditions, where the direction of the incident beam is chosen along a certain crystal zone axis. An example of the surface structure determination of a Si(111) surface at high temperatures is reported. We discuss effects of terraces and antiphase domains of surfaces in structure determinations by RHEED.

Crystal Surface Symmetry from Zone-Axis Patterns in Reflection High-Energy-Electron Diffraction

1984 ◽  
Vol 53 (22) ◽  
pp. 2125-2128 ◽  
Author(s):  
M. D. Shannon ◽  
J. A. Eades ◽  
M. E. Meichle ◽  
P. S. Turner ◽  
B. F. Buxton
Keyword(s):  
Electron Diffraction ◽  
Crystal Surface ◽  
High Energy ◽  
Zone Axis ◽  
Energy Electron ◽  
High Energy Electron
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